Geometric Image Manipulation

Transcription

1 Bruce A. Draper J. Ross Beveridge, January 24, 204 Geometric Image Manipulation Lecture (part a) January 24, 204

2 Bruce A. Draper J. Ross Beveridge, January 24, 204 Status Update Programming assignment The OpenCV warm-up exercise Is now on the web site It is due Friday, Jan 3 st (one week) Let s go over it

3 Bruce A. Draper J. Ross Beveridge, January 24, 204 Image Manipulation: Context To start with the obvious, an image is a 2D array of pixels Pixel locations represent points on the image plane Pixel values represent measurements of light Color images : energies by frequency ranges (RGB: three overlapping ranges) Intensity images : average energy across the visible range Your CS40 ray tracers should have taught you about image formation To directly compare two images, they should be registered Geometrically : image should line up with image 2 Photometrically : equal pixel values should imply equal energy

4 Bruce A. Draper J. Ross Beveridge, January 24, 204 Geometric Registration It s not enough for two matching images to have the same set of pixel values They have to be in the same relative positions Image from CalTech256 data set Otherwise, these two images match

5 Bruce A. Draper J. Ross Beveridge, January 24, 204 Geometric Registration (II) Geometric registration finds a mapping that maps one image onto the other We will limit ourselves to linear transformation We should be able to register these

6 Bruce A. Draper J. Ross Beveridge, January 24, 204 Registration formalism We denote an image as a 2D function: I x, y ( ) Or, in homogeneous coordinates: I x, y, w ( ) The goal is to find the transformation matrix G such that: I i ( * ( u, v, w) = I j * G * ) x y ,

7 Bruce A. Draper J. Ross Beveridge, January 24, 204 Interpolation (foreshadow ) Seldom get integer-tointeger mapping. Geometry part computes real-valued positions of pixel centers. We will worry about how to interpolate values later.

8 Bruce A. Draper J. Ross Beveridge, January 24, 204 Image Transformations The simplest set of transformations are translation, rotation, and scale Together these are called the similarity transform. Similarity transforms have 4 degrees of freedom. In matrix form these are: u v = s s scale x y and... u v = ( ) (sin() 0 ( ) cos () 0 cos sin 0 0 rotation x y

9 Bruce A. Draper J. Ross Beveridge, January 24, 204 Image Transformations : Translation = y x t t v u y x Translation (note the 2D homogeneous coordinates)

10 Bruce A. Draper J. Ross Beveridge, January 24, 204 Translation Applied to Images Translate 20 in x Translate -20 in x 0 20 ( 0 0 ( 0 0 (

11 Bruce A. Draper J. Ross Beveridge, January 24, 204 Scale Applied to Images Note the origin Scale Uniformly by 2 Scale Uniformly by

12 Bruce A. Draper J. Ross Beveridge, January 24, 204 Rotation Applied to Images Rotate by 5 Rotate by -5 Note that a positive rotation rotates the positive X axis toward the positive Y axis

13 Bruce A. Draper J. Ross Beveridge, January 24, 204 Combining Transformations To rotate (or scale) around a point (x,y), break it into three steps. Translate by (-x, -y) to make (x,y) the origin 0 x 0 y Rotate (or scale) around the (new) origin cos sin 0 0 ( ) sin() 0 ( ) cos () 0 ( ( ( ( 3. Translate back 0 x 0 y 0 0

14 Bruce A. Draper J. Ross Beveridge, January 24, 204 Composition of Matrices These three steps are combined into one ( ) )sin(( ) 0 0 x cos ( 0 )x 0 y sin (( ) cos (( ) 0 0 )y = ( ) )sin(( ) y sin (( ) ) x cos (( ) ( ) ) y cos (( ) 0 x cos ( 0 y sin (( ) cos (( ) )x sin ( ( ) )sin(( ) y sin (( ) ) x cos (( ) + x ( ) ) y cos (( ) + y cos ( sin (( ) cos (( ) )x sin ( 0 0 =

15 Bruce A. Draper J. Ross Beveridge, January 24, 204 Similarity Affine Transformations All the similarity transforms can be combined into one generic matrix: u a b c x Hint: diagonal v = d e f y terms are not 0 0 equal, and b -d. But This matrix does more. What? hint: two more transformation types included. hint: 6 degrees of freedom (DOF) How can you specify this matrix? This is equivalent to adding two shear parameters (or unequal scaling one shear).

16 Bruce A. Draper J. Ross Beveridge, January 24, 204 Affine Examples

17 Bruce A. Draper J. Ross Beveridge, January 24, 204 Similarity vs Affine Matrices Similarity : 4 DOF Affine : 6 DOF = 0 0 y x f e d c b a v u u v = a b c b a d 0 0 x y

18 Bruce A. Draper J. Ross Beveridge, January 24, 204 Specifying Affine Transformations There are six unknowns in the matrix (a through f) If you specify one point in the source image and a corresponding point in the target image, that yields two equations: u i = ax i + by i + c v i = dx i + ey i + f So providing three point-to-point correspondences specifies an affine matrix

19 Bruce A. Draper J. Ross Beveridge, January 24, 204 To be continued