IDL 图像处理第2讲

DIP Lecture 2
1
Spatial Transformation
In a spatial transformation each point (x, y) of image A is mapped to a point (u, v) in a new coordinate system. u = f1(x, y) v = f2(x, y)
DIP Lecture 2 7
Transform Operations
The transformation matrices below can be used as building blocks. 1 0 x0 T = 0 1 y0 Translation by (x0, y0) 0 0 1 s1 0 0 Scale by s1 and s2 T = 0 s2 0 0 0 1 cos θ sin θ 0 T = sin θ cos θ 0 Rotate by θ 0 0 1 You will usually want to translate the center of the image to the origin of the coordinate system, do any rotations and scalings, and then translate it back.
DIP Lecture 2
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Combinations of Transforms
Complex ane transforms can be constructed by a sequence of basic ane transforms. Transform combinations are most easily described in terms of matrix operations. To use matrix operations we introduce homogeneous coordinates. These enable all ane operations to be expressed as a matrix multiplication. Otherwise, translation is an exception. The ane equations are expressed as u a b c x v = d e f y 1 0 0 1 1 An equivalent expression using matrix notation is q = Tp where q, T and q are the dened above.
cos θ sin θ 0 s0 0 0 1 0 x0 H = RST = sin θ cos θ 0 0 s1 0 0 1 x1 0 0 1 0 0 1 0 0 1 = s0 cos θ s1 sin θ s0x0 cos θ + s1x1 sin θ s0 sin θ s1 cos θ s1x1 cos θ s0x0 sin θ
Image A
Image B
DIP Lecture 2
12
Point Matching
Find a number of points {p0, p1, . . . , pn1} in image A that match points {q0, q1, . . . , qn1} in image B. Use the homogeneous coordinate representation of each point as a column in matrices P and Q: x0 x1 . . . P = y0 y1 . . . 1 1 ... u0 u1 . . . Q = v0 v1 . . . 1 1 ... Then q = Hp becomes Q = HP The solution for H that provides the minimum mean-squared error is H = QPT (PPT )1 = QP where P = PT (PPT )1 is the (right) pseudo-inverse of P.
c13 and c23 aect translations, c11 and c22 aect magnications, and the combination aects rotations and shears.
DIP Lecture 2
3
Ane Transformation
A shear in the x direction is produced by u = x + 0.2y v=y
DIP Lecture 2
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Combined Transform Operations
Operation Translate to Origin Expression 1.00 0.00 5.00 T1 = 0.00 1.00 5.00 0.00 0.00 1.00 0.92 0.39 0.00 T2 = 0.39 0.92 0.00 0.00 0.00 1.00 Result
Mapping from (x, y) to (u, v) coordinates. A digital image array has an implicit grid that is mapped to discrete points in the new domain. These points may not fall on grid points in the new domain.
DIP Lecture 2 2
Ane Transformation
An ane transformation is any transformation that preserves collinearity (i.e., all points lying on a line initially still lie on a line after transformation) and ratios of distances (e.g., the midpoint of a line segment remains the midpoint after transformation). In general, an ane transformation is a composition of rotations, translations, magnications, and shears. u = c11x + c12y + c13 v = c21x + c22y + c23
Lecture 2: Geometric Image Transformations
Harvey Rhody Chester F. Carlson Center for Imaging Science Rochester Institute of Technology rhody@ September 8, 2005 Abstract Geometric transformations are widely used for image registration and the removal of geometric distortion. Common applications include construction of mosaics, geographical mapping, stereo and video.
1 . . . 1 un1 vn1 = q0 q1 . . . 1
pn1
qn1
Point Matching
The transformation used in the previous example can be found from a few matching points chosen randomly in each image.
Rotate by 23 degrees
Translate to original location
1.00 0.00 5.00 T3 = 0.00 1.00 5.00 0.00 0.00 1.00
DIP Lecture 2
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Composite Ane Transformation
The transformation matrix of a sequence of ane transformations, say T1 then T2 then T3 is T = T3T2T3 The composite transformation for the example above is 0.92 0.39 1.56 T = T3T2T1 = 0.39 0.92 2.35 0.00 0.00 1.00 Any combination of ane transformations formed in this way is an ane transformation. The inverse transform is T1 = T1T1T1 1 2 3 If we nd the transform in one direction, we can invert it to go the other way.
DIP Lecture 2
Spatial Transformations of Images
A spatial transformation of an image is a geometric transformation of the image coordinate system. It is often necessary to perform a spatial transformation to: Align images that were taken at dierent times or with dierent sensors Correct images for lens distortion Correct eects of camera orientation Image morphing or other special eects