Hom o om o or o phi p c Processing

Transcription

1 Homomorphic o o Processing Motivation: Image with a large dynamic range, e.g. natural scene on a bright sunny day, recorded on a medium with a small dynamic range, e.g. a film image contrast significantly reduced especially in the dark and bright regions. One Approach: Enhance image by reducing its dynamic range and increase its local contrast prior to recording it on medium with a small dynamic range. use a homomorphic processing system which h operates in log intensity it domain. EEE 50

2 Homomorphic o o Processing Strategy based on: x(n,n ) = i(n,n ) r(n,n ) Image formed by Illumination: Reflectance recording light assumed to be reflected from slow varying and objects which are main contributor to illuminated by dynamic range some light source Reflectance component: represents details of objects and assumed to very rapidly and the primary contributor to local contrast To decrease dynamic range, decrease i(n,n ) To increase local contrast, increase r(n,n ) EEE 50

3 Homomorphic o o Processing To separate the two components i(n,n ) and r(n,n ), apply a logarithmic operation low frequency usually high/mid frequency log x(n,n ) = log [i(n,n )] + log [r(n,n )] y (n,n ) assumed to remain slowly varying γ < assumed to remain rapidly varying LPF log [i(n,n )] x(n,n ) log y y + exp x 0 (n,,n ) log [r(n,n )] HPF H(ω, ω ) γ > EEE 50 3

4 Homomorphic o o Processing H(ω, ω ) x(n,n ) log γ y (n,nn ) y (n,nn ) γ exp x 0 (n,,n ) ω + ω y (n,n ) γ log [i(n,n )] + γ log [r(n, n )] x 0 (n,n ) = [i(n,n )] γ [r(n,n )] γ γ and γ allow for control over illumination and contrast. Note: System with log operation followed by a linear operation followed by exponentiation operation is called a homomorphic system for multiplication or multiplicative homomorphic processing. System above can be viewed as a filtering system in the log intensity domain. Note: Log intensity domain also considered as a perceptual domain since image intensities appear to be modified at the peripheral level of HVS by some form of nonlinearity such as a logarithmic operation. EEE 50 4

5 Objective: Enhance image by reducing degradation that may be present. Will discuss methods that attempt to reduce additive random noise and/or salt and pepper (0 or 55) type, also known as impulsive noise. Simple technique: Low-pass filtering (LPF) Motivation: Image energy primarily concentrated in the low frequency components (due to high correlation between neighboring pixels in a natural scene). Noise energy is typically more spread out in frequency domain. Low pass filtering reduces large amount of noise at the expense of reducing a small amount of signal. Example: simple D case Noise Edge spike LPF will smooth both noise spikes and edges. EEE 50 5

6 Simple D case: Convolve with finite and small impulse response (also called Mask) of LPF. Mask small because larger window will cause more blurring can devise adaptive filtering with variable window size depending on local statistics of image. With 3 3 3Mask k( (window with weights), go through the image and replace the center value of window with a weighted average. Compute weighted average and place it in the center edges will be blurred. EEE 50 6

7 Mask Example: Needed to preserve DC value of image. 9 n n h ( n, n ) = = ( 0,0) H This filter alters also values of non-noisy pixels. To impruve performance:. Use selective averaging center value in mask y ( n, n ) = x Avg ( n n ), if x ( n, n ) Avg threshold Otherwise i.e., if pixel value is really out of range, use the avg; otherwise do nothing Avg = average of neighbors (not including processed pixel) EEE 50 7

8 Note: Threshold determined by trial and error. Example: Threshold = 4; 3 3 Mask Avg = = ; = 7 > 4 replace center sample with the neighbor average Avg = ; = < 4 keep original EEE 50

9 . Use directional filtering Directional averaging filters (selective with respect to direction) Examine the average based on several directionally oriented masks. θ 3 θ θ center sample = x(n θ,n ) 4 Compute averages in θ, θ, θ 3 and θ 4 directions Avg i; i =,, 3, 4. y[n,n ] = Avg i where Avg i is the one closest in amplitude to x(n,n ), i.e. x(n,n )-Avg i minimum. has a tendency not to destroy boundaries Note: directional analysis can also be used to check if a pixel belongs to a directional edge (leave unchanged) or is noise (remove noise). EEE 50 9

10 3. Use non-linear filtering Median filtering (Special case) Rank-ordered filtering (General case) Median filtering: As before, slide window along the image. Consider pixels in a window around input pixels being processed. Replace input pixels by median of pixels in the window. Median obtained by sorting all pixels in the window (including input pixel) in increasing or decreasing order of amplitudes and picking the middle value if odd number of pixels in window, or Avg of values in middle if even number of pixels in window. x o ( n,n ) = { x ( n k,n l ) ; ( k, l ) W } median sort values in window + pick middle value chosen window Note: Typical size of windows W are 3 3, 5 5 or 7 7. EEE 50 0

11 Properties of median filters. Nonlinear { x( m) + y( m) } median{ x( m) } median{ y( m) } median +. Very useful for reducing impulsive (salt and pepper) noise and removing isolated lines and pixels while preserving spatial resolution (reduced blurring) and edges. Poorer performance in case of additive noise: image blurred but sharp edges are not blurred. 3. Poor performance if the majority of pixels in the window are due to noise. 4. A D median filter (window is D) has better performance in preserving edges (discontinuities) than a D median filter (window is D). EEE 50

12 Example: 3-points median filter x o ( n) = median { x( n ), x( n), x( n + ) } med= med(,,)= edge preserved x (n) x o (n) n n med(,,)= noise eliminated EEE 50

13 o se S oot g Example: D median filtering, 3 3 window. median(0,0,0,0,0,,,,)=0 corners distorted and not preserved 0 Since D median filter tends to preserve edges of D sequence b tt t bl di filt i better to use separable median filtering. 3 EEE 50

14 Doing separable median filtering Median filter the rows of x(n,n ) obtain t(n,n ) Median filter the columns of t(n,n ) obtain x o (n,n ) Remark: Since median filter is non-linear, the obtained output depends on the order in which the D horizontal and vertical filters are applied. Rank-order filtering (Generalization of median filtering) Can improve performance when number of noise pixels in window is large. x ( n) N x( n) R, k { }; o = N is the window size k is the rank order = parameter specifying which sample to pick in window x(m) n N m EEE 50 4

15 Procedure for Rank-Order Filtering: (same as median but pick the k th value in window) Take the sliding window and shifted to position n Take all pixels in the window and order them in descending order (highest to lowest) or ascending order. Choose the k th sample in the list x o (n). Example: Median k = N + (if N odd middle value) EEE 50 5