Module 3

Image Restoration and Reconstruction

A Model of the image Degradation/ Restoration Process

The degradation process is model ed as a degradation function that, together within a additive noise term ,operates on an input image f(x,y) to produce a degraded image g(x,y),some knowledge about the degradation function H,and some knowledge about the additive noise term ɳ(x,y), the objective of restoration is to obtain an estimate (x,y)of the original image.We want the estimate to be as close as possible to the original input image and, in general, the more we know about H and ɳ,the closer (x,y) will be to f(x,y).

If H is a linear,position-invariant process,then the degraded image is given in the spatial domain by

g(x,y)=h(x,y)*f(x,y)+ɳ(x,y)

Where h(x,y) is the spatial representation of the degradation function and, the symbol “ * “ indicates convolution. We know from the discussion that convolution in the spatial domain is analogous to multiplication in the frequency domain, so we may write the model in an equivalent frequency domain representation:

G(u,v) = H(u,v)F(u,v)+N(u,v)

Where terms in capital letters are the Fourier transforms of the corresponding terms.

Noise Models

The principal sources of noise in digital images arise during image acquisition and /or transmission. The performance of imaging sensors is affected by a variety of factors , such as environmental conditions during image acquisition, and by the quality of the sensing elements themselves. For instance, in acquiring images with a CCD camera, light levels and sensor temperature are major factors affecting the amount of noise in the resulting image. Images are corrupted during transmission due to interference in the channel used for transmission.

Department of ISE, NCET. 1 Spatial and Frequency properties of Noise

Frequency properties refer to the frequency content of noise in the fourier sense .for ex. When the fourier spectrum of noise is constant the noise usually is called White noise.

Some important noise probability density function

● Gaussian Noise

Because of its mathematical tractability in both the spatial and frequency domains , guassian noise models are used frequently in practice. In fact this tractability is so convenient that it often results in gussaian models being used in situations in which they are marginally applicable at best.

where z represents intensity, is the (average) value of z , and σ is its standard deviation σ2 is variance of z.

● Rayleigh noise

The probability density function of Rayleigh noise is given by

Department of ISE, NCET. 2 The plot of Rayleigh noise

Is given

● Erlang (gamma) noise

The probability density function of Erlang noise is given by

where a > 0 and b is a positive integer ! indicates the factorial. The mean and variance of this density are given by

Department of ISE, NCET. 3 The plot is aften reffered to as gamma density . it is correct only when the denominator is gamma function Г(b). when the denominator is as shown then the density is Erlang density.

● Exponential noise

The PDF of exponential noise is given by

where a > 0 . The mean and variance of this density are given by

Department of ISE, NCET. 4 ● Uniform noise

The PDF of uniform noise is given by

The mean and variance of this density function are given by

● Impulse (salt-and-pepper) noise

The PDF of impulse noise is given by

If b >a , intensity b appears as a light dot in the image. Conversely, intensity a will appear like a dark dot.

Department of ISE, NCET. 5 If either Pa or Pb is zero , the impulse noise is called unipolar. If neither probability is zero and especially if they are approximately equal, the impulse noise values will resemble salt and pepper granules randomly distributed over the image.

Example 5.1: Noisy images and their histograms Figure 5.3 shows a test pattern.

Department of ISE, NCET. 6 Department of ISE, NCET. 7 ● Periodic Noise

Periodic noise in an image arises typically from electrical or electromechanical interference during image acquisition. The periodic noise can be reduced significantly via frequency domain filtering,

Estimation of Noise Parameters

The parameters of periodic noise can be estimated by inspection of the Fourier spectrum of the image. Periodic noise tends to produce frequency spikes, which are detectable even by visual analysis. In simplistic cases, it is also possible to infer the periodicity of noise components directly from the image. Automated analysis is possible if the noise spikes are either exceptionally pronounced, or when knowledge is available about the general location of the frequency components of the interference.

It is often necessary to estimate the noise probability density functions for a particular imaging arrangement. When images already generated by a sensor are available, it may be possible to estimate the parameters of the probability density functions from small patches of reasonably constant background intensity.

Department of ISE, NCET. 8 The histograms shown in Figure 5.6 were calculated using image data from these small stripes. We can see that the shapes of these histograms correspond closely to the shapes shown in (d), (e), and (k) of Figure 5.4.

The simplest use of the data from the image strips is for calculating the mean and variance of intensity levels. Let S denote a stripe and PS (Zi), i = 0,1,2,...,L- 1, denote the probability estimates of the intensities of the pixels in S , then the mean and variance of the pixels in S are

The shape of the histogram identifies the closest probability density function match. The Gaussian probability density function is completely specified by these two parameters. For the other shapes discussed previously, we can use the mean and variance to solve the parameters a and b. Impulse noise is handled differently because the estimate needed is of the actual probability of occurrence of the white and black pixels.

Restoration in the Presence of Noise Only – Spatial Filtering When the only degradation present in an image is noise, and

Become

Department of ISE, NCET. 9 and

Since the noise terms are unknown, subtracting them from g(x ,y ) or G(u,v) is not a realistic option. In the case of periodic noise, it usually is possible to estimate N(u,v) from the spectrum of G(u ,v).

Mean Filters ● filter

Let Sxy represent the set of coordinates in a subimage window of size m x n, centered at (x ,y ). The arithmetic mean filter computes the average value of the corrupted image g(x ,y ) in Sxy . The value of the restored image at point (x ,y ) is the arithmetic mean computed in the region Sxy :

This operation can be implemented using a spatial filter of size m x n in which the coefficients have value . A mean filter smooths local variation in an image, and noise is reduced as a result of blurring.

● filter Using a geometric mean filter, an image is restored by

Here, each restored pixel is given by the product of the pixels in the subimage window, raised to the power.

● filter The harmonic mean filter is given by the expression

which works well for some types of noise like Gaussian noise and salt noise, but fails for pepper noise.

● Contraharmonic mean filter The contraharmonic mean filter yields a restored image based on the expression

Department of ISE, NCET. 10

Where Q is called the order of the filter. The contraharmonic mean filter is well suited for reducing or eliminating the effects of salt-and-pepper noise. For positive values of Q, it eliminates pepper noise. For negative values of Q , it eliminates salt noise. When Q = 0, the contraharmonic mean filter reduces to the arithmetic mean filter. When Q = −1, the contraharmonic mean filter becomes the harmonic mean filter.

Order-statistic filters Response based on ordering or ranking the pixel contained in the image area encompassed by the filter.

● Median filter Replace the value of the pixel by the median of the intensity levels in the neighborhood of the pixel

▪ Median filters are quite popular because for certain types of random noise, they Provide noise reduction with considerably less blurring ▪ Effective in the presence of bipolar and unipolar impulse noise.

● Max and min filters

▪ Max filter finds the brightest points in the image; reduces pepper noise ▪ Min filter finds the darkest points in the image; reduces salt noise.

● Midpoint filter Computes the midpoint between the maximum and minimum values in the area encompassed by the filter.

Department of ISE, NCET. 11 ▪ Combines order statistics and averaging

▪ Good for randomly distributed noise, like Gaussian noise and uniform noise.

● Alpha-trimmed mean filter ▪ Delete d/2 lowest and d/2 highest values in the neighborhood

▪ Average the remaining mn − d pixels, denoted by gr(s, t) ▪ Given by

d can range from 0 to mn − 1 , When d = 0, the filter is arithmetic mean filter , When d = mn − 1, the filter is the median filter.

Adaptive filters Adaptive filters Change behavior based on statistical characteristics of image inside the filter Better performance but increase in filters complexity Adaptive, local noise reduction filters ▪ Mean gives a measure of average intensity in the region while variance quantifies contrast

▪ Response of filter on local region Sxy based on four quantities g(x, y) – value of noisy image at (x, y) 2 ση – variance of corrupting noise

mL – local mean of pixels in Sxy

2 σL – local variance in the neighborhood ▪ Behavior of the filter should be

2 1. No noise case: If σ η is zero, return the value of g(x, y) 2 2. Edges:If local variance is high relative to σ η , return a value close to g(x, y) 3. If two variances are equal : if σ 2 L ≈ σ 2 η , reduce local noise by averaging 4. An adaptive expression capturing the above is:

2 ▪ Need to know the variance of overall noise σ η 2 2 ▪ The ratio is set to 1 if the condition σ η > σ L, this makes this filter nonlinear.

Adaptive median filter

Department of ISE, NCET. 12 ▪ Performs well if the spatial density impulse noiseis not larger spatial density than very little (Pa,

Pb > 0.2) ▪ Preserves detail while smoothing nonimpulse noise; median filter unable to achieve that

▪ Works with an adaptive neighborhood, by changing the size of Sxy

Notation zmin Minimum intensity value in Sxy

zmax Maximum intensity value in Sxy

zmed Median of intensity values in Sxy

zxy Intensity value at coordinates (x, y)

Smax Maximum allowed size of Sxy Works in two stages:

Stage A A1 = z med – z min

A2 = z med – z max If A1 > 0 && A2 < 0 go to Stage B Else increase the window size

If window size ≤ S max repeat stage A else output z med

Stage B B1 = z xy – z min

B2 = z xy – z max

If B1 > 0 && B2 < 0 output z xy

Else output z med Three goals of algorithm 1. Remove salt-and-pepper (impulse) noise 2. Provide smoothing of non-impulsive noise 3. Reduce distortion, such as excessive thickening or thinning of object boundaries

● Z min and Z max are considered to be impulse-like components ● Stage A checks whether the median filter output zmed is an impulse

● If Z min < Z med < Z max, Z med cannot be an impulse

● Stage B checks if the point at the center of window Z x y itself is an impulse ● Adaptive median filter does not necessarily replace each point by the median, preserving detail in the process ● Effect of small value of Pa and Pb · As density of impulses increases, we need a larger window to clean up the noise spikes

Inverse Filtering The simplest approach to obtain the restoration of images is direct inverse filtering, where we compute an estimate (u, v), of the transform of the original image simply by dividing the transform of the degraded image, G (u, v), by the degradation function. (u, v)= -- eq1

Department of ISE, NCET. 13 Substituting the RHS by

G(u,v) = H(u,v)F(u,v)+N(u,v)

Then eq1 becomes

(u, v)= F(u,v)+ eq2

This equation tells that even if we know the degradation function , we cannot recover the undegraded image exactly because N(u,v) is not known.

If the degradation function has zero or small values , ythen the ration N(u,v)/H(u,v) could easily dominate the estimate (u, v).

Minimum Mean square Error (Wiener)Filtering

This approach incorporates both the Degradation function and statistical characteristics of noise into the restoration process. The main obi=jective is to find an estimate of the uncorrupted image f such that the mean square error between them is minimized. This error measure is given by

e2 = E {(f- )2} eq 3 where E{.}is the expected value of the argument.itv is assumed that the noise and image are uncorrelated: that one or the other has zero mean and the intensity levels in the estimate are a linear function of the levels in the degraded image.

Based on these conditions the minimum of the error function in the eq3 is given in the frequency domain by the expression

(u, v) = G(u,v)

= G(u,v)

G(u,v)

Where H(u,v) = degradation function

= Complex conjugate of H(u,v)

= H(u,v)

=|N(u,v)|2 = power spectrum of the noise

= |F(u,v)|2 = power spectrum of the undegraded image

Department of ISE, NCET. 14 Department of ISE, NCET. 15