Transformation Functions for Image Registration
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1 Transformation Functions for Image Registration A. Goshtasby Wright State University 6/16/2011 CVPR 2011 Tutorial 6, Introduction 1
2 Problem Definition Given n corresponding points in two images: find a transformation function with two components to satisfy: 6/16/2011 CVPR 2011 Tutorial 6, Introduction 2
3 Categorization of Transformation Functions Transformations with afixednumber of parameters -- Translation -- Rigid -- Similarity -- Affine -- Projective -- Transformations with a varying number of parameters -- Explicit -- Parametric -- Implicit 6/16/2011 CVPR 2011 Tutorial 6, Introduction 3
4 Translation (X,Y): Point in reference image (x,y): Point in sensed image Knowing a pair of corresponding points in the images, parameters h and k can be determined. d Having more than one pair of corresponding points in images, h and k should be determined by a robust estimator. 6/16/2011 CVPR 2011 Tutorial 6, Introduction 4
5 Robust Estimator RANSAC: Already covered in the earlier talk by Matas. Least Median of Squares: Similar il to RANSAC except that t instead of counting the number of points that fall within a required distance of each other with a guessed h and k, do the following: 1. For each point in the reference image find its distance to the point closest to it in the sensed image after translation by (h,k). 2. Order the (square) distances for all points in the reference image. 3. If the median value is within the required tolerance, stop. This means that the obtained (h,k) can match at least 50% of the points with the required tolerance. Otherwise, if time allows, select another pair of points from the two images, determine h and k, and go to Step 1. 6/16/2011 CVPR 2011 Tutorial 6, Introduction 5
6 Rigid Transformation Also known as the Euclidean transformation, it preserves distances and angles. Minimum 2 corresponding points are required to determine the rigid transformation parameters. 6/16/2011 CVPR 2011 Tutorial 6, Introduction 6
7 Similarity Transformation Also known as the transformation of the Cartesian coordinate system, it preserves angles. Minimum 2 corresponding points are required to determine the parameters of the similarity transformation. 6/16/2011 CVPR 2011 Tutorial 6, Introduction 7
8 Affine Transformation It preserves parallelism. Minimum 3 corresponding gpoints are required to determine the affine parameters. 6/16/2011 CVPR 2011 Tutorial 6, Introduction 8
9 Projective Transformation Also known as homography, it preserves straightness. Minimum 4 corresponding points are required to determine the projective transformation parameters. 6/16/2011 CVPR 2011 Tutorial 6, Introduction 9
10 Adaptive Transformation Functions Explicit Parametric weighted- mean Multiquadric Surface (thin-plate) spline Compactly supported RBF Local weighted linear Moving LSQ Piecewise polynomials Subdivision surfaces Weighted linear Parametric Implicit Interpolating RBF 6/16/2011 CVPR 2011 Tutorial 6, Introduction 10
11 Multiquadric Each component of the transformation is a single-valued surface defined by: d is a measure of stiffness of the surface. The larger its value, the less detailed the obtained surface. Ai: i=1,,n are determine knowing n corresponding points in images. Monotonically increasing radial basis functions. 6/16/2011 CVPR 2011 Tutorial 6, Introduction 11
12 Reference Image Sensed Image Red corresponding points are used to register images. Blue corresponding points are used to evaluate registration. Resampled Sensed Image, d=12 Overlaid Reference and Resampled Images 6/16/2011 CVPR 2011 Tutorial 6, Introduction 12
13 Surface Spline Also known as thin-plate spline (TPS): d shows stiffness of surface. A1, A2, A3, Bi: i=1,,n are determine knowing n corresponding points. Monotonically increasing radial basis functions. 6/16/2011 CVPR 2011 Tutorial 6, Introduction 13
14 Images registered by surface spline transformation function, d=0. 6/16/2011 CVPR 2011 Tutorial 6, Introduction 14
15 Compactly Supported RBF Wendland s Ai:i=1 i 1,,n are determine knowing n corresponding points in images. Monotonically decreasing radial basis functions. 6/16/2011 CVPR 2011 Tutorial 6, Introduction 15
16 Images registered by Wendland s compactly supported RBF, a= /16/2011 CVPR 2011 Tutorial 6, Introduction 16
17 Local Weighted Linear Given points {(xi,yi,fi): i=1,,n} find f (x,y) approximating the points. Maude s Rk shows distance of (x,y) y)tothekth the closest data point. Li(x,y) is a linear function that evaluates to Fi at (xi,yi) and fits the k points closest to (x,y) by the least-squares. Computation of f (x,y) does not involve solution of a large system of equations. f (x,y) is obtained by a weighted mean of the planes. 6/16/2011 CVPR 2011 Tutorial 6, Introduction 17
18 Images registered by Maude s local weighted linear, k=10. 6/16/2011 CVPR 2011 Tutorial 6, Introduction 18
19 Moving Least Squares Given points {pi = (xi,yi): i=1,,n}, } with associating data values {Fi: i=1,,n}, find function f (p) that minimizes: at each p = (x,y). Wi(p) ( )is a non-negative monotonically decreasing radial function centered at pi, such as: Function f (p) is a low-degree polynomial in x and y. 6/16/2011 CVPR 2011 Tutorial 6, Introduction 19
20 Images registered by moving least-squares with polynomials of degree 1. 6/16/2011 CVPR 2011 Tutorial 6, Introduction 20
21 Piecewise Linear Interpolation 1. Triangulate points in reference image. 2. Knowing correspondence between points in images, find corresponding triangles in sensed image. 3. Register corresponding triangles in images by an affine transformation function (two polynomials of degree one). Note that this only registers the area within the convex hull of the control points in images. 6/16/2011 CVPR 2011 Tutorial 6, Introduction 21
22 Images registered by a piecewise-linear transformation function. 6/16/2011 CVPR 2011 Tutorial 6, Introduction 22
23 Subdivision Surfaces These are finer piecewise linear approximations to piecewise polynomial functions. Loop subdivision: Produces an approximating surface. 6/16/2011 CVPR 2011 Tutorial 6, Introduction 23
24 Images registered using the Loop subdivision surface as a component of Transformation 6/16/2011 CVPR 2011 Tutorial 6, Introduction 24
25 Weighted Mean Interpolation Note the weights are rational, positive, and have sum of 1 everywhere. Also note Wi(xi,yi) ( ) = 1, so f (x,y) interpolates data. 6/16/2011 CVPR 2011 Tutorial 6, Introduction 25
26 Weakness: It produces flat spots at interpolating i points. Ideal surface Obtained surface 6/16/2011 CVPR 2011 Tutorial 6, Introduction 26
27 Weighted-Linear Transformation Instead of a point, consider a plane that passes through the point and provides the desired gradient. Use a weighted sum of the planes as the surface interpolating the points. 6/16/2011 CVPR 2011 Tutorial 6, Introduction 27
28 Images registered by weighted linear transformation function. 6/16/2011 CVPR 2011 Tutorial 6, Introduction 28
29 Parametric Weighted-Mean nc, nr: number of columns and rows in reference image 6/16/2011 CVPR 2011 Tutorial 6, Introduction 29
30 Image s registered using parametric weighted-mean interpolation. 6/16/2011 CVPR 2011 Tutorial 6, Introduction 30
31 Interpolating Implicit Surface Function interpolates {pi = (xi,yi,fi): i=1 1,,n} n}iff f (pi) = 0 for i=1 1,,n. ( p pi ) is a monotonically increasing radial function centered at pi, such as ( p pi ) = p pi, and L(p) is an optional linear polynomial. To avoid trivial solution Ai = 0, i=1,,n, when L(p) = 0, virtual point pn+1 is added below and virtual point pn+2 is added above the surface to be determined and f (pn+1) is set to a very large negative value and f (pn+2) is set to a very large positive value. Quantize f to an appropriate volume and threshold at 0. The obtained zero-crossings represent the desired surface. 6/16/2011 CVPR 2011 Tutorial 6, Introduction 31
32 Images registered by implicit interpolating transformation function. 6/16/2011 CVPR 2011 Tutorial 6, Introduction 32
33 Evaluation Mountain Rock Face 6/16/2011 CVPR 2011 Tutorial 6, Introduction 33
34 Evaluation Terrain Aerial Parking 6/16/2011 CVPR 2011 Tutorial 6, Introduction 34
35 6/16/2011 CVPR 2011 Tutorial 6, Introduction 35
36 Registration by Moving LQ 6/16/2011 CVPR 2011 Tutorial 6, Introduction 36
37 6/16/2011 CVPR 2011 Tutorial 6, Introduction 37
38 References 1. A. Goshtasby, 2-D and 3-D Image Registration for Medical, Remote Sensing, and dindustrial lapplications, i Wiley Press, P. J. Rousseeuw, Least median of squares regression, J. American Statistical Association, vol. 79, no. 388, pp , R. L. Hardy, Theory and applications of the multiquadricbiharmonic method 20 years of discovery , Computers and Mathematics with Applications, vol. 19, no. 8/9, pp , R. L. Harder and R. N. Desmarais, Interpolation using surface splines, J. Aircraft, vol. 9, no. 2, pp , H. Wendland, Piecewise polynomial, positive definite and compactly supported radial lfunctions of minimal i ldegree, Advances in Computational Mathematics, vol. 4, pp , A. D. Maude, Interpolation mainly for graph plotters, The Computer Journal, vol. 16, no. 1, pp , /16/2011 CVPR 2011 Tutorial 6, Introduction 38
39 References 7. P. Lancaster and K. Salkauskas, Surfaces generated by moving least squares methods, Mathematics ti of Computation, ti vol. 37, no. 155, pp , A. Goshtasby, Piecewise linear mapping functions for image registration, Pattern Recognition, vol. 19, no. 6, pp , A. Goshtasby, A weighted linear method for approximation of irregularly spaced data, Geometric Modeling and Computing, M. M. Lucian and M. Neamtu (Eds.), Nashboro Press, Brentwood, TN, pp , D. Shepard, A two-dimensional interpolation function for irregularly spaced data, Proc. 23rd Nat l Conf. ACM, pp , J.C.Carr,R.K.Beatson, Carr, J. B. Cherrie, T. J. Mitchell, W. R. Fright, B. C. McCallum, T. R. Evans, Reconstruction and representation of 3D objects with radial basis functions, Proc. SIGGRAPH 01, pp , /16/2011 CVPR 2011 Tutorial 6, Introduction 39
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