De-noising of Gaussian noise affected images by Non-Local Means algorithm

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微小技术（Microsemi）产品用户指南：图像去噪滤波器50200643版本3.0说明书

UG0643User Guide Image De-Noising FilterMicrosemi HeadquartersOne Enterprise, Aliso Viejo,CA 92656 USAWithin the USA: +1 (800) 713-4113 Outside the USA: +1 (949) 380-6100 Sales: +1 (949) 380-6136Fax: +1 (949) 215-4996Email: *************************** ©2020 Microsemi, a wholly owned subsidiary of Microchip Technology Inc. All rights reserved. Microsemi and the Microsemi logo are registered trademarks of Microsemi Corporation. All other trademarks and service marks are the property of their respective owners. Microsemi makes no warranty, representation, or guarantee regarding the information contained herein or the suitability of its products and services for any particular purpose, nor does Microsemi assume any liability whatsoever arising out of the application or use of any product or circuit. The products sold hereunder and any other products sold by Microsemi have been subject to limited testing and should not be used in conjunction with mission-critical equipment or applications. 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Information provided in this document is proprietary to Microsemi, and Microsemi reserves the right to make any changes to the information in this document or to any products and services at any time without notice.About MicrosemiMicrosemi, a wholly owned subsidiary of Microchip T echnology Inc. (Nasdaq: MCHP), offers a comprehensive portfolio of semiconductor and system solutions for aerospace & defense, communications, data center and industrial markets. Products include high-performance and radiation-hardened analog mixed-signal integrated circuits, FPGAs, SoCs and ASICs; power management products; timing and synchronization devices and precise time solutions, setting the world's standard for time; voice processing devices; RF solutions; discrete components; enterprise storage and communication solutions, security technologies and scalable anti-tamper products; Ethernet solutions; Power-over-Ethernet ICs andmidspans; as well as custom design capabilities and services. Learn more at .Contents1Revision History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11.1Revision 3.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2Revision2.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3Revision 1.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 3Image De-Noising Filter Hardware Implementation . . . . . . . . . . . . . . . . . . . . . . . . . .33.1Inputs and Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2Configuration Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.3Testbench . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.4Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.5Resource Utilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10FiguresFigure 1Median-Based Denoising Filter Effect on a Noisy Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Figure 2Image De-Noising Filter Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Figure 3Create SmartDesign Testbench . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Figure 4Create New SmartDesign Testbench Dialog Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Figure 5Image De-Noise Filter Core in Libero SoC Catalog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Figure 6Image De-Noise Filter Core on SmartDesign Testbench Canvas . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Figure 7Promote to Top Level Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Figure 8Image De-Noise Filter Core Ports Promoted to Top Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Figure 9Generate Component Icon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Figure 10Import Files Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Figure 11Input Image File Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Figure 12Input Image File in Simulation Directory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Figure 13Open Interactively Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Figure 14ModelSim Tool with Image De-Noising Filter Testbench File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Figure 15Image De-Noising Filter Effect on a Noisy Image 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Figure 16Image De-Noising Filter Effect on a Noisy Image 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10TablesTable 1Image De-Noising Filter Ports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Table 2Design Configuration Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Table 3Testbench Configuration Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Table 4Resource Utilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Revision History1Revision HistoryThe revision history describes the changes that were implemented in the document. The changes arelisted by revision, starting with the most current publication.1.1Revision 3.0The following is a summary of the changes in revision 3.0 of this document.•Input Data_In_i is replaced with R_I, G_I and B_I to support RGB color format.•Output Data_Out_o is replaced with R_O, G_O and B_O.•Median Filter design logic is redesigned to support for (n x n) resolution with pipelined logics whereas previous design is implemented with Sequential FSM.1.2Revision2.0The following is a summary of the changes in revision 2.0 of this document.•In Image De-Noising Filter Hardware Implementation, page3:•YCbCr in signal names was replaced with Data.•The following text was deleted: The median filtering is only applied on the Y channel. The C B and C R signals are passed through the required pipe-lining registers to synchronize with Ychannel. For the Y channel, three pixels from each of the three video lines are read into threeshift-registers.•Details about the Image De-noising Filter testbench were added. For more information, see Testbench, page5.•The Timing Diagrams section and the appendix were deleted.•The number of buffers in the hardware was updated from four to five. For more information, see Image De-Noising Filter Hardware Implementation, page3.•Information about port widths was added. For more information, see Inputs and Outputs, page4.•Resource utilization data was updated. For more information, see Resource Utilization, page10.1.3Revision 1.0The first publication of this document.Introduction2IntroductionImages captured from image sensors are affected by noise. Impulse noise is the most common type ofnoise, also called salt-and-pepper noise. It is caused by malfunctioning pixels in camera sensors, faultymemory locations in the hardware, or errors in data transmission.Image denoising plays a vital role in digital image processing. Many schemes are available for removingnoise from images. A good denoising scheme retrieves a clearer image even if the image is highlyaffected by noise.Image denoising may either be linear or non-linear. A mean filter is an example of linear filtering, and amedian filter is an example of non-linear filtering. While the linear model has traditionally been preferredfor image denoising because of its speed, the limitation of this model is that it does not preserve theedges of the image. The non-linear model preserves the edges well compared to the linear model, but itis relatively slow.Despite the slowness, non-linear filtering is a good alternative to linear filtering because it effectivelysuppresses impulse noise while preserving the edge information. The median filter ensures that eachpixel in the image fits in with the pixels around it. It filters out samples that are not representative of theirsurroundings—the impulses. Therefore, it is very useful in filtering out missing or damaged pixels.For 2D images, standard median operation is implemented by sliding a window over the image. The 3 ×3 window size, considered to be effective for the most commonly used images, is implemented in the IP.At each position of the window, the nine pixel values inside the window are copied and sorted. The valueof the central pixel is replaced with the median value of the nine pixels in the window. The window slidesright by one column after every clock cycle until the end of the line. The following illustration shows theeffect of a median-based denoising filter on a noisy image.Figure 1 • Median-Based Denoising Filter Effect on a Noisy Image3Image De-Noising Filter HardwareImplementationMicrosemi Image De-noising Filter IP core—a part of Microsemi’s imaging and video solutions IP suite—supports 3 × 3 2D median filtering and effectively removes impulse noise from images.The Image De-noising Filter hardware contains three one-line buffers storing one horizontal video lineeach. The incoming data stream fills these three buffers, one by one. In the design illustrated in thisdocument, the median filter is implemented on 3 × 3 matrix, so three lines of video form the 3 × 3 windowfor the median. When the third buffer contains three pixel values, the read process is initiated.Three shift registers form the 3 × 3 2D array for median calculation. These shift registers are applied asinput to the median finder, which contains 8-bit comparators that sort the nine input values in increasingorder of magnitude and produce the median value, which is then updated into the output register. Thenew pixel column is shifted into the shift register, with the oldest data being shifted out. The 3 × 3 windowmoves from the left to right and from top to bottom for each frame.The following illustration shows the block diagram of the Image De-noising Filter hardware with defaultRGB888 input.Figure 2 • Image De-Noising Filter Hardware3.1Inputs and OutputsThe following table lists the input and output ports of the Image De-noising Filter.3.2Configuration ParametersThe following table lists the configuration parameters for the Image De-noising Filter design.Note:These are generic parameters that vary based on the application requirements.Table 1 • Image De-Noising Filter PortsPort Name Direction WidthDescriptionRESETN_I Input Active-low asynchronous reset signal to design SYS_CLK_I Input System clockR_I Input [(g_DATAWIDTH–1):0]Data input – Red Pixel G_I Input [(g_DATAWIDTH–1):0]Data input – Green Pixel B_IInput [(g_DATAWIDTH–1):0]Data input – Blue Pixel DATA_VALID_I Input Input data valid signal R_O Output [(g_DATAWIDTH-1):0]Data output - Red Pixel G_O Output [(g_DATAWIDTH-1):0]Data output - Green Pixel B_OOutput [(g_DATAWIDTH-1):0]Data output - Blue Pixel DATA_VALID_OOutputOutput data valid signalTable 2 • Design Configuration ParametersNameDescription Default G_DATA_WIDTH Data bit width 8G_RAM_SIZEBuffer size of RAM2048 (for horizontal resolution of 1920)3.3TestbenchT o demonstrate the functionality of the Image De-Noise Filter core, a sample testbench file (image-denoise_test ) is available in the Stimulus Hierarchy (View > Windows > Stimulus Hierarchy), and a sample testbench input image file (RGB_input.txt ) is available in the Libero ® SoC Files window (View > Windows > Files).The following table lists the testbench parameters that can be configured according to the application, if necessary.The following steps describe how to simulate the core using the testbench.1.In the Libero SoC Design Flow window, expand Create Design , and double-click Create SmartDesign Testbench, as shown in the following figure.Figure 3 •Create SmartDesign Testbench2.Enter a name for the SmartDesign testbench and click OK .Figure 4 •Create New SmartDesign Testbench Dialog BoxA SmartDesign testbench is created, and a canvas appears to the right of the Design Flow pane.3.In the Libero SoC Catalog (View > Windows > Catalog), expand Solutions-Video, and drag the Image De-Noise Filter IP core onto the SmartDesign testbench canvas.Table 3 • Testbench Configuration ParametersName Description CLKPERIOD Clock period HEIGHT Height of the image WIDTH Width of the image g_DATAWIDTH Data bit widthWAITNumber of clock cycles of delay between the transmission of one line of the input image and the next IMAGE_FILE_NAMEInput image nameFigure 5 • Image De-Noise Filter Core in Libero SoC CatalogThe core appears on the canvas, as shown in the following figure.Figure 6 • Image De-Noise Filter Core on SmartDesign Testbench Canvas4.Select all the ports of the core, right-click, and click Promote to Top Level, as shown in the followingfigure.Figure 7 • Promote to Top Level OptionThe ports are promoted to the top level, as shown in the following figure.Figure 8 • Image De-Noise Filter Core Ports Promoted to Top Level5.T o generate the Image De-noising Filter SmartDesign component, click Generate Component iconon the SmartDesign Toolbar, as shown in the following figure.Figure 9 • Generate Component IconA sample testbench input image file is created at:…\Project_name\component\Microsemi\SolutionCore\Image_Denoising_Fil-ter\1.2.0\Stimulus6.In the Libero SoC Files window, right-click the simulation directory, and click Import Files..., asshown in the following figure.Figure 10 • Import Files Option7.Do one of the following:•To import the sample testbench input image, browse to the sample testbench input image file, and click Open, as shown in the following figure.•To import a different image, browse to the desired image file, and click Open.Figure 11 • Input Image File SelectionThe input image file appears in the simulation directory, as shown in the following figure.Figure 12 • Input Image File in Simulation Directory8.In the Stimulus Hierarchy, expand Work, and right-click the Image De-noising Filter testbench file(image_denoise_test.v).9.Click Simulate Pre-Synth Design, and then click Open Interactively.Figure 13 • Open Interactively OptionThe ModelSim tool appears with the testbench file loaded on to it, as shown in the following figure. Figure 14 • ModelSim Tool with Image De-Noising Filter Testbench File10.If the simulation is interrupted because of the runtime limit in the DO file, use the run -all command tocomplete the simulation.After the simulation is completed, the testbench output image file (.txt) appears in the simulationfolder.3.4Simulation ResultsThe following illustration shows the effect of the Image De-noising Filter on a noisy image.Figure 15 • Image De-Noising Filter Effect on a Noisy Image 1Figure 16 • Image De-Noising Filter Effect on a Noisy Image 23.5Resource UtilizationIn this design, the Image De-noising Filter is implemented on an MPF300TS-1FCG1152I PolarFireSystem-on-Chip (SoC) FPGA. The following table provides resource utilization data for a 24-bit datawidth design after synthesis.Note:Image De-noising Filter supports for SmartFusion2 and PolarFire FPGAs.Table 4 • Resource UtilizationResource UtilizationDFFs19614_input LUTs2417MACC0RAM1Kx1815RAM64x180。

M,

2 Below in Part 6.3) we apply our techniques to prove that the algebraic number of all contractible solutions of (3) also equals χ(N ) 3 a solution u of (1), (2) with f = f is called nondegenerate if for f close to f the equation 0 0 0 (1), (2) has the only solution u close to u0 and u smoothly depends on f . 4 but the dimension is always zero if the bundle T is trivial, e.g. if is a torus - see in [6] 2
uz ¯)u = ¯ = (∂/∂ z
We are mostly concerned with contractible solutions, u : T2 → M is contractible. (2)
Equation (1) linearized about a conFredholm operator. Essentially for this reason for a typical f the equation has a ﬁnite number of solutions. Our goal is to estimate this number from below by investigating the manifold formed by all pairs (u, f ) where u solves (1), (2) with f = f .

Hybrid image segmentation using watersheds and fast region merging

Hybrid Image Segmentation Using Watersheds and Fast Region Merging Kostas Haris,Seraﬁm N.Efstratiadis,Member,IEEE,Nicos Maglaveras,Member,IEEE,and Aggelos K.Katsaggelos,Fellow,IEEEAbstract—A hybrid multidimensional image segmentation al-gorithm is proposed,which combines edge and region-based techniques through the morphological algorithm of watersheds. An edge-preserving statistical noise reduction approach is used as a preprocessing stage in order to compute an accurate estimate of the image gradient.Then,an initial partitioning of the image into primitive regions is produced by applying the watershed trans-form on the image gradient magnitude.This initial segmentation is the input to a computationally efﬁcient hierarchical(bottom-up)region merging process that produces theﬁnal segmentation. The latter process uses the region adjacency graph(RAG)repre-sentation of the image regions.At each step,the most similar pair of regions is determined(minimum cost RAG edge),the regions are merged and the RAG is updated.Traditionally,the above is implemented by storing all RAG edges in a priority queue. We propose a signiﬁcantly faster algorithm,which additionally maintains the so-called nearest neighbor graph,due to which the priority queue size and processing time are drastically reduced. Theﬁnal segmentation provides,due to the RAG,one-pixel wide, closed,and accurately localized contours/surfaces.Experimental results obtained with two-dimensional/three-dimensional(2-D/3-D)magnetic resonance images are presented.Index Terms—Image segmentation,nearest neighbor region merging,noise reduction,watershed transform.I.I NTRODUCTIONI MAGE segmentation is an essential process for most sub-sequent image analysis tasks.In particular,many of the existing techniques for image description and recognition[1], [2],image visualization[3],[4],and object based image com-pression[5]–[7]highly depend on the segmentation results. The general segmentation problem involves the partitioning of a given image into a number of homogeneous segments (spatially connected groups of pixels),such that the union of any two neighboring segments yields a heterogeneousManuscript received July13,1996;revised October20,1997.This work was supported in part by the I4C project of the Health Telematics programme of the CEC.The associate editor coordinating the review of this manuscript and approving it for publication was Prof.Jeffrey J.Rodriguez.K.Haris is with the Laboratory of Medical Informatics,Faculty of Medicine,Aristotle University,Thessaloniki54006,Greece,and with the Department of Informatics,School of Technological Applications, Technological Educational Institution of Thessaloniki,Sindos54101,Greece (e-mail:haris@med.auth.gr).S.N.Efstratiadis and N.Maglaveras are with the Laboratory of Medical Informatics,Faculty of Medicine,Aristotle University,Thessaloniki54006, Greece(e-mail:seraﬁm@med.auth.gr;nicmag@med.auth.gr).A.K.Katsaggelos is with the Department of Electrical and Com-puter Engineering,McCormick School of Engineering and Applied Science,Northwestern University,Evanston,IL60208-3118USA(e-mail: aggk@).Publisher Item Identiﬁer S1057-7149(98)08714-4.segment.Alternatively,segmentation can be considered as a pixel labeling process in the sense that all pixels that belong to the same homogeneous region are assigned the same label. There are several ways to deﬁne homogeneity of a region based on the particular objective of the segmentation process. However,independently of the homogeneity criteria,the noise corrupting almost all acquired images is likely to prohibit the generation of error-free image partitions[8].Many techniques have been proposed to deal with the image segmentation problem[9],[10].They can be broadly grouped into the following categories.Histogram-Based Techniques:The image is assumed to be composed of a number of constant intensity objects in a well-separated background.The image histogram is usually considered as being the sample probability density function (pdf)of a Gaussian mixture and,thus,the segmentation prob-lem is reformulated as one of parameter estimation followed by pixel classiﬁcation[10].However,these methods work well only under very strict conditions,such as small noise variance or few and nearly equal size regions.Another problem is the determination of the number of classes,which is usually assumed to be known.Better results have been obtained by the application of spatial smoothness constraints[11].Edge-Based Techniques:The image edges are detected and then grouped(linked)into contours/surfaces that represent the boundaries of image objects[12],[13].Most techniques use a differentiationﬁlter in order to approximate theﬁrst-order image gradient or the image Laplacian[14],[15].Then, candidate edges are extracted by thresholding the gradient or Laplacian magnitude.During the edge grouping stage,the detected edge pixels are grouped in order to form continuous, one-pixel wide contours as expected[16].A very successful method was proposed by Canny[15]according to which the image isﬁrst convolved by the Gaussian derivatives,the candidate edge pixels are isolated by the method of nonmax-imum suppression and then they are grouped by hysteresis thresholding.The method has been accelerated by the use of recursiveﬁltering[17]and extended successfully to3D images[18].However,the edge grouping process presents serious difﬁculties in producing connected,one-pixel wide contours/surfaces.Region-Based Techniques:The goal is the detection of re-gions(connected sets of pixels)that satisfy certain predeﬁned homogeneity criteria.In region-growing or merging tech-niques,the input image isﬁrst tessellated into a set of homo-geneous primitive regions.Then,using an iterative merging1057–7149/98$10.00©1998IEEEprocess,similar neighboring regions are merged according to a certain decision rule[12],[19]–[21].In splitting techniques, the entire image is initially considered as one rectangular region.In each step,each heterogeneous image region of the image is divided into four rectangular segments and the process is terminated when all regions are homogeneous.In split-and-merge techniques,after the splitting stage a merging process is applied for unifying the resulting similar neigh-boring regions[22],[23].However,the splitting technique tends to produce boundaries consisting of long horizontal and vertical segments(i.e.,distorted boundaries).The heart of the above techniques is the region homogeneity test,usually formulated as a hypothesis testing problem[23],[24]. Markov Random Field-Based Techniques:The true image is assumed to be a realization of a Markov or Gibbs random ﬁeld with a distribution that captures the spatial context of the scene[25].Given the prior distribution of the true image and the observed noisy one,the segmentation problem is formulated as an optimization problem.The commonly used estimation principles are maximum a posteriori(MAP) estimation,maximization of the marginal probabilities(ICM) [26]and maximization of the posterior marginals[27].How-ever,these methods require fairly accurate knowledge of the prior true image distribution and most of them are quite computationally expensive.Hybrid Techniques:The aim here is offering an improved solution to the segmentation problem by combining techniques of the previous categories.Most of them are based on the inte-gration of edge-and region-based methods.In[20],the image is initially partitioned into regions using surface curvature-sign and,then,a variable-order surfaceﬁtting iterative region merging process is initiated.In[28],the image is initially segmented using the region-based split-and-merge technique and,then,the detected contours are reﬁned using edge in-formation.In[29],an initial image partition is obtained by detecting ridges and troughs in the gradient magnitude image through maximum gradient paths connecting singular points. Then,region merging is applied through the elimination of ridges and troughs via similarity/dissimilarity measures.The algorithm proposed in this paper belongs to the category of hybrid techniques,since it results from the integration of edge-and region-based techniques through the morphological watershed transform.Many morphological segmentation ap-proaches using the watershed transform have been proposed in the literature[30],[31].Watersheds have also been used in multiresolution methods for producing resolution hierarchies of image ridges and valleys[3],[32].Although these methods were successful in segmenting certain classes of images, they require signiﬁcant interactive user guidance or accurate prior knowledge on the image structure.By improving and extending earlier work on this problem[8],[33],[34],the pro-posed algorithm delivers accurately localized,one pixel wide and closed object contours/surfaces while it requires a small number of input parameters(semiautomatic segmentation). Initially,the noise corrupting the image is reduced by a novel noise reduction technique that is based on local homogeneity testing followed by local classiﬁcation[35].This technique is applied to the original image and preserves edges remarkably well,while reducing the noise quite effectively.At the second stage,this noise suppression allows a more accurate calculation of the image gradient and reduction of the number of the detected false edges.Then,the gradient magnitude is input to the watershed detection algorithm,which produces an initial image tessellation into a large number of primitive regions [31].This initial oversegmentation is due to the high sensitivity of the watershed algorithm to the gradient image intensity variations,and,consequently,depends on the performance of the noise reduction algorithm.Oversegmentation is further reduced by thresholding the gradient magnitude prior to the application of the watershed transform.The output of the watershed transform is the starting point of a bottom-up hierarchical merging approach,where at each step the most similar pair of adjacent regions is detected and merged.Here, the region adjacency graph(RAG)is used to represent the image partitions and is combined with a newly introduced nearest neighbor graph(NNG),in order to accelerate the region merging process.Our experimental results indicate a remarkable acceleration of the merging process in comparison to the RAG based merging.Finally,a merging stopping rule may be adopted for unsupervised segmentation.In Section II,the segmentation problem is formulated and the algorithm outline is presented.In Section III,a novel edge-preserving noise reduction technique is presented as a preprocessing step,followed by the proposed gradient ap-proximation method.In Section IV,the watershed algorithm used and an oversegmentation reduction technique are brieﬂy described.In Section V,the proposed accelerated bottom-up hierarchical merging process is presented and analyzed. Results are presented in Section VI on two-dimensional/three-dimensional(2-D/3-D)synthetic and real magnetic resonance (MR)images.Finally,conclusions and possible extensions of the algorithm are discussed in Section VII.II.P ROBLEM F ORMULATION AND A LGORITHM O UTLINELet be the set of intensitiesandbe the spatial coordinates of a pixel inaneighborhood ofpixel is deﬁned asfollows:where are odd and denotes the largest integer not greater than its argument.In the3-D case,the neighborhood ofpointis corrupted by additive independent identically distributed Gaussian noise. Hence,the observedimage(1)where.It is also assumed that the true image is piecewise constant.More speciﬁcally,there is a partitionof,forsome natural number(2)whereif(3)It is reminded that two regions are adjacent if they share a common boundary,that is,if there is at least one pixel in one region,such that,its3ofof the observed image,for oddis considered to be a sample of sizeand variance.Aheterogeneous is considered to be a sample of sizewhere(9)for(10)whereand th order sample momentof,for.Experimental comparisons of the moment es-timator with the ML estimator have shown that,when theclasses are well-separated,the estimators yield nearly identicalestimates[8].Provided that the original image follows theadopted piecewise constant model and the noise is above acertain level,the performance of the proposed noise reductionmethod is superior to that of other methods,such as linearﬁltering,medianﬁltering and anisotropic diffusion[36].Theperformance of the noise reduction stage depends on theaccurate estimation of the noisevariance in the observedimage.Several noise variance estimation methods have beenproposed in the literature[37].Also,the noise reduction stagedepends on the value ofparameteris computed.Among the known gradient operators,namely,classical(Sobel,Prewitt),Gaussian or morphological,the Gaussian derivativeshave been extensively studied in the literature[12].Providedthat the original noise level is not high or the noise hasbeen effectively reduced in theﬁrst stage,then all the aboveoperators may perform well.However,if the original noiselevel is high or the noise has not been effectively reduced intheﬁrst stage,the use of small scale Gaussian derivativeﬁltersmay further reduce noise.Finally,the gradient magnitudeimagebe a greyscale digital image.Watersheds are deﬁnedas the lines separating the so-called catchment basins,whichbelong to different minima.More speciﬁcally,aminimumis a connected set of pixelswithintensity,where.The catchmentbasinis a set of pixels,such that,if a dropof water falls at any pixelin.The watersheds computation algorithmused here is based on immersion simulations[31],that is,on the recursive detection and fast labeling of the differentcatchment basins using queues.The algorithm consists of twosteps:sorting andﬂooding.At theﬁrst step,the image pixelsare sorted in increasing order according to their intensities.Using the image intensity histogram,a hash table is allocatedin memory,where the.Then,this hash table isﬁlled byscanning the image.Therefore,sorting requires scanning theimage twice using only constant memory.At theﬂooding step,the pixels are quickly accessed in increasing intensity order(immersion)using the sorted image and labels are assigned tocatchment basins.The label propagation is based on queuesconstructed using neighborhoods[31].The output of the watersheds algorithm is a tessellationof the input image into its different catchment basins,eachone characterized by a unique label.Among the image water-shed points,only these located exactly half-way between twocatchment basins are given a special label[31].In order toobtain theﬁnal image tessellation,the watersheds are removedby assigning their corresponding points to the neighboringcatchment basins.The input to the watersheds algorithm is the gradientmagnitudeimageFig.1.Flow diagram of the proposed segmentation algorithm.[30],[31],do not use all the regional minima of the input image in the ﬂooding step but only a small number of them.These selected regional minima are referred to as markers .Prior to the application of the watershed transform,the intensity image can be modiﬁed so that its regional minima are identical to a predetermined set of markers by the homotopy modiﬁcation method [30].This method achieves the suppression of the minima not related to the markers by applying geodesic reconstruction techniques and can be implemented efﬁciently using queues of pixels [38].Although markers have been successfully used in segmenting many types of images,their selection requires either careful user intervention or explicit prior knowledge on the image structure.In our approach,image oversegmentation is regarded as an initial image partition to which a fast region-merging procedure is applied (see Section V).As explained in Section V,the larger the initial oversegmentation,the higher the probability of false region merges during merging.In addition,the computational overhead of region merging clearly depends on the size of this initial partition,and consequently the smallest possible oversegmentation size is sought.One way to limit the size of the initial image partition is to prevent oversegmentation in homogeneous (ﬂat)regions,where the gradient magnitude is low since it is generated by the residual noise of the ﬁrst stage (see Fig.1).The watershed transform is applied to the thresholded gradient magnitudeimagehaving value smaller than a giventhresholdlocated in homogeneous regions arereplaced by fewer zero-valued regional minimainmay cause merging of regional minimainin the thresholding process,thatis,.A candidate edge pixel is deﬁned as havingintensity valueinin (11)may be determined directly based on the esti-mated noise variancewhich produce satisfactory initial oversegmentation reduction in almost all experimental cases considered are lessthansevere oversegmentation is avoided through edge preserving noise reduction and gradient magnitude thresholding(Section III).B.Region Dissimilarity FunctionsThe objective cost function used in this work is the square error of the piecewise constant approximation of the ob-servedimagebeabe the set of pixels belongingtoregion.In the piecewise constant approximationof,ofpartitionand isequal to the mean valueof.The correspondingsquare errorisis theoptimal,whichminimizes the following dissimilarity function[41],[42]:.If-p a r t i t i o ni s d eﬁn e d a s a n u n d i r e c t e d g r a p h,i s t h e s e t ofF i g.2.S i x-p a r t i t i o n o f a n i m a g e(l e f t),a n d t h e c o r r e s p o n d i n g R A G(r i g h t).F i g.3.M e r g i n g o f t w o R AG n o d e s.F i g.4.R A G(l e f t)a n d o n e o f i t s p o s s i b l e N N G’s(r i g h t).e d g e s.E a c h r e g i o n i s r e p r e s e n t e d b y a g r a p h n ot w o r e g i o n s(n o d e s )e x i s t s t h e e d g e-p a r t i t i o n i m a g e i s u s e d f o r t h e c o n s t r u ci n i t i a l R A G ((a)(b)(c)Fig.5.Examples of the three possible NNG-cycle modiﬁcation types due to merging.(a)NNG-cycle cancellation.(b),(c)NNG-cycle creation with (b)and without (c)the participation of the node resulting from merging.Notation:RAG edge (111),NNG edge (!)and NNG cycle ($).Fig.6.Two examples of NNG-edge modiﬁcation due to merging.Given the RAG of theinitial -RAG)and the heap of its edges,the RAG of thesuboptimal-RAG)is constructed by the following algorithm,which implements the stepwise optimization procedure de-scribed above.Input:RAG ofthe-RAG).Iteration:ForFind the minimum cost edge inthe-partition(time and the cor-responding nodes are merged.The merging operation causes changes in both the RAG and the heap.All RAG nodes that neighbored a node of the merged node pair must restructuretheir neighbor lists.Also,the dissimilarity values (costs)of the neighboring nodes with the node resulting from the merging stage change and must be recalculated using (12).The positions of the changed-cost edges in the heap must be updated,requiring time for each update.In addition,a few edges must be removed since they are canceled due to merging.This is illustrated in Fig.3,where a merging example of two RAG nodes is given.Before the merging of nodes a and b ,node e is a common neighbor to a and b .After their merging,one of the edges (a ,e ),(b ,e )must be removed from the RAG and the heap.Then,the positions of the changed-cost edges in the heap must be updated (edges (ab ,c ),(ab ,d ),(ab ,e )in Fig.3).However,since these positions are unknown,a linear search operationrequiringtime resultsin time for each merge,whereFig.7.Synthetic image (left)and real medical MR image(right).Fig.8.Result of the noise reduction stage on the images of Fig.7.considerably increased.This is particularly true in 3-D images where the initial partition usually contains a very large number of regions.D.Fast Nearest Neighbor MergingThe proposed solution to accelerate region merging is based on the observation that it is not necessary to keep all RAG edges in the heap but only a small portion of them [8].Speciﬁcally,we introduce the NNG,which is deﬁned as follows.For a givenRAG,,the NNG,namely,,is a directed graphwithand thedirectededgebelongsto,the edge is directed toward thenode with the minimum label.The above deﬁnition implies that the out-degree of each node is equal to one.The edge starting at a node is directed toward its most similar neighbor.A cycle in the NNG is deﬁned as a sequence of connected graph nodes (path)in which the starting and ending nodes coincide (see Fig.4).By deﬁnition,the NNG containsFig.9.Initial segmentation results of the images in Fig.8after applying the Gaussianﬁlter( =0:7)and thresholding.(a)T=0(2672regions).(b) T=0(3782regions).(c)T=5(1376regions).(d)T=5(1997regions).edges and has the following properties[8].Property1:The NNG contains at least one cycle.Property2:The maximum length of a cycle is two.The regions of the most similar pair are con-nected by a cycle.Property4:A node can participate at most one cycle.Property5:The maximum number of cycles isIteration:ForFind the minimum cost edge in the-partition.Fig.10.Intermediate segmentation results.Top:1000regions.Middle:500regions.Bottom:100regions.Fig.11.Final segmentation results overlaid on the original images.Left:7regions.Right:25regions.During the merging operation,the NNG is updated as follows.When the nodes of a cycle are merged,the costs of the neighboring RAG edges and,consequently,the structure of the NNG are modiﬁed.Two NNG cycles are deﬁned as neighbors if there is at least one RAG edge connecting two of their nodes.For example,in Fig.5(a),thecyclesandandofcycle,resulting in the cancellation of cycleandtime for each merge,whereis the number of NNG cycles modiﬁed by the merge,and256,8b/pixel)images shown in Fig.7were used in order to illustrate the stages of the segmentation algorithm and visually assess the quality of the segmentation results.The synthetic image [Fig.7(a)]is piecewise constant,the background intensity level is 80,the object intensity level is 110and contains simulated additive white Gaussian noise with standarddeviation13neighborhood.In theabove MR image the estimated noise standard deviationwas.The window size was set to119for the MR image,and it affectsFig.12.Number of RAG edges(solid line)and NNG cycles(dotted line)as a function of the merge number for the image in Fig.8(right).Fig.13.Histogram of the RAG node degree for the image in Fig.8(right).the performance of the noise reduction algorithm as follows. For large window sizes,the power of the homogeneity test (i.e.,the probability of correctly accepting heterogeneity)is large in the case of step edges,while it is relatively small in the case of bar edges.Therefore,the thin features of the image(lines,corners)are oversmoothed.For small window sizes,the power of the homogeneity test is small and the variance of the mixture parameter estimates is large.Therefore, the resulting noise reduction is small.However,the above phenomena occur for very noisy images.In Fig.8,it is clear that the noise is sufﬁciently reduced while the image context is preserved and enhanced.Note that the proposed noise reduction algorithm does not impose any smoothness constraints and,therefore,when the noise level is not high, the image structure is preserved remarkably well.However,we believe that the lack of smoothness constraints is the source of the nonrobust behavior of the algorithm on very noisy images. In addition,the adopted image model does not handle more complex structures such as smooth intensity transitions(ramp edges)and junctions.At the second stage,the gradient magnitude of the smoothed image is calculated using the Gaussianﬁlter derivatives withscale.Then,the gradient magnitude was thresh-olded using(11),where the smoothed gradientmagnitude3neighborhood averaging of noncandidate edge pixels.At the third stage,the watershed detection algorithm was applied to the thresholded image gradient magnitude.Fig.9shows the initial tessellations of the images produced by the application of the watershed detection algorithm on the image gradient magnitude for various thresholds.It is clear that the larger the threshold the smaller the number of the regions produced by the watershed(a)(b) (c)(d) Fig.14.Segmentation of a natural image.(a)Original image(“MIT”).(b)Result of the noise reduction stage.(c)Initial segmentation after gradient thresholding(T=2,2347regions).(d)Final segmentation(80regions).detection algorithm.However,the use of high thresholds may destroy part of the image contours,which cannot be recovered at the merging stage of the segmentation algorithm.More speciﬁcally,it was observed that when the noise is not high, the choice for the threshold value close to the noise standard deviation is safe.However,when noise is high,small threshold values should be used.This is justiﬁed from the fact that when noise is high,the noise reduction algorithm may oversmooth part of the image intensity discontinuities resulting in low gradient magnitudes.Therefore,the use of high threshold values in(11)may destroy part of the object boundaries. The initial tessellations are used at the last stage of the algorithm for the construction of the RAG’s and NNG’s,and then the merging process begins.Fig.10shows several inter-mediate results of the merging process using the corresponding initial segmentation results shown in Fig.9(c)and(d).The ﬁnal segmentation results are given in Fig.11with seven and 25regions,respectively.The number of regions of the initial image tessellation determines the computational and memory requirements for the construction and processing(merging)of the RAG and NNG.The number of the RAG-edges and the number of NNG-cycles are shown in Fig.12as a function of the number of merges.The size of the cycle heap is nearly one order of magnitude smaller than the size of the heap of RAG edges.As explained in Section V,the additional computational effort for manipulating the NNG at each merge of region pair depends on the distribution of the second order neighborhood size in the RAG.In Fig.13,a typical histogram of the RAG degree at an intermediate stage of merging is shown.As expected,the RAG is a graph with low mean degree and this explains the low additional computational effort for the NNG maintenance.TABLE IIT YPICAL E XECUTION T IMES OF THE P ROPOSED S EGMENTATION A LGORITHM AND I TS S TAGES WITH AND W ITHOUT THE USE OF THENNGThe proposed segmentation algorithm was also applied to natural images,such as,the standard “MIT”image(256.The result of the noise reduction stageusing a5,2347regions)and the ﬁnal segmentation result (80regions)are given in Fig.14(c)and (d),respectively.Note that,despite the simplicity of both the underlying image model and the dissimilarity function used,the majority of important image regions were successfully extracted.The 3-D version of the algorithm was applied to a16256MR cardiac image,a slice of which is shown inFig.15(a).Fig.15(b)shows the result of the noise reductionstage,where a35window was used.Fig.15(c)shows the initial segmentation which resulted from the watershed detection stage on the thresholded gradient magnitude image,where the scale of the Gaussian ﬁlter was 0.7and thethreshold.Lastly,Fig.15(d)shows the ﬁnal segmentation result containing 403-D regions.Based on our experiments we concluded,that the smaller the number of the initial (correct)partition segments,the better the ﬁnal segmentation results.On the other hand,the use of thresholds producing initial partitions with small number of segments may cause the disappearance of a few signiﬁcant contours.The 2-D and 3-D version of the proposed image segmentation algorithm were implemented in the C program-ming language on a Silicon Graphics (R4000)computer.Table II shows typical execution times and percentages with respect to the total time for each stage of the proposed algorithm with and without the use of NNG.Note that the noise reduction stage requires a great percentage of the total execution time.This is due to the current implementation in which the required parameters are computed at each window position separately.The noise reduction algorithm may be accelerated by consid-ering a faster implementation,namely,using the separability property in order to compute the sample moments [12].Finally,regarding the memory requirements of the proposed algorithm,they are high due primarily to the watershed al-gorithm [31].At the merging step,the memory requiredfor(a)(b)(c)(d)Fig.15.Three-dimensional image segmentation results.(a)Raw 3-D MR image (slice 5).(b)Smoothed image.(c)Initial oversegmentation (3058regions).(d)Final segmentation (40regions).。

阈值分割技术

摘要小波分析理论是一种新兴的信号处理理论，它在时间上和频率上都有很好的局部性，这使得小波分析非常适合于时-频分析，借助时-频局部分析特性，小波分析理论已经成为信号去噪中的一种重要的工具。

利用小波方法去噪，是小波分析应用于实际的重要方面。

小波去噪的关键是如何选择阈值和如何利用阈值来处理小波系数，通过对小波阈值化去噪的原理介绍,运用MATLAB 中的小波工具箱，对一个含噪信号进行阈值去噪，实例验证理论的实际效果，证实了理论的可靠性。

本文简述了几种小波去噪方法，其中的阈值去噪的方法是一种实现简单、效果较好的小波去噪方法。

小波分析的出现是信号处理领域的一次重大革命，然后由于传统的硬阈值函数滤波存在使信号会产生振荡不具有同原始信号一样的光滑性，而软阈值函数存在丢失信号的某些特征。

根据需要构造了一个新的阈值函数，从而克服了软、硬阈值函数的缺点，由仿真模拟可以看出效果良好。

本文根据已有的阈值处理函数的优缺点,提出了一种新的阈值处理函数,用于图像的小波阈值去噪.实验表明,该方法比传统的硬阈值函数与软阈值函数具有更好的去噪效果。

关键词：小波阈值去噪，阈值函数,小波分析, 滤波Title Threshold based on wavelet signal denoising algorithmAbstractThe wavelet analysis theory is a new signal processing theory. It has a very good topicality in time and frequency, which makes the wavelet analysis very suitable for the time - frequency analysis. With the time - frequency‟s local analysis characteristics, the wavelet analysis theory has become an important tool in the signal de-noising. Using wavelet methods in de-noising, is an important aspect in the application of wavelet analysis. The key of wavelet de-noising is how to choose a threshold and how to use thresholds to deal with wavelet coefficients. It confirms the reliability of the theory through the wavelet threshold de-noising principle, the use of the wavelet toolbox in MA TLAB, carrying on threshold de-noising for a signal with noise and actual results of the example confirmation theory. This paper has summarized several methods about the wavelet de-noising, in which the threshold de-noising is a simple, effective method of wavelet de-noising.The emergence of wavelet analysis is a signal processing field of a major revolution, then the traditional hard threshold function filter the signal will be generated oscillations do not have the same original signal the same smoothness, and the soft threshold function has lost signal characteristics. According to the need to construct a new threshold function, thereby overcoming the disadvantages of soft, hard threshold function, the effect can be seen well by simulation.In this paper, on the basis of the existing threshold processing function of the advantages and disadvantages, this paper puts forward a new thresholding function, is used to image the wavelet threshold denoising method. Experimental results show that, the method is better than the traditional hard threshold function and soft threshold function has better denoising effectKeywords：Wavelet threshold denoising，threshold function，Wavelet filtering threshold function目录引言 (1)1．小波去噪原理分析 (2)1.1. 小波域阈值去噪的基本原理 (2)1.2．小波阈值去噪的基本思路 (2)2．阈值的处理和选取 (3)2.1 小波阈值处理 (3)2.2 软阈值和硬阈值 (3)2.3 阈值函数的选取 (4)3．新阈值函数滤波仿真模拟 (10)4．小波消噪的MATLAB实现 (13)4.1 matlab去噪及语言特点 (13)4.2 小波去噪函数集合 (15)4.3 小波去噪验证仿真 (15)4.4 小波去噪的MATLAB 仿真对比试验 (16)结论 (18)致谢 (19)参考文献 (20)附录 (21)引言图像是信息社会人们获取信息的重要来源之一。

Cubature Kalman Filters

1254IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 6, JUNE 2009Cubature Kalman FiltersIenkaran Arasaratnam and Simon Haykin, Life Fellow, IEEEAbstract—In this paper, we present a new nonlinear ﬁlter for high-dimensional state estimation, which we have named the cubature Kalman ﬁlter (CKF). The heart of the CKF is a spherical-radial cubature rule, which makes it possible to numerically compute multivariate moment integrals encountered in the nonlinear Bayesian ﬁlter. Speciﬁcally, we derive a third-degree spherical-radial cubature rule that provides a set of cubature points scaling linearly with the state-vector dimension. The CKF may therefore provide a systematic solution for high-dimensional nonlinear ﬁltering problems. The paper also includes the derivation of a square-root version of the CKF for improved numerical stability. The CKF is tested experimentally in two nonlinear state estimation problems. In the ﬁrst problem, the proposed cubature rule is used to compute the second-order statistics of a nonlinearly transformed Gaussian random variable. The second problem addresses the use of the CKF for tracking a maneuvering aircraft. The results of both experiments demonstrate the improved performance of the CKF over conventional nonlinear ﬁlters. Index Terms—Bayesian ﬁlters, cubature rules, Gaussian quadrature rules, invariant theory, Kalman ﬁlter, nonlinear ﬁltering.• Time update, which involves computing the predictive density(3)where denotes the history of input; is the measurement pairs up to time and the state transition old posterior density at time is obtained from (1). density • Measurement update, which involves computing the posterior density of the current stateI. INTRODUCTIONUsing the state-space model (1), (2) and Bayes’ rule we have (4) where the normalizing constant is given byIN this paper, we consider the ﬁltering problem of a nonlinear dynamic system with additive noise, whose statespace model is deﬁned by the pair of difference equations in discrete-time [1] (1) (2)is the state of the dynamic system at discrete where and are time ; is the known control input, some known functions; which may be derived from a compensator as in Fig. 1; is the measurement; and are independent process and measurement Gaussian noise sequences with zero and , respectively. means and covariances In the Bayesian ﬁltering paradigm, the posterior density of the state provides a complete statistical description of the state at that time. On the receipt of a new measurement at time , we in update the old posterior density of the state at time two basic steps:Manuscript received July 02, 2008; revised July 02, 2008, August 29, 2008, and September 16, 2008. First published May 27, 2009; current version published June 10, 2009. This work was supported by the Natural Sciences & Engineering Research Council (NSERC) of Canada. Recommended by Associate Editor S. Celikovsky. The authors are with the Cognitive Systems Laboratory, Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON L8S 4K1, Canada (e-mail: aienkaran@grads.ece.mcmaster.ca; haykin@mcmaster. ca). Color versions of one or more of the ﬁgures in this paper are available online at . Digital Object Identiﬁer 10.1109/TAC.2009.2019800To develop a recursive relationship between the predictive density and the posterior density in (4), the inputs have to satisfy the relationshipwhich is also called the natural condition of control [2]. has sufﬁcient This condition therefore suggests that information to generate the input . To be speciﬁc, the can be generated using . Under this condiinput tion, we may equivalently write (5) Hence, substituting (5) into (4) yields (6) as desired, where (7) and the measurement likelihood function obtained from (2). is0018-9286/$25.00 © 2009 IEEEARASARATNAM AND HAYKIN: CUBATURE KALMAN FILTERS1255Fig. 1. Signal-ﬂow diagram of a dynamic state-space model driven by the feedback control input. The observer may employ a Bayesian ﬁlter. The label denotes the unit delay.The Bayesian ﬁlter solution given by (3), (6), and (7) provides a uniﬁed recursive approach for nonlinear ﬁltering problems, at least conceptually. From a practical perspective, however, we ﬁnd that the multi-dimensional integrals involved in (3) and (7) are typically intractable. Notable exceptions arise in the following restricted cases: 1) A linear-Gaussian dynamic system, the optimal solution for which is given by the celebrated Kalman ﬁlter [3]. 2) A discrete-valued state-space with a ﬁxed number of states, the optimal solution for which is given by the grid ﬁlter (Hidden-Markov model ﬁlter) [4]. 3) A “Benes type” of nonlinearity, the optimal solution for which is also tractable [5]. In general, when we are confronted with a nonlinear ﬁltering problem, we have to abandon the idea of seeking an optimal or analytical solution and be content with a suboptimal solution to the Bayesian ﬁlter [6]. In computational terms, suboptimal solutions to the posterior density can be obtained using one of two approximate approaches: 1) Local approach. Here, we derive nonlinear ﬁlters by ﬁxing the posterior density to take a priori form. For example, we may assume it to be Gaussian; the nonlinear ﬁlters, namely, the extended Kalman ﬁlter (EKF) [7], the central-difference Kalman ﬁlter (CDKF) [8], [9], the unscented Kalman ﬁlter (UKF) [10], and the quadrature Kalman ﬁlter (QKF) [11], [12], fall under this ﬁrst category. The emphasis on locality makes the design of the ﬁlter simple and fast to execute. 2) Global approach. Here, we do not make any explicit assumption about the posterior density form. For example, the point-mass ﬁlter using adaptive grids [13], the Gaussian mixture ﬁlter [14], and particle ﬁlters using Monte Carlo integrations with the importance sampling [15], [16] fall under this second category. Typically, the global methods suffer from enormous computational demands. Unfortunately, the presently known nonlinear ﬁlters mentioned above suffer from the curse of dimensionality [17] or divergence or both. The effect of curse of dimensionality may often become detrimental in high-dimensional state-space models with state-vectors of size 20 or more. The divergence may occur for several reasons including i) inaccurate or incomplete model of the underlying physical system, ii) informationloss in capturing the true evolving posterior density completely, e.g., a nonlinear ﬁlter designed under the Gaussian assumption may fail to capture the key features of a multi-modal posterior density, iii) high degree of nonlinearities in the equations that describe the state-space model, and iv) numerical errors. Indeed, each of the above-mentioned ﬁlters has its own domain of applicability and it is doubtful that a single ﬁlter exists that would be considered effective for a complete range of applications. For example, the EKF, which has been the method of choice for nonlinear ﬁltering problems in many practical applications for the last four decades, works well only in a ‘mild’ nonlinear environment owing to the ﬁrst-order Taylor series approximation for nonlinear functions. The motivation for this paper has been to derive a more accurate nonlinear ﬁlter that could be applied to solve a wide range (from low to high dimensions) of nonlinear ﬁltering problems. Here, we take the local approach to build a new ﬁlter, which we have named the cubature Kalman ﬁlter (CKF). It is known that the Bayesian ﬁlter is rendered tractable when all conditional densities are assumed to be Gaussian. In this case, the Bayesian ﬁlter solution reduces to computing multi-dimensional integrals, whose integrands are all of the form nonlinear function Gaussian. The CKF exploits the properties of highly efﬁcient numerical integration methods known as cubature rules for those multi-dimensional integrals [18]. With the cubature rules at our disposal, we may describe the underlying philosophy behind the derivation of the new ﬁlter as nonlinear ﬁltering through linear estimation theory, hence the name “cubature Kalman ﬁlter.” The CKF is numerically accurate and easily extendable to high-dimensional problems. The rest of the paper is organized as follows: Section II derives the Bayesian ﬁlter theory in the Gaussian domain. Section III describes numerical methods available for moment integrals encountered in the Bayesian ﬁlter. The cubature Kalman ﬁlter, using a third-degree spherical-radial cubature rule, is derived in Section IV. Our argument for choosing a third-degree rule is articulated in Section V. We go on to derive a square-root version of the CKF for improved numerical stability in Section VI. The existing sigma-point approach is compared with the cubature method in Section VII. We apply the CKF in two nonlinear state estimation problems in Section VIII. Section IX concludes the paper with a possible extension of the CKF algorithm for a more general setting.1256IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 6, JUNE 2009II. BAYESIAN FILTER THEORY IN THE GAUSSIAN DOMAIN The key approximation taken to develop the Bayesian ﬁlter theory under the Gaussian domain is that the predictive density and the ﬁlter likelihood density are both Gaussian, which eventually leads to a Gaussian posterior den. The Gaussian is the most convenient and widely sity used density function for the following reasons: • It has many distinctive mathematical properties. — The Gaussian family is closed under linear transformation and conditioning. — Uncorrelated jointly Gaussian random variables are independent. • It approximates many physical random phenomena by virtue of the central limit theorem of probability theory (see Sections 5.7 and 6.7 in [19] for more details). Under the Gaussian approximation, the functional recursion of the Bayesian ﬁlter reduces to an algebraic recursion operating only on means and covariances of various conditional densities encountered in the time and the measurement updates. A. Time Update In the time update, the Bayesian ﬁlter computes the mean and the associated covariance of the Gaussian predictive density as follows: (8) where is the statistical expectation operator. Substituting (1) into (8) yieldsTABLE I KALMAN FILTERING FRAMEWORKB. Measurement Update It is well known that the errors in the predicted measurements are zero-mean white sequences [2], [20]. Under the assumption that these errors can be well approximated by the Gaussian, we write the ﬁlter likelihood density (12) where the predicted measurement (13) and the associated covariance(14) Hence, we write the conditional Gaussian density of the joint state and the measurement(15) (9) where the cross-covariance is assumed to be zero-mean and uncorrelated Because with the past measurements, we get (16) On the receipt of a new measurement , the Bayesian ﬁlter from (15) yielding computes the posterior density (17) (10) where is the conventional symbol for a Gaussian density. Similarly, we obtain the error covariance where (18) (19) (20) If and are linear functions of the state, the Bayesian ﬁlter under the Gaussian assumption reduces to the Kalman ﬁlter. Table I shows how quantities derived above are called in the Kalman ﬁltering framework. The signal-ﬂow diagram in Fig. 2 summarizes the steps involved in the recursion cycle of the Bayesian ﬁlter. The heart of the Bayesian ﬁlter is therefore how to compute Gaussian(11)ARASARATNAM AND HAYKIN: CUBATURE KALMAN FILTERS1257Fig. 2. Signal-ﬂow diagram of the recursive Bayesian ﬁlter under the Gaussian assumption, where “G-” stands for “Gaussian-.”weighted integrals whose integrands are all of the form nonGaussian density that are present in (10), linear function (11), (13), (14) and (16). The next section describes numerical integration methods to compute multi-dimensional weighted integrals. III. REVIEW ON NUMERICAL METHODS FOR MOMENT INTEGRALS Consider a multi-dimensional weighted integral of the form (21) is some arbitrary function, is the region of where for all integration, and the known weighting function . In a Gaussian-weighted integral, for example, is a Gaussian density and satisﬁes the nonnegativity condition in the entire region. If the solution to the above integral (21) is difﬁcult to obtain, we may seek numerical integration methods to compute it. The basic task of numerically computing the integral (21) is to ﬁnd a set of points and weights that approximates by a weighted sum of function evaluations the integral (22) The methods used to ﬁnd can be divided into product rules and non-product rules, as described next. A. Product Rules ), we For the simplest one-dimensional case (that is, may apply the quadrature rule to compute the integral (21) numerically [21], [22]. In the context of the Bayesian ﬁlter, we mention the Gauss-Hermite quadrature rule; when the is in the form of a Gaussian density weighting functionis well approximated by a polynomial and the integrand in , the Gauss-Hermite quadrature rule is used to compute the Gaussian-weighted integral efﬁciently [12]. The quadrature rule may be extended to compute multidimensional integrals by successively applying it in a tensorproduct of one-dimensional integrals. Consider an -point per dimension quadrature rule that is exact for polynomials of points for functional degree up to . We set up a grid of evaluations and numerically compute an -dimensional integral while retaining the accuracy for polynomials of degree up to only. Hence, the computational complexity of the product quadrature rule increases exponentially with , and therefore , suffers from the curse of dimensionality. Typically for the product Gauss-Hermite quadrature rule is not a reasonable choice to approximate a recursive optimal Bayesian ﬁlter. B. Non-Product Rules To mitigate the curse of dimensionality issue in the product rules, we may seek non-product rules for integrals of arbitrary dimensions by choosing points directly from the domain of integration [18], [23]. Some of the well-known non-product rules include randomized Monte Carlo methods [4], quasi-Monte Carlo methods [24], [25], lattice rules [26] and sparse grids [27]–[29]. The randomized Monte Carlo methods evaluate the integration using a set of equally-weighted sample points drawn randomly, whereas in quasi-Monte Carlo methods and lattice rules the points are generated from a unit hyper-cube region using deterministically deﬁned mechanisms. On the other hand, the sparse grids based on Smolyak formula in principle, combine a quadrature (univariate) routine for high-dimensional integrals more sophisticatedly; they detect important dimensions automatically and place more grid points there. Although the non-product methods mentioned here are powerful numerical integration tools to compute a given integral with a prescribed accuracy, they do suffer from the curse of dimensionality to certain extent [30].1258IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 6, JUNE 2009C. Proposed Method In the recursive Bayesian estimation paradigm, we are interested in non-product rules that i) yield reasonable accuracy, ii) require small number of function evaluations, and iii) are easily extendable to arbitrarily high dimensions. In this paper we derive an efﬁcient non-product cubature rule for Gaussianweighted integrals. Speciﬁcally, we obtain a third-degree fullysymmetric cubature rule, whose complexity in terms of function evaluations increases linearly with the dimension . Typically, a set of cubature points and weights are chosen so that the cubature rule is exact for a set of monomials of degree or less, as shown by (23)Gaussian density. Speciﬁcally, we consider an integral of the form (24)deﬁned in the Cartesian coordinate system. To compute the above integral numerically we take the following two steps: i) We transform it into a more familiar spherical-radial integration form ii) subsequently, we propose a third-degree spherical-radial rule. A. Transformation In the spherical-radial transformation, the key step is a change of variable from the Cartesian vector to a radius and with , so direction vector as follows: Let for . Then the integral (24) can be that rewritten in a spherical-radial coordinate system as (25) is the surface of the sphere deﬁned by and is the spherical surface measure or the area element on . We may thus write the radial integral (26) is deﬁned by the spherical integral with the unit where weighting function (27) The spherical and the radial integrals are numerically computed by the spherical cubature rule (Section IV-B below) and the Gaussian quadrature rule (Section IV-C below), respectively. Before proceeding further, we introduce a number of notations and deﬁnitions when constructing such rules as follows: • A cubature rule is said to be fully symmetric if the following two conditions hold: implies , where is any point obtainable 1) from by permutations and/or sign changes of the coordinates of . on the region . That is, all points in 2) the fully symmetric set yield the same weight value. For example, in the one-dimensional space, a point in the fully symmetric set implies that and . • In a fully symmetric region, we call a point as a generator , where if , . The new should not be confused with the control input . zero coordinates and use • For brevity, we suppress to represent a complete fully the notation symmetric set of points that can be obtained by permutating and changing the sign of the generator in all possible ways. Of course, the complete set entails where; are non-negative integers and . Here, an important quality criterion of a cubature rule is its degree; the higher the degree of the cubature rule is, the more accurate solution it yields. To ﬁnd the unknowns of the cubature rule of degree , we solve a set of moment equations. However, solving the system of moment equations may be more tedious with increasing polynomial degree and/or dimension of the integration domain. For example, an -point cubature rule entails unknown parameters from its points and weights. In general, we may form a system of equations with respect to unknowns from distinct monomials of degree up to . For the nonlinear system to have at least one solution (in this case, the system is said to be consistent), we use at least as many unknowns as equations [31]. That is, we choose to be . Suppose we obtain a cu. In this case, we solve bature rule of degree three for nonlinear moment equations; the re) sulting rule may consist of more than 85 ( weighted cubature points. To reduce the size of the system of algebraically independent equations or equivalently the number of cubature points markedly, Sobolev proposed the invariant theory in 1962 [32] (see also [31] and the references therein for a recent account of the invariant theory). The invariant theory, in principle, discusses how to restrict the structure of a cubature rule by exploiting symmetries of the region of integration and the weighting function. For example, integration regions such as the unit hypercube, the unit hypersphere, and the unit simplex exhibit symmetry. Hence, it is reasonable to look for cubature rules sharing the same symmetry. For the case considered above and ), using the invariant theory, we may con( cubature points struct a cubature rule consisting of by solving only a pair of moment equations (see Section IV). Note that the points and weights of the cubature rule are in. Hence, they can be computed dependent of the integrand off-line and stored in advance to speed up the ﬁlter execution. where IV. CUBATURE KALMAN FILTER As described in Section II, nonlinear ﬁltering in the Gaussian domain reduces to a problem of how to compute integrals, whose integrands are all of the form nonlinear functionARASARATNAM AND HAYKIN: CUBATURE KALMAN FILTERS1259points when are all distinct. For example, represents the following set of points:Here, the generator is • We use . set B. Spherical Cubature Rule. to denote the -th point from theWe ﬁrst postulate a third-degree spherical cubature rule that takes the simplest structure due to the invariant theory (28) The point set due to is invariant under permutations and sign changes. For the above choice of the rule (28), the monomials with being an odd integer, are integrated exactly. In order that this rule is exact for all monomials of degree up to three, it remains to require that the rule is exact , 2. Equivalently, to for all monomials for which ﬁnd the unknown parameters and , it sufﬁces to consider , and due to the fully symmonomials metric cubature rule (29) (30) where the surface area of the unit sphere with . Solving (29) and (30) , and . Hence, the cubature points are yields located at the intersection of the unit sphere and its axes. C. Radial Rule We next propose a Gaussian quadrature for the radial integration. The Gaussian quadrature is known to be the most efﬁcient numerical method to compute a one-dimensional integration [21], [22]. An -point Gaussian quadrature is exact and constructed as up to polynomials of degree follows: (31) where is a known weighting function and non-negative on ; the points and the associated weights the interval are unknowns to be determined uniquely. In our case, a comparison of (26) and (31) yields the weighting function and and , respecthe interval to be tively. To transform this integral into an integral for which the solution is familiar, we make another change of variable via yielding. The integral on the right-hand side of where (32) is now in the form of the well-known generalized GaussLaguerre formula. The points and weights for the generalized Gauss-Laguerre quadrature are readily obtained as discussed elsewhere [21]. A ﬁrst-degree Gauss-Laguerre rule is exact for . Equivalently, the rule is exact for ; it . is not exact for odd degree polynomials such as Fortunately, when the radial-rule is combined with the spherical rule to compute the integral (24), the (combined) spherical-radial rule vanishes for all odd-degree polynomials; the reason is that the spherical rule vanishes by symmetry for any odd-degree polynomial (see (25)). Hence, the spherical-radial rule for (24) is exact for all odd degrees. Following this argument, for a spherical-radial rule to be exact for all third-degree polyno, it sufﬁces to consider the ﬁrst-degree genermials in alized Gauss-Laguerre rule entailing a single point and weight. We may thus write (33) where the point is chosen to be the square-root of the root of the ﬁrst-order generalized Laguerre polynomial, which is orthogonal with respect to the modiﬁed weighting function ; subsequently, we ﬁnd by solving the zeroth-order moment equation appropriately. In this case, we , and . A detailed account have of computing the points and weights of a Gaussian quadrature with the classical and nonclassical weighting function is presented in [33]. D. Spherical-Radial Rule In this subsection, we describe two useful results that are used to i) combine the spherical and radial rule obtained separately, and ii) extend the spherical-radial rule for a Gaussian weighted integral. The respective results are presented as two propositions: Proposition 4.1: Let the radial integral be computed numer-point Gaussian quadrature rule ically by theLet the spherical integral be computed numerically by the -point spherical ruleThen, an by-point spherical-radial cubature rule is given(34) Proof: Because cubature rules are devised to be exact for a subspace of monomials of some degree, we consider an integrand of the form(32)1260IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 6, JUNE 2009where are some positive integers. Hence, we write the integral of interestwhereFor the moment, we assume the above integrand to be a mono. Making the mial of degree exactly; that is, change of variable as described in Section IV-A, we getWe use the cubature-point set to numerically compute integrals (10), (11), and (13)–(16) and obtain the CKF algorithm, details of which are presented in Appendix A. Note that the above cubature-point set is now deﬁned in the Cartesian coordinate system. V. IS THERE A NEED FOR HIGHER-DEGREE CUBATURE RULES? In this section, we emphasize the importance of third-degree cubature rules over higher-degree rules (degree more than three), when they are embedded into the cubature Kalman ﬁltering framework for the following reasons: • Sufﬁcient approximation. The CKF recursively propagates the ﬁrst two-order moments, namely, the mean and covariance of the state variable. A third-degree cubature rule is also constructed using up to the second-order moment. Moreover, a natural assumption for a nonlinearly transformed variable to be closed in the Gaussian domain is that the nonlinear function involved is reasonably smooth. In this case, it may be reasonable to assume that the given nonlinear function can be well-approximated by a quadratic function near the prior mean. Because the third-degree rule is exact up to third-degree polynomials, it computes the posterior mean accurately in this case. However, it computes the error covariance approximately; for the covariance estimate to be more accurate, a cubature rule is required to be exact at least up to a fourth degree polynomial. Nevertheless, a higher-degree rule will translate to higher accuracy only if the integrand is well-behaved in the sense of being approximated by a higher-degree polynomial, and the weighting function is known to be a Gaussian density exactly. In practice, these two requirements are hardly met. However, considering in the cubature Kalman ﬁltering framework, our experience with higher-degree rules has indicated that they yield no improvement or make the performance worse. • Efﬁcient and robust computation. The theoretical lower bound for the number of cubature points of a third-degree centrally symmetric cubature rule is given by twice the dimension of an integration region [34]. Hence, the proposed spherical-radial cubature rule is considered to be the most efﬁcient third-degree cubature rule. Because the number of points or function evaluations in the proposed cubature rules scales linearly with the dimension, it may be considered as a practical step for easing the curse of dimensionality. According to [35] and Section 1.5 in [18], a ‘good’ cubature rule has the following two properties: (i) all the cubature points lie inside the region of integration, and (ii) all the cubature weights are positive. The proposed rule equal, positive weights for an -dimensional entails unbounded region and hence belongs to a good cubature family. Of course, we hardly ﬁnd higher-degree cubature rules belonging to a good cubature family especially for high-dimensional integrations.Decomposing the above integration into the radial and spherical integrals yieldsApplying the numerical rules appropriately, we haveas desired. As we may extend the above results for monomials of degree less than , the proposition holds for any arbitrary integrand that can be written as a linear combination of monomials of degree up to (see also [18, Section 2.8]). Proposition 4.2: Let the weighting functions and be and . such that , we Then for every square matrix have (35) Proof: Consider the left-hand side of (35). Because a positive deﬁnite matrix, we factorize to be , we get Making a change of variable via is .which proves the proposition. For the third-degree spherical-radial rule, and . Hence, it entails a total of cubature points. Using the above propositions, we extend this third-degree spherical-radial rule to compute a standard Gaussian weighted integral as follows:ARASARATNAM AND HAYKIN: CUBATURE KALMAN FILTERS1261In the ﬁnal analysis, the use of higher-degree cubature rules in the design of the CKF may marginally improve its performance at the expense of a reduced numerical stability and an increased computational cost. VI. SQUARE-ROOT CUBATURE KALMAN FILTER This section addresses i) the rationale for why we need a square-root extension of the standard CKF and ii) how the square-root solution can be developed systematically. The two basic properties of an error covariance matrix are i) symmetry and ii) positive deﬁniteness. It is important that we preserve these two properties in each update cycle. The reason is that the use of a forced symmetry on the solution of the matrix Ricatti equation improves the numerical stability of the Kalman ﬁlter [36], whereas the underlying meaning of the covariance is embedded in the positive deﬁniteness. In practice, due to errors introduced by arithmetic operations performed on ﬁnite word-length digital computers, these two properties are often lost. Speciﬁcally, the loss of the positive deﬁniteness may probably be more hazardous as it stops the CKF to run continuously. In each update cycle of the CKF, we mention the following numerically sensitive operations that may catalyze to destroy the properties of the covariance: • Matrix square-rooting [see (38) and (43)]. • Matrix inversion [see (49)]. • Matrix squared-form amplifying roundoff errors [see (42), (47) and (48)]. • Substraction of the two positive deﬁnite matrices present in the covariant update [see (51)]. Moreover, some nonlinear ﬁltering problems may be numerically ill-conditioned. For example, the covariance is likely to turn out to be non-positive deﬁnite when i) very accurate measurements are processed, or ii) a linear combination of state vector components is known with greater accuracy while other combinations are essentially unobservable [37]. As a systematic solution to mitigate ill effects that may eventually lead to an unstable or even divergent behavior, the logical procedure is to go for a square-root version of the CKF, hereafter called square-root cubature Kalman ﬁlter (SCKF). The SCKF essentially propagates square-root factors of the predictive and posterior error covariances. Hence, we avoid matrix square-rooting operations. In addition, the SCKF offers the following beneﬁts [38]: • Preservation of symmetry and positive (semi)deﬁniteness of the covariance. Improved numerical accuracy owing to the fact that , where the symbol denotes the condition number. • Doubled-order precision. To develop the SCKF, we use (i) the least-squares method for the Kalman gain and (ii) matrix triangular factorizations or triangularizations (e.g., the QR decomposition) for covariance updates. The least-squares method avoids to compute a matrix inversion explicitly, whereas the triangularization essentially computes a triangular square-root factor of the covariance without square-rooting a squared-matrix form of the covariance. Appendix B presents the SCKF algorithm, where all of the steps can be deduced directly from the CKF except for the update of the posterior error covariance; hence we derive it in a squared-equivalent form of the covariance in the appendix.The computational complexity of the SCKF in terms of ﬂops, grows as the cube of the state dimension, hence it is comparable to that of the CKF or the EKF. We may reduce the complexity signiﬁcantly by (i) manipulating sparsity of the square-root covariance carefully and (ii) coding triangularization algorithms for distributed processor-memory architectures. VII. A COMPARISON OF UKF WITH CKF Similarly to the CKF, the unscented Kalman ﬁlter (UKF) is another approximate Bayesian ﬁlter built in the Gaussian domain, but uses a completely different set of deterministic weighted points [10], [39]. To elaborate the approach taken in the UKF, consider an -dimensional random variable having with mean and covariance a symmetric prior density , within which the Gaussian is a special case. Then a set of sample points and weights, are chosen to satisfy the following moment-matching conditions:Among many candidate sets, one symmetrically distributed sample point set, hereafter called the sigma-point set, is picked up as follows:where and the -th column of a matrix is denoted ; the parameter is used to scale the spread of sigma points by from the prior mean , hence the name “scaling parameter”. Due to its symmetry, the sigma-point set matches the skewness. Moreover, to capture the kurtosis of the prior density closely, it is sug(Appendix I of [10], gested that we choose to be [39]). This choice preserves moments up to the ﬁfth order exactly in the simple one-dimensional Gaussian case. In summary, the sigma-point set is chosen to capture a number as correctly as of low-order moments of the prior density possible. Then the unscented transformation is introduced as a method that are related to of computing posterior statistics of by a nonlinear transformation . It approximates the mean and the covariance of by a weighted sum of projected space, as shown by sigma points in the(36)(37)。

信号处理中英文对照外文翻译文献

信号处理中英文对照外文翻译文献（文档含英文原文和中文翻译）译文：一小波研究的意义与背景在实际应用中，针对不同性质的信号和干扰，寻找最佳的处理方法降低噪声，一直是信号处理领域广泛讨论的重要问题。

目前有很多方法可用于信号降噪，如中值滤波，低通滤波，傅立叶变换等，但它们都滤掉了信号细节中的有用部分。

传统的信号去噪方法以信号的平稳性为前提，仅从时域或频域分别给出统计平均结果。

根据有效信号的时域或频域特性去除噪声，而不能同时兼顾信号在时域和频域的局部和全貌。

更多的实践证明，经典的方法基于傅里叶变换的滤波，并不能对非平稳信号进行有效的分析和处理，去噪效果已不能很好地满足工程应用发展的要求。

常用的硬阈值法则和软阈值法则采用设置高频小波系数为零的方法从信号中滤除噪声。

实践证明，这些小波阈值去噪方法具有近似优化特性，在非平稳信号领域中具有良好表现。

小波理论是在傅立叶变换和短时傅立叶变换的基础上发展起来的，它具有多分辨分析的特点，在时域和频域上都具有表征信号局部特征的能力，是信号时频分析的优良工具。

小波变换具有多分辨性、时频局部化特性及计算的快速性等属性，这使得小波变换在地球物理领域有着广泛的应用。

随着技术的发展，小波包分析 (Wavelet Packet Analysis) 方法产生并发展起来，小波包分析是小波分析的拓展，具有十分广泛的应用价值。

它能够为信号提供一种更加精细的分析方法，它将频带进行多层次划分，对离散小波变换没有细分的高频部分进一步分析，并能够根据被分析信号的特征，自适应选择相应的频带，使之与信号匹配，从而提高了时频分辨率。

小波包分析 (wavelet packet analysis) 能够为信号提供一种更加精细的分析方法，它将频带进行多层次划分，对小波分析没有细分的高频部分进一步分解，并能够根据被分析信号的特征，自适应地选择相应频带 , 使之与信号频谱相匹配，因而小波包具有更广泛的应用价值。

利用小波包分析进行信号降噪，一种直观而有效的小波包去噪方法就是直接对小波包分解系数取阈值，选择相关的滤波因子，利用保留下来的系数进行信号的重构，最终达到降噪的目的。

noise的意思用法总结

noise的意思用法总结noise有噪音,嘈杂声，吵闹声,声音，声响,杂音的意思。

那你们想知道noise的用法吗?今日我给大家带来了noise的用法 ,期望能够帮忙到大家，一起来学习吧。

noise的意思n. 噪音,嘈杂声，吵闹声,声音，声响,杂音vt. 谣传,哄传,传奇vi. 发出声音,大声谈论变形：过去式: noised; 现在分词：noising; 过去分词：noised;noise用法noise可以用作名词noise的基本意思是“噪声”“吵闹声”,指刺耳、尖锐的声音,有时是混合的或多种声音夹杂在一起,含有使人不开心之义。

noise作“吵闹声”讲一般只用其单数形式。

在表示各种不同的声音时可用noises。

noise也可泛指一般“声音”,既可用作可数名词,也可用作不行数名词。

noise用作名词的用法例句The loud noise from the nearby factory chafed him.四周工厂的噪声使他烦燥。

The dog perked its ears at the noise.一听到噪声，狗就竖起了耳朵There is so much noise in this restaurant; I can hardly hear you talking.这个餐厅里太嘈杂了，我几乎听不到你说话。

noise用法例句1、Sightseers may be a little overwhelmed by the crowds and noise.拥挤的人群和吵闹的噪音可能会让游客有些茫然不知所措。

2、Flying at 1,000 ft. he heard a peculiar noise from the rotors.在1,000英尺的高度飞行时，他听到旋翼发出一种惊奇的噪音。

3、With a low-pitched rumbling noise, the propeller began to rotate.伴随着隆隆的低沉噪声，螺旋桨开头旋转起来。

基于正交小波变换的目标检测方法

基于正交小波变换的目标检测方法孙宏岩【摘要】理论分析表明,独立高斯噪声经过正交小波变换后保持方差和独立性不变.基于Mallat的小波多分辨分析,通过对小波系数进行平方律处理,建立了基于正交小波变换的恒虚警率检测器模型,推导了相应的虚警和检测概率公式,分析了噪声未知情况下小波系数序列长度对检测性能的影响,并给出了合适的长度值.实验结果表明,所提出的检测器能满足不同虚警概率和杂波背景的要求,具有较好自适应性.【期刊名称】《海军航空工程学院学报》【年(卷),期】2015(030)005【总页数】5页(P414-418)【关键词】信号检测;正交小波变换;多分辨分析;高斯噪声【作者】孙宏岩【作者单位】海军装备部航空技术保障部,北京100071【正文语种】中文【中图分类】TN957具有自动检测的智能化雷达是现代雷达的发展趋势，而自动检测过程的一个重要部分是恒虚警率（CFAR）处理[1-3]。

CFAR设计的目的是提供相对来说可以避免噪声背景杂波和干扰变化影响的检测阈值，并且当与到达的样本进行比较时，使目标检测具有恒定的虚警概率。

另一方面，自从Donoho提出基于小波变换的软阈值消噪方法[4]以来，小波变换在信号检测方面显示了巨大的潜力，至今已在信号分析、图像处理、量子力学、雷达、计算机分类与识别、数据压缩、边缘检测等方面得到了广泛的应用。

传统的CFAR方法多在时域对信号进行处理[5-7]，关于频域CFAR处理也提出过一定的方法[8]。

小波变换是一种时间窗和频率窗都可改变的时频局部分析方法，在处理非平稳信号方面有着独特的优势。

针对雷达信号的非平稳性，研究小波域CFAR处理方法具有重要意义。

本文首先对含高斯噪声信号进行正交小波多尺度分析，基于高斯噪声在小波域中的特性，通过对小波系数进行平方律处理，建立了基于正交小波变换的恒虚警率（OW-CFAR）检测器模型，并给出了相应的虚警和检测概率公式，分析了信号小波系数序列长度对检测性能的影响。

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Take a window centered in x and size (2m+1 2m+ 1), A(x,m) Take a window centered in x and size (2n+1 X 2n+ 1), W(x,n).
wmax=O;
X
For each pixel y in A(x,m) and y different from x, compute the difference between W(x,n) and W(y,n) as d(x,y). Compute the weight from the distance d(x,y) as w(x,y) = exp (- d(x,y) / h); If w(x,y) is bigger than Wmax then Wmax = w(x,y); Compute the average average + = w(x,y) * u(y); Carry the sum of the weights, totalweight + = w(x, y); Step 4. Give to x the maximum of the other weights, average + = Wmax * u(x); totalweight + = Wmax; Compute the restored value, rest(x) = average / totalweight; Step 5. The distance is calculated as follows: function distance (x,y,n) { distancetotal = 0 ; distance = (u (x) - u (y)r2; for k= 1 until n {
Stepl. Step2.
I.
INTRODUCTION
Efforts from most de-noising methods are to separate the image into the oscillation free part (true image contents) and the oscillatory part (noisy contents) by separating the high frequencies from the low frequencies. Images can contain high frequency components like fine details and structures. The moment, the high frequencies are filtered, the high frequency contents of the true image will be filtered along with the high frequency noise since the techniques cannot distinguish between the noise and true image. It will lead to a loss of fine detail in the de-noised image. Also, the issue to filter the low frequency noise from the image is a concern. Low frequency noise will continue to be there in the image even after de noising. As a remedy on the loss of image details after noise filtering, Baudes A., Coll B., Morel I.M. have developed the Non-Local Means (NL-Means) algorithm [4]. In this paper, NL-Means algorithm for standard database images and natural photograph images captured by general digital camera is implemented. The paper is organised as follows: Section I gives brief introduction, section II deals with the NL-Means algorithm, its Pseudo code and the post-processing, section III with experimentation and results while conclusions are given in section IV and future scope in section V.
DixitA. A. Phadke A. C.
Department of Electronics and Telecommunication Maharashtra Institute of Technology Pune- 411038, India aarya dixit@
II. BASIC NL-MEANS ALGORITHM [4],
A.
PSEUDO CODE AND
POST PROCESSING FILTER
[5]
Basic NL-Means algorithm
The self-similarity assumption can be exploited to de-noise an image. Image pixels with similar neighborhoods can be used to determine the de-noised value of a pixel. Weights are assigned to pixels on the basis of their similarity with the pixel being reconstructed. While assessing the similarity, the pixel under consideration as well as its neighbourhood is taken into account. Mathematically, it can be expressed as:
(2)
Cex) is a normalizing constant. Ga is a Gaussian kernel and h is a filtering parameter [4].
B. Pseudo code for NL-Means algorithm- For each pixel x
Keywords-de-noising; Noise Standard Deviation; Non-local Means; PSNR; visual quality
Step3.
978-1-4673-4922-2/13/$31.00 ©2013 IEEE
1215
NL[u](x)=-feC(x)
1
l (x+. )-u(y+. )12) (0) (Ga' u uy dy () hZ
(1)
The integration is carried out over all the pixels in the search window. Where
c(x)= fe-
l (x+. )-u(y+. )IZ) (0) (Ga' u dz h2
_
Department of Electronics and Telecommunication Maharashtra Institute of Technology Pune- 411038, India anuradha.phadke@se removal and image enhancement are the important tasks addressed by many Image Processing algorithms, especially, when the images are corrupted by high noise level e.g. in the case of remote imaging, thermal imaging, night vision etc. The noise makes the image recognition more difficult as it gives a grainy, snowy or textured appearance to the image. So there exists a need for efficient image de-noising method without introducing any artifacts in the original image. The Images with a Noise Standard Deviation (Sigma) greater than 25 are considered as high noise images. The dark images, e.g. night shoots, have very low dynamic range of brightness. The darkness and the high noise needs to be carefully tackled by the image processing algorithm for acceptable visual quality e.g. surveillance applications. Furthermore, de-noising is often necessary as a pre-processing step in image compression, segmentation, recognition etc. Basically, the image de-noising methods are divided into two types: local and non local. A non local method called as Non-Local Means [4J estimates a noise-free pixel intensity as a weighted average of all pixel intensities in the image, and the weights are proportional to the similarity between the local neighbourhood of the pixel being processed and local neighbourhoods of surrounding pixels. The method is quite spontaneous that results in PSNR and visual quality comparable with other de-noising methods.