Image Warping: A Review. Prof. George Wolberg Dept. of Computer Science City College of New York

Transcription

1 Image Warping: A Review Prof. George Wolberg Dept. of Computer Science City College of New York

2 Objectives In this lecture we review digital image warping: - Geometric transformations - Forward inverse mapping - Sampling - Image reconstruction - Interpolation kernels - Separable transforms - Fant s resampling algorithm 2

3 Definition Image warping deals with the geometric transformation of digital images. 3

4 Geometric Transformations Affine Perspective Bilinear Polynomial Splines Elastic (local deformations) 4

5 Spatial Transformations Forward Mapping v)] Inverse Mapping y), V(x, y)] [x, y] = [X(u, v), Y(u, [u, v] = [U(x, 5

6 Forward / Inverse Mapping Input image Output (accumulator) image v y u Input image x Output image v y u x 6

7 Sampling Point Sampling Area Sampling LPF 7

8 Area Sampling Treats pixels as finite areas Avoids aliasing (undersampling) artifacts Approximated by supersampling 8

9 Supersampling Average of projected subpixels LPF

10 Image Reconstruction Pixel values are known at integer positions Samples can project to real-valued positions How do we evaluate the image values at these real-valued positions? Reconstruction Reconstruct 10

11 Interpolation Reconstruction interpolates the input In practice, interpolation is performed at points of interest only, not entire function Interpolation is achieved by convolution 11

12 Convolution g( x) N k 0 f ( x k )h(x x k ) g(x) 12

13 Interpolation Functions Interpolation functions/kernels include: Box filter Triangle filter Cubic convolution Windowed sinc functions 13

14 Box Filter Nearest neighbor interpolation Blocky artifacts may occur 14

15 Triangle Filter Linear interpolation Popular for use with small deformations 15

16 Cubic Convolution Local cubic interpolation algorithm Advanced feature in digital cameras 16

17 Windowed Sinc Function Smoothly tapered ideal sinc function 17

18 Inverse Mapping Input image Output image v y u x Visit output in scanline order Supersampling approximates area sampling Popular in computer graphics 18

19 Forward Mapping Input image Output (accumulator) image v y u x Visit input in scanline order Use output accumulator array 2D antialiasing is difficult Separable transforms facilitate efficient solution 19

20 Separable Transforms [ X ( u, v), Y ( u, v)] F ( u, v) G( x, v) F(u, v) is a row-preserving transformation that maps all input points to their final column positions, i.e., [x, v]. G(x, v) is a column-preserving transformation that maps the [x, v] points to their final row positions, i.e., [x, y]. 20

21 Catmull-Smith Algorithm First pass Maps image S(u,v) into intermediate image I(x,v) I(x,v) = S(X(u,v), v) Second pass Maps I(x,v) into target image T(x,y) T(x,y) = I(x, Y(H x (v), v)) where H x is the solution to x=x(u,v) for u 21

22 2-Pass Rotation F G X, Y 22

23 2-Pass Perspective F G X, Y 23

24 Fant s Algorithm Forward mapping intensity resampling Scanline order in input and output Amenable to hardware implementation 24

25 Fant s algorithm: Example (1) XLUT YLUT I YLUT x I x

26 Fant s algorithm: Example (2) I x ( 0) ( 100)((. 4)) I x ( 1) ( 100) 1 ( ) (( 1)) I x ( 2) ( 100) 1 14 ( 106) 14 ((. 3)) ( 106)((. 7)) I x ( 3) ( 106) 1. 7 ( 92). 7 ((. 2)) ( 92)((. 1)) ( 90)((. 6))

27 Bibliography Catmull, E. and A.R. Smith, 3-D Transformations of Images in Scanline Order, Proc. Siggraph 80, pp , Fant, Karl M., A Nonaliasing, Real-Time Spatial Transform Technique, IEEE Computer Graphics and Applications, vol. 6, no. 3, pp , Wolberg, George, Digital Image Warping, IEEE Computer Society Press, Los Alamitos, CA