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

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Unformatted text preview: CAP5415 Computer Vision Assignment # 2 Let I be an image, gσ (x ) = 1 1 x2 x2 + y2 exp − , g σ ( x, y ) = , exp − 2σ 2 2πσ 2 2σ 2 2π σ ∂ I ∆I = ∂x ∂ I ∂y T ∂2 ∂2 is the Gradient of image I and ∆ I = 2 I + 2 I is the ∂x ∂y 2 Laplacian of image I. f * g denotes the convolution of f with g . 1. Proof the following results ( ) 1. g σ1 ( x )* g σ 2 ( x )* I = g σ ( x )* I . Also, find the value of σ in terms of σ 1 and σ 2 . 2. gσ ( x, y ) = gσ (x )gσ ( y ) . 3. gσ ( x, y ) * I = gσ ( x ) * ( g σ ( y ) * I ) . 4. ∆( gσ (x, y ) * I ) = (∆gσ ( x, y )) * I . 5. ∆ is isotropic (rotation invariant). 6. ∆2 is linear and isotropic (rotation invariant). 7. ∆2 (g σ ( x, y ) * I ) = (∆2 g σ ( x, y ))* I . You may want to use the following results or facts. 1D Convolution: f * g = f ( x − x′)g ( x′)dx′ 2D Convolution: f * g = f ( x − x′, y − y ′)g ( x′, y ′)dx′dy ′ Convolution Theorem: F ( f * g ) = F ( f )F ( g ) and F ( fg ) = F ( f ) * F ( g ) , where F ( f ) is Fourier transform of function f . Fourier Transform of Gaussian: F ( g σ ( x )) = 2π σ g 1 (u ) , F ( g σ ( x, y )) = σ 2π σ2 g 1 (u , v ) σ ...
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