Geometric Image Manipulation. Lecture #4 Wednesday, January 24, 2018

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1 Geometric Image Maniplation Lectre 4 Wednesda, Janar 4, 08

2 Programming Assignment

3 Image Maniplation: Contet To start with the obvios, an image is a D arra of piels Piel locations represent points on the image plane Piel vales represent measrements of light Color images : energies b freqenc ranges (RGB: three overlapping ranges) Intensit images : average energ across the visible range Bilding ra tracers shold have taght o abot image formation To directl compare two images, the shold be registered Geometricall : image shold lines p with image Photometricall : eqal piel vales shold impl eqal energ

The have to be in the same relative positions Image

4 Geometric Registration It s not enogh for two matching images to have the same set of piel vales The have to be in the same relative positions Image from CalTech56 data set Otherwise, these two images match

5 Geometric Registration (II) Geometric registration finds a mapping that maps one image onto the other We will limit orselves to linear transformation We shold be able to register these

6 Registration formalism We denote an image as a D fnction, Or, in homogeneos coordinates,,, Given two image I i and I j Matching points {(,v),...} {(,), } Find G sch that ', (, * +,

7 Interpolation (foreshadow ) Seldom get integer-tointeger mappings. Geometr part comptes real-valed positions of piel centers. We will worr abot how to interpolate vales later.

8 Classes of Image Transformations Rigid transformations Combine rotation and translation Preserve relative distances and angles Degrees of freedom Similarit transformations Add scaling to rotation and translation Preserves relative angles 4 Degrees of freedom

9 Bilding Blocks: Rotation Trigonometric version v cos Θ ( ) sin Θ ( ) 0 sin Θ ( ) cos Θ ( ) Projection onto basis vectors v ˆ U ˆ U 0 ˆ V ˆ V ˆ U ˆ U ˆ V ˆ V ˆ U ˆ V 0

10 Bilding Blocks: Scaling Uniform scaling v s s Non-niform scaling v s s 0 0 0

11 Bilding Blocks: Translation v 0 t 0 t 0 0 Recall how homogenos coordinates formlates translation as a matri mltipl Q M P

12 Translation Applied to Images Wh Black? Translate 0 in 0 0 ' 0 0 ' 0 0 ' Translate -0 in 0 0 ( 0 0 ( 0 0 '(

13 Scale Applied to Images Note the origin Scale Uniforml b Scale Uniforml b ' 0 0' 0 0 ' ' ' 0 0 '

14 Rotation Applied to Images Rotate b 5 Rotate b -5 Note that a positive rotation rotates the positive X ais toward the positive Y ais

15 Composition of Matrices To rotate b θ arond a point (,): cos θ sin θ cos θ sin θ ( ) sin( θ) 0 ( ) cos( θ) cos θ sin θ ( ) sin( θ) sin( θ) cos( θ) ( ) cos( θ) sin( θ) cos( θ) 0 0 ( ) sin( θ) sin( θ) cos( θ) + ( ) cos( θ) sin( θ) cos( θ) + 0 0

16 Affine Transformations All the similarit transforms can be combined into one generic matri: a v d 0 b e 0 c f Hint: diagonal terms are not eqal, and b ¹ -d. Bt This matri does more. What? hint: more transformations. hint: 6 degrees of freedom. How can o specif this matri? Eqivalent to adding shear parameters, or neqal scaling shear parameter.

17 Affine Eamples: Shear ' 0 0' 0 0 ' 0 ' 0 0' 0 0 '

18 Similarit vs. Affine Matrices Similarit : 4 DOF Affine : 6 DOF 0 0 f e d c b a v v a b c b a d 0 0

19 Specifing Affine Transformations There are si nknowns in the matri (a throgh f) If o specif one point in the sorce image and a corresponding point in the target image, that ields two eqations: i a i + b i + c v i d i + e i + f So providing three point-to-point correspondences specifies an affine matri

20 Affine Specification: Eample There is one affine transformation that will map the green point on the right to the green point on the left, and align the red and ble points also.

21 Solving Affine Transformations These linear eqations can be easil solved: WLOG, assme 0 then c and v f so: ( ) ( ) ( ) ( ) ( ) b b b b b a b a b a ' Calclation of a, b c is independent of calclation of e, f g.

22 Solving Affine (cont.) This can be sbstitted in to solve for a The same process with s solves for d,e,f Abot the WLOG: It was tre becase o can translate the original coordinate sstem b (-, - ) So what do o do to compensate? Alternativel, set p a sstem of linear eqations and solve... Will show this for a harder case shortl.

Bruce A. Draper & J. Ross Beveridge, January 25, Geometric Image Manipulation. Lecture #1 January 25, 2013

Bruce A. Draper & J. Ross Beveridge, January 25, Geometric Image Manipulation. Lecture #1 January 25, 2013 Brce A. Draper & J. Ross Beerdge, Janar 5, Geometrc Image Manplaton Lectre # Janar 5, Brce A. Draper & J. Ross Beerdge, Janar 5, Image Manplaton: Contet To start wth the obos, an mage s a D arra of pels