SIFT的讲解

This is a 3x3 linear system
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Localization
∂2 D ∂σ 2 2 ∂ D ∂σy ∂2 D ∂σx ∂2D ∂σy ∂2D ∂y 2 ∂2D ∂yx ∂2D ∂D ∂σ ∂σx σ ∂D ∂2 D y = − ∂yx ∂y x ∂D ∂2 D ∂x ∂x 2
Derivatives approximated by finite differences,
example:
∂D Dki ,+j1 − Dki ,−j1 = ∂σ 2 ∂ 2 D Dki ,−j1 − 2 Dki , j + Dki ,+j1 = ∂σ 2 1 ∂ 2 D ( Dki +1, j − Dki +1, j ) − ( Dki −1, j − Dki −1, j ) +1 −1 +1 −1 = ∂σy 4
Multiple orientations assigned to keypoints from an orientation histogram
Significantly improve stability of matching
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Choosing the right image descriptors
Distinctive image features using local descriptors
Mudit Agrawal Aniruddha Kembhavi
CMSC828J: Object recognition and representation
1
Contents
Overall picture Region detectors Scale invariant detection Localization Orientation assignment Region description SIFT approach Local Jet Storing and matching State of the art - Video Google
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Scale Invariant Detection
The problem: how do we choose corresponding circles independently in each image?
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Scale Invariant Detection
Distribution-Based
Luminance based approaches
Histograms of pixel intensities and location SIFT – Based on gradient distribution in the region
Geometric based approaches
m( x, y ) = ( L( x + 1, y ) − L( x − 1, y )) 2 + ( L( x, y + 1) − L( x, y − 1)) 2
Hale Waihona Puke θ ( x, y ) = a tan 2(( L( x, y + 1) − L( x, y − 1)) /( L( x + 1, y ) − L( x − 1, y )))
= L( x, y, kσ ) − L( x, y,σ )
Efficient to compute Close approximation to scale-normalized Laplacian of Gaussian,
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σ 2∇ 2G
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Image space to scale space
Which is the most popularly used image descriptor ?
a. Local intensity histogram
b. SIFT
c. Local Jets
d. Fourier transform based
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Local image descriptor - SIFT
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Our goal
Detecting repeatable image regions
Obtaining reliable and distinctive descriptors
?
Searching an image database for an object efficiently
If X’ is > 0.5 in any dimension, process repeated
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Being picky!
Contrast (use prev. equation):
If | D(X) | < 0.03, throw it out
1 ∂DT ˆ ˆ D( Χ) = D + X 2 ∂Χ
Two significant eigenvalues indicate an interest point
Harris-Laplace – Invariant to rotation and scale Uses Laplacian of Gaussian operator
SIFT - Scale space extrema using Difference of Gaussian
k4σ k3σ k2σ kσ σ
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Relationship of D to σ ∇ G
2 2
Diffusion equation: ∂G = σ∇ 2G
∂σ
Approximate ∂G/∂σ:
giving,
∂G G ( x, y, kσ ) − G ( x, y,σ ) ≈ ∂σ kσ − σ
Miscellaneous
Using generalized moment invariants – characterize shape and intensity distribution
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Who wants to be a Millionaire?
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Scale Invariant Detection
Consider regions (e.g. circles) of different sizes around a point Regions of corresponding sizes will look the same in both images
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Localization
3D quadratic function is fit to the local sample points Start with Taylor expansion with sample point as the origin
where
∂DT 1 T ∂2D D(Χ) = D + Χ+ Χ Χ 2 ∂Χ 2 ∂Χ
Edge-iness:
Use ratio of principal curvatures to throw out poorly defined peaks
Dxx Dxy Curvatures come from Hessian: H = D D yy xy No need to explicitly calculate eigenvalues. We only need
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Invariance vital
Scale Rotation Orientation Illumination Noise Affine
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Region detectors
Harris points - Invariant to rotation
Shape context
Spatial-Frequency Techniques
Fourier transform based – No spatial information Gabor filters and wavelets – Large number of filters
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their ratio!
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Orientation assignment
Descriptor computed relative to keypoint’s orientation achieves rotation invariance Precomputed along with mag. for all levels (useful in descriptor computation)
Choosing the right image descriptors
Differential descriptors
Local Jets - Set of image derivatives Steerable filters – steering derivatives in the direction of the gradient
where
G ( x, y , σ ) =
1 2πσ
2
e
−( x 2 + y 2 ) / 2σ 2