Fourier Transform and Texture Filtering
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1 Fourier Transform and Texture Filtering Lucas J. van Vliet
2 Image Analysis Paradigm scene Image formation sensor pre-processing Image enhancement Image restoration Texture filtering Texture properties: scale orientation Local variance regularity segmentation analysis classification CBP course: Fourier & Texture
3 Texture properties Texture is domain property rather than a pixel property. Fourier analysis offers a powerful tool to analyze and describe domain properties Properties Local power = local variance Scale (from fine detail to course structures) Anisotropy (orientation preference) CBP course: Fourier & Texture 3
4 Discrete Fourier Transform Each image can be decomposed in weighted sum of complex exponentials (sines and cosines) of frequency f and angle φ. (or two frequency components u and v) image size NxN N 1N 1 π j ( ux+ vy ) N 1 g( x, y) = G( u, v) e N u= 0v= 0 N 1N 1 π j ( ux+ vy ) N 1 G( u, v) = g( x, y) e N u= 0v= 0 For real-valued images: Ev { g( x, y) } Re { G( u, v) } F F Od { g( x, y) } j Im { G( u, v) } CBP course: Fourier & Texture 4
5 Getting used to Fourier () An image is a weighted sum of cos (even) and sin (odd) images. Fourier domain with complex amplitude: a+jb a+jb CBP course: Fourier & Texture 5
6 Getting used to Fourier (1) Graphite surface by Scanning Tunneling Microscopy Atomic structure of graphite shows a hexagonal surface 0 column N-1 (c,r) = (0,0) (u,v) = (-½N,-½N) (c,r) = (½N,½N) (u,v) = (0,0) 0 row x u N-1 y v (c,r) = (N-1,N-1) (u,v) = (½N-1, ½N-1) CBP course: Fourier & Texture 6
7 Superposition Fourier spectrum = F F F F F = CBP course: Fourier & Texture 7
8 Two domains The coordinate systems (x,y) and (u,v) have their origin at the position (column,row) = (½N, ½N) 0 column N-1 column (u,v) = (0,0) 0 row x row u N-1 y v CBP course: Fourier & Texture 8
9 Local variance filter Recipe: local variance filter(filter size = n) Compute the local mean (blurring filter of size n) Subtract (if necessary) the local mean. Compute the square of each pixel value. Suppress the double response by local averaging (blurring filter of size n). Local variance is a measure for the local squared-contrast. CBP course: Fourier & Texture 9
10 Segmenting textured objects Simple thresholding fails! The objects differ in texture from the background Local variance followed by a (morphological) closing yields CBP course: Fourier & Texture 10
11 Scaling: local vs global Problem: Choosing the proper scale is an important, but tedious task. Scale too small: Texture characteristics are missed which yields an incomplete data description. Scale too large: Confusion (mixing) of adjacent texture characteristics, lack of localization, and blindness for small and subtle textured regions. Solution: Multi-scale analysis. Analyze the image as function of scale: from fine detail to course image-filling objects. π ( u π x y ) ( v N N ) + + σ F g( x, y 1 ) = e σ G( u, v) = e πσ CBP course: Fourier & Texture 11
12 Multi-Scale Series of images filtered of decreasing scale: Scale-space Sample the scales logarithmically using filters of size = base scale yields n scales per octave base,,,..., 1 n Input image Scale space scale 0 scale 1 scale scale 3 scale 4 scale 5 Scale difference scale derivative var (scale 1) var (scale ) var (scale 3) var (scale 4) var (scale 5) { } Local variance between scales n and n-1. Scale space CBP course: Fourier & Texture 1
13 Scale-spaces Morphological scale-space: Use openings (closings) Gaussian scale-space: Use Gaussian filters Increasing scales Fourier domain with footprints of Gaussian filters of increasing scale Filter size is inversely proportional to footprint in Fourier domain CBP course: Fourier & Texture 13
14 Chirp example Scale derivative Spatial variance Scale-space Scale-space CBP course: Fourier & Texture 14
15 Gabor: scale and orientation Problem: Crossing and touching of oriented patterns. Isotropic scale cannot discriminate based on orientation. Solution: Multi-orientation analysis Orientation selective filtering Multi-scale and multi-orientation analysis by Gabor filtering. Orientation selectivity jπ( u x+ v y) i i g 1 ( x, y N ) = e e u, v i i πσ x + y σ Selected frequency (u i,v i ) Gaussian window of size σ CBP course: Fourier & Texture 15
16 Gabor filter bank Linear sampling of orientation Logarithmic sampling of scale (Remember: reciprocal relation of size in space and freq. A single input image yields many output images. Question: Which output image(s) discriminate between textures? (u i,v i ) σ 1 G ( u v) = e u, v, i i π ( ( u u ) π ) ( ( v v ) N i + N i ) σ = G( u u, v v ) i CBP course: Fourier & Texture 16 i
17 Gabor filtering Each Gabor filter selects a (Gaussian-shaped) region of the Fourier spectrum of the image. A Gabor filter can be implemented by convolution! [ ( i i ) ( ( i i ))] ( ) ( ) g x, y = cos ux+ vy + jsin ux+ vy 1 e u, v x y π π σ πσ i i N N + CBP course: Fourier & Texture 17
18 How to go further? Problem: How to segment textured objects from background adjacent textures Each pixel is described by many texture attributes: a feature vector Pixels from a single texture yield a cluster in feature space. Solutions: Unsupervised: Cluster analysis Supervised: Pattern recognition allows training a classifier based on examples CBP course: Fourier & Texture 18
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