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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.

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

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

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The problem of adding noise to an image is identical to that of adding a random number to the gray level of each pixel. 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. Histogram matching.

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Rayleigh noise Rayleigh noise = a+( b/4) 1/2 = a+( b/4) 1/2 = b(4- )/4 = b(4- )/4

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Erlang (gamma) noise Erlang (gamma) noise =b/a; =b/a 2 =b/a; =b/a 2

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Exponential noise Exponential noise =1/a; =1/a 2 =1/a; =1/a 2 A special case Erlang noise model when b = 1.A special case Erlang noise model when b = 1.

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Uniform noise Uniform noise =(a+b)/2; =(b-a) 2 /12 =(a+b)/2; =(b-a) 2 /12

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Impulse noise (salt and pepper noise) Impulse noise (salt and pepper noise)

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Example

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Results of adding noise

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

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Impulse noise is caused by Malfunctioning pixels in camera sensors Fault memory locations in hardware Transmisison in a noisy channel Two types: Salt-and-pepper Noise Uniformly-Distrubuted Random Noise

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How do we know which noise model adaptive to the currently available imaging tool? 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.

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Gaussian n: Gaussian n: Find the mean and standard deviation of the histogram (Gaussian noise). Rayleigh, Erlang, and uniform noise: Rayleigh, Erlang, and uniform noise: Calculate the a and b from and . Impulse noise: Impulse noise: Compute the height of peaks at gray levels 0 and 255 to find P a and P b.

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Estimation of Noise Parameters

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The recently proposed Nonlocal Means Algorithm (NLmeans) has offered remarkably promising results. Unlike previous denoising methods that rely on the local regularity assumption, the NL-means exploits spatial correlation in the entire image for noise removal. It adjusts each pixel value with a weighted average of other pixels whose neighborhood has a similar geometrical configuration. Since image pixels are highly correlated while noise is typically independently and identically distributed (i.i.d.), averaging of these pixels results in noise cancellation and yields a pixel that is similar to its original value. Non-Local Means Algorithm (Buades 2005)

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New Idea: NL-Means Filter (Buades 2005) Same goals: Smooth within Similar Regions Same goals: Smooth within Similar Regions KEY INSIGHT: Generalize, extend Similarity KEY INSIGHT: Generalize, extend Similarity Bilateral: Averages neighbors with similar intensities; NL-Means: Averages neighbors with similar neighborhoods!

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NL-Means Method: Buades (2005) For each and For each and every pixel p: every pixel p:

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NL-Means Method: Buades (2005) For each and For each and every pixel p: every pixel p: Define a small, simple fixed size neighborhood;

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NL-Means Method: Buades (2005) For each and For each and every pixel p: every pixel p: Define a small, simple fixed size neighborhood; Define vector V p : a list of neighboring pixel values. V p = 0.74 0.32 0.41 0.55

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NL-Means Method: Buades (2005) Similar pixels p, q SMALL vector distance; || V p V q || 2 p q

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NL-Means Method: Buades (2005) Dissimilar pixels p, q LARGE vector distance; || V p V q || 2 p q q

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NL-Means Method: Buades (2005) Dissimilar pixels p, q LARGE vector distance; Filter with this! || V p V q || 2 p q

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NL-Means Method: Buades (2005) p, q neighbors define p, q neighbors define a vector distance; Filter with this: No spatial term! || V p V q || 2 p q

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NL-Means Method: Buades (2005) pixels p, q neighbors Set a vector distance; Vector Distance to p sets weight for each pixel q || V p V q || 2 p q

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NL-Means Filter (Buades 2005) Noisy source image: Noisy source image:

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NL-Means Filter (Buades 2005) Gaussian Filter Gaussian Filter Low noise, Low detail

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NL-Means Filter (Buades 2005) Anisotropic Diffusion Anisotropic Diffusion (Note stairsteps: ~ piecewise constant)

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NL-Means Filter (Buades 2005) Bilateral Filter Bilateral Filter (better, but similar stairsteps:

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NL-Means Filter (Buades 2005) NL-Means: NL-Means:Sharp, Low noise, Few artifacts.

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

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Median filter

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The median filter is used for removing noise. It can remove isolated impulsive noise and at the same time it preserves the edges and other structures in the image. Contrary to average filtering it does not smooth the edges.

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Unlike the mean filter, the median filter is non- linear. This means that for two images A(x) and B(x):mean filter

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Removal of Line Artifacts by Median Filtering

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The original image. b) Original image corrupted by salt and pepper noise (p=5 % that a bit is flipped. c) After smoothing with a 3 x 3 filter most of the noise has been eliminated.

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d) If we smooth the image with a larger median filter, e.g. 7 x 7, all the noise pixels disappear. e) Alternatively, we can pass a 3 x 3 filter over the image 3 times in order to remove the noise with less loss of detail.

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Salt and pepper noise (5 %)Salt and pepper noise (20 %) Smoothed by 3 x 3 window

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Salt and pepper noise (5 %)Salt and pepper noise (20 %) Median filtered by 3 x 3 window

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

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Alpha-trimmed mean filter is windowed filter of nonlinear class, by its nature is hybrid of the mean and median filters. The basic idea behind filter is for any element of the signal (image) look at its neighborhood, discard the most atypical elements and calculate mean value using the rest of them.

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Combining the advantages of mean filter and order-statistics 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 S xy and average the remaining mn-d pixel, denoted by g r (s, t). where d=0 ~ mn-1

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Filters whose behavior changes based on statistical characteristics of the image. Filters whose behavior changes based on statistical characteristics of the image. Two adaptive filters are considered: Two adaptive filters are considered: (1)Adaptive, local noise reduction filter. (2)Adaptive median filter.

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Two parameters are considered: Two parameters are considered: Mean: measure of average gray level. Variance: measure of average contrast. Four measurements: Four measurements: (1)noisy image at (x, y ): g(x, y ) (2)The variance of noise 2 (3)The local mean m L in S xy (4)The local variance 2 L

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

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Adaptive, Local Noise Reduction Filter

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Adaptive median filter can handle impulse noise with larger probability (P a and P b are large). Adaptive median filter can handle impulse noise with larger probability (P a and P b are large). This approach changes window size during operation (according to certain criteria). This approach changes window size during operation (according to certain criteria). First, define the following notations: First, define the following notations: z min =minimum gray-level value in S xy z max =maximum gray-level value in S xy z med =median gray-level value in S xy z xy = gray-level at (x,y) S max =maximum allowed size of S xy

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The adaptive median filter algorithm works in two levels: A and B The adaptive median filter algorithm works in two levels: A and B Level A: A1=z med -z min Level A: A1=z med -z min A2=z med -z max If A1>0 and A2 0 and A20 AND B2 0 AND B2