Geometric Image Transformations and Related Topics

Transcription

1 Geometric Image Transformations and Related Topics 9 th Lesson on Image Processing Martina Mudrová 2004

2 Topics What will be the topic of the following lesson? Geometric image transformations Interpolation in the image Spatial image transformations Warping and morphing Image registration 2

3 Geometric Transformations translation scaling rotation around the origin shearing ' e' 2 e' 1 X ' more complicated transformation are composed from these basic operations P e 2 e 1 P' Carthesian coordinate sstem r r r ( P, e1, e2, e3) r r r e, e e 1 2, 3 linear non-dependent perpendicular one to each other Affine coordinate sstem r r r ( P, e1, e2, e3) r r r e, e e 1 2, 3 linear non-dependent 3

4 Translation of an Image the simplest geometric transformation [,] piel position [,] ' [','] ' = ' = + 0' + 0' [0',0'] ' [0,0] 4

5 Scaling of an Image (Zooming) (1) ' 0.5 [,] s scale coefficient in the direction of ais s scale coefficient in the direction of ais [0,0] [','] 1.5 ' ' = ' =. s. s - scaling process changes image size!! = image resizing 0< s <1 contraction s >1 dilatation s<0 opposite direction 5

6 Scaling of an Image (Zooming) (2) What happens, if the coefficients s, are not integer? ' [0,0] ' - Appropriate interpolation method is required 6

7 Rotation of an Image around the Origin ' ' ' = ' =.cos( α).sin( α) +.sin( α).cos( α) [0,0] Rotation around the point [X,Y] : -composition of translation and rotation around the origin ' = ' = Y X + ( X ).cos( α) ( Y).sin( α) + ( X ).sin( α) + ( Y).cos( α) Piels do not map eactl onto another piel interpolation is required!! 7

8 Rotation of an Image As a new image must be rectangular, how should I solve a surrounding area? 1. to cut the resulting image according to the original size (in piels) 2. to enlarge the size of resulting image so as new image can fit into new area, some value must be selected for the undefined piels 8

9 Matlab Commands for Basic Geometric Image Transformations imresize imrotate 9

10 Image Interpolation Methods Interpolation methods are used to compute the value in the unmeasured position Input: points A, B, C, A(a0,a1), B(b0,b1),, known values of an image function h(a), h(b), coordinates [i,j] of point Q in which we would like to know value of function h Output: h(q) h grascale function in the case of intensit image 3 functions of RGB components, the same method must be used for each of them The most common interpolation methods: nearest neighbor method bilinear method bicubic method spline interpolation 10

11 Nearest Neighbor Method -the piel value is replaced b the same value as the nearest known point = interpolation of the 0-th order - ver rough method but: - acceptable for all image tpes - the onl method for the indeed images and BW images 11

12 Bilinear Interpolation Method What is a difference between bi-linear and linear interpolation? Bi-linear means application the same linear principle twice Linear interpolation h() Bilinear interpolation A h(b) h(q)=? h(a) j B P R Q D h( Q) A Q B Q = h( A) + ( h( B) h( A)). B A A C i = interpolation of the 1-st order 12

13 Interpolation of Higher Orders -bicubic, spline interpolation: - use polnomials of the 2-nd or 3-rd order instead of the linear function - need large number of surrounding points - higher computation requirements 13

14 2D Interpolation Method Use What can happens in the case of improper interpolation method use? Eamples: 1. Rotation of BW image with bicubic int. method use > loosing sharpness 2. Resizing of indeed images (with palette) > colour nonsense 3. Which method is the best for true-colour images? Interpolation methods provides basic tools in the case of picture damage What are other methods of reconstruction in this case? - methods of signal modelling and prediction in 2D AR models, NN, 14

15 Matlab Commands for 2D Interpolation griddata meshgrid (interp2) 15

16 Spatial Transformations = advanced methods of geometric transformations -are used to remove various tpes of image distortions caused b - using the various optical sstems (lense) - taking the photos from the various points of view - projection of the spatial 3D objects into the 2D space onl (aerial photos of the surface of the Earth,...) -... Eamples of image distortion: original Pincushion distortion Barrel distortion 16

17 Image Registration Principle -due to reasons described in the previous slide the piels are shifted from their correct position: -the goal of image registration: Geometric restoration = find such a transformation and its parameters which leads from one coordinate sstem to other: ' ' ), ( ' ), ( ' T T = = ' ' c c c c c c c c = = for eample: 17

18 Mapping Methods Principle Forward mapping: - we are going through the piels of the input image A and finding their position in the output image B according to the selected transformation - the appropriate interpolation method must be used to fill the empt areas Backward mapping: - we are going through the piels of the output image B and finding the corresponding points in the input image 18

19 The Most Common Spatial Transformations What are the most common spatial transformations? What are their properties? Linear conformal (translation, rotation, scaling) - streight lines remain straight, parallel remain parallel, angles are saved Affine (shearing) streight lines remain straight, parallel remain parallel, but angles are changed Projective streight lines remain straight, parallel lines converges toward vanishing points Polnomial - new sstem is described using curves (polnomial ot order 2,3 or higher)... 19

20 Estimation of Transformation Parameters How can be estimated the parameters of the selected transformation? -the parameters c 1, c 2,... can be determined from the transformation of known points control point pairs using appropriate criteria (LSM) For eample: based image... image which we suppose to be correct input image...image which should be corrected 20

21 Corresponding Terms Image warping image is printed on a sheet of rubber -image is drawn on a 3D net projected into 2D, the same method of the net distortion is used for ech part of teture - b changing the net the resulting image is changed - tr to use Matlab command warp Image morphing -involves two steps of warping with a smooth (spline) interpolation between initial and the resulting image - it is animated warping 21

22 Algorithm of Image Registration in Matlab Goal: Registration of an input image according to the base image Algorithm and Matlab commands: 1. Read the input and base images (imread) 2. Select the control point pairs (cpselect) and save them into the Matlab environment 3. Correct the control points pairs using correlation analsis (cpcorr) 4. Perform the selected spatial transform, estimate its parameters (cp2tform, imtransform) 22