Digital Image Processing Chapter 5 Image Restoration

Introduction Image enhancement Image restoration Subjective process Objective process to recover the original image. Involving formula or criteria that will yield an optimal result.

5. 1 Model of Image degradation and restoration g(x,y)=h(x,y)f(x,y)+(x,y) Note: H is a linear, position-invariant process.

5.2 Noise Model Spatial and frequency property of noise White noise (random noise) A sequence of random positive/negative numbers whose mean is zero. Independent of spatial coordinates and the image itself. In the frequency domain, all the frequencies are the same. All the frequencies are corrupted by an additional constant frequency. Periodic noise

Noise Probability Density Functions (1) Gaussian noise  : mean;  : variance 70% [(-), (+)] 95 % [(-2), (+2)] Because of its tractability, Gaussian (normal) noise model is often applicable at best.

Implementation The problem of adding noise to an image is identical to that of adding a random number to the gray level of each pixel. Noise models describe the distribution (probability density function, PDF) of these random numbers. How to match the PDF of a group of random numbers to a specific noise model? Histogram matching.

Noise Probability Density Functions (2) Rayleigh noise  = a+(b/4)1/2  = b(4-)/4

Noise Probability Density Functions (3) Erlang (gamma) noise =b/a;  =b/a2

Noise Probability Density Functions (4) Exponential noise =1/a;  =1/a2 A special case Erlang noise model when b = 1.

Noise Probability Density Functions (5) Uniform noise =(a+b)/2;  =(b-a)2/12

Noise Probability Density Functions (6) Impulse noise (salt and pepper noise)

Example

Results of adding noise

Results of adding noise

The Principle Use of Noise Models Gaussian noise: electronic circuit noise and sensor noise due to poor illumination or high temperature. Rayleigh noise: Noise in range imaging. Erlang noise: Noise in laser imaging. Impulse noise: Quick transients take place during imaging. Uniform noise: Used in simulations.

5.2.4 Estimation of Noise Parameters How do we know which noise model adaptive to the currently available imaging tool? Image a solid gray board that is illuminated uniformly. Crop a small patch of constant grey level and analyze its histogram to see which model matches.

5.2.4 Estimation of Noise Parameters Gaussian n: Find the mean  and standard deviation  of the histogram (Gaussian noise). Rayleigh, Erlang, and uniform noise: Calculate the a and b from  and . Impulse noise: Compute the height of peaks at gray levels 0 and 255 to find Pa and Pb.

Estimation of Noise Parameters

5.3 Restoration using spatial filter Used when only additive noise is present. Mean filters Arithmetic mean filter: The new pixel value is resulted from averaging the pixels in the area by Sxy. Geometric mean filter: Achieves smoothing comparable to arithmetic approach, but tends to lose less image detail.

Order-Statistics filters Median filter Max filter: find the brightest points to reduce the pepper noise Min filter: find the darkest point to reduce the salt noise Midpoint filter: combining statistics and averaging.

Example 5.3 Iteratively applying median filter to an image corrupted by impulse noise.

Alpha-Trimmed Mean Filter Combining the advantages of mean filter and order-statistics filter. Suppose delete d/2 lowest and d/2 highest gray- level value in the neighborhood of Sxy and average the remaining mn-d pixel, denoted by gr(s, t). where d=0 ~ mn-1

Adaptive Filter Filters whose behavior changes based on statistical characteristics of the image. Two adaptive filters are considered: Adaptive, local noise reduction filter. Adaptive median filter.

Adaptive, Local Noise Reduction Filter Two parameters are considered: Mean: measure of average gray level. Variance: measure of average contrast. Four measurements: noisy image at (x, y ): g(x, y ) The variance of noise 2 The local mean mL in Sxy The local variance 2L

Adaptive, Local Noise Reduction Filter Given the corrupted image g(x, y), find f(x, y). Conditions: 2 is zero (Zero-noise case) Simply return the value of g(x, y). If 2L is higher than 2 Could be edge and should be preserved. Return value close to g(x, y). If 2L = 2 when the local area has similar properties with the overall image. Return arithmetic mean value of the pixels in Sxy. General expression:

Adaptive, Local Noise Reduction Filter

Adaptive Median Filter Adaptive median filter can handle impulse noise with larger probability (Pa and Pb are large). This approach changes window size during operation (according to certain criteria). First, define the following notations: zmin=minimum gray-level value in Sxy zmax=maximum gray-level value in Sxy zmed=median gray-level value in Sxy zxy= gray-level at (x,y) Smax=maximum allowed size of Sxy

Adaptive Median Filter The adaptive median filter algorithm works in two levels: A and B Level A: A1=zmed-zmin A2=zmed-zmax If A1>0 and A2<0 goto level B else increase the window size If window size  Smax repeat level A else output zxy Level B: B1=zxy-zmin B2=zxy-zmax If B1>0 AND B2<0, output zxy Else output zmed.

Example 5.5