Hw2 - CAP5415 Computer Vision Assignment 2 Let I be an...

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CAP5415 Computer Vision Assignment # 2 Let I be an image, ( 29 ° ° ± ² ³ ³ ´ µ - = 2 2 2 exp 2 1 σ σ π σ x x g , ( 29 ° ° ± ² ³ ³ ´ µ + - = 2 2 2 2 2 exp 2 1 , σ πσ σ y x y x g , T I y I x I · ¸ ¹ º » = is the Gradient of image I and I y I x I 2 2 2 2 2 + = is the Laplacian of image I. g f * denotes the convolution of f with g . 1. Proof the following results 1. ( 29 ( 29 ( 29 ( I x g I x g x g * * * 2 1 σ σ σ = . Also, find the value of σ in terms of 1 σ and 2 σ . 2. ( 29 ( 29 ( y g x g y x g σ σ σ = , . 3. ( 29 ( 29 ( 29 ( 29 I y g x g I y x g * * * , σ σ σ = . 4. ( 29 ( 29 ( 29 ( 29 I y x g I y x g * , * , σ σ = . 5. is isotropic (rotation invariant). 6. 2 is linear and isotropic (rotation invariant). 7. ( 29 ( 29 ( 29 ( 29 I y x g I y x g * , * , 2 2 σ σ = . You may want to use the following results or facts.
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