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Chapter 4

Image Denoising using Interpolated Median Filter

4.1 Introduction

The image denoising methods fall into two broad categories: spatial

domain methods and frequency domain methods. The term spatial domain refers

to the image plane itself, and approaches in this category are based on direct

manipulation of pixels in an image. The frequency domain processing techniques

are based on modifying the Fourier transform of an image.

There is no general theory for image enhancement. When an image is

processed for visual interpretation, the viewer is the ultimate judge of how well a

particular method works. Visual evaluation of an image quality is a highly

subjective process, thus making the definition of a “good image” an elusive

standard by which to compare algorithm performance. When the problem is one

of the processing images for machine perception, the evaluation task is somewhat

easier. For example, in dealing with a character recognition application, and

leaving aside other issues such as computational requirements, the best image

processing method would be the one yielding the best machine recognition results.

However, even in situations when a clear-cut criterion of performance can be

imposed on the problem, a certain amount of trial and error is usually required

before a particular image enhancement approach is selected /1,2/.

As explained previously, the term spatial domain refers to the aggregate of

pixels composing an image. Spatial domain methods are procedures that operate

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directly on these pixels. Spatial domain processes will be denoted by the

expression

)],([),( yxfTyxg (4.1)

where ),( yxf is the input image and ),( yxg is the processed image, and T is an

operator on f , defined over some neighborhood of ),( yx . In addition, the operator

T can operate on a single image or on a set of images, such as performing the

pixel-by-pixel sum of a sequence of images for noise reduction. Figure 4.1 shows

the basic implementation of equation 4.1 on a single image. The point (x, y)

shown is an arbitrary location in the image, and the small region shown containing

the point is a neighborhood of (x, y), and is much smaller in size than the image.

Figure 4.1: A 3 x 3 neighborhood about a point (x, y) in an image in the spatial domain

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The process illustrated in figure 4.1 consists of moving the origin of the

neighborhood from pixel to pixel and applying the operator T to the pixels in the

neighborhood to yield the output at that location. Thus, for any specific location

(x, y), the value of the output image ‘g’ at those coordinates is equal to the result

of applying T to the neighborhood with origin at (x, y) in f.

For example, if the neighborhood is a square of size 3x3, and that operator

T is defined to compute the average intensity of the neighborhood. Consider an

arbitrary location in an image, say (100,150). Assuming that the origin of the

neighborhood is at its center, the result, g (100, 150), at that location is computed

as the sum of f (100, 150) and its 8-neighbors, divided by 9 (i.e., the average

intensity of the pixels encompassed by the neighborhood). The origin of the

neighborhood is then moved to the next location and the procedure is repeated to

generate the next value of the output image g. Typically, the process starts at the

top left of the input image and proceeds pixel by pixel in a horizontal scan, one

row at a time. When the origin of the neighborhood is at the border of the image,

part of the neighborhood will reside outside the image.

The procedure is either to ignore the outside neighbors in the computations

specified by T, or to pad the image with a border of 0s or some other specified

intensity values. The thickness of the padded border depends on the size of the

neighborhood. This whole process is called as spatial filtering, in which the

neighborhood, along with a predefined operation, is called a spatial filter (also

referred as a spatial mask, kernel, template, or window). The type of operation

performed on the neighborhood determines the nature of the filtering process/1/.

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4.2 Fundamentals of spatial filtering

The filtering is a fundamental operation in image processing. It can be used

for image enhancement, noise reduction, edge detection, and sharpening. The

concept of filtering has been applied in the frequency domain, where it rejects

some frequency components while accepting others. In spatial domain, filtering is

a pixel neighborhood operation. The commonly used spatial filtering techniques

include median filtering, average filtering, Gaussian filtering, etc. The filtering

function sometimes called filter mask, or filter kernel. They can be broadly

classified into two different categories: linear filtering and order-static filters. In

the case of linear filtering, the operation can be accomplished by convolution, i.e.,

the value of any given pixel in the output image is represented by the weighted

sum of the pixel values of its neighborhood (a linear combination) in the input

image.

For order-static filter, the value of a given pixel in the output image is

represented by some statistics within its support neighborhood in the original

image, such as the median filter. Those filters are normally non-linear and cannot

be easily implemented in the frequency domain. However, the common elements

of a filter are: (1) A neighborhood (2) An operation on the neighborhood

including the pixel itself. Typically the neighborhood is a rectangular of different

size, for example it may be 3x3, 5x5…. and smaller than the image itself /1,2/.

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4.2.1 The mechanics of spatial filtering

The spatial filters as explained in the previous section, consists of a

neighborhood, and a predefined operation that is performed on the image pixels

encompassed by the neighborhood. Filtering creates a new pixel with coordinates

equal to the coordinates of the center of the neighborhood, and whose value is the

result of the filtering operation.

A processed (filtered) image is generated, as the center of the filter visits

each pixel in the input image. If the operation performed is on the image pixels is

linear, then the filter is said to be linear spatial filter. Otherwise, the filter is

nonlinear.

4.2.2 Spatial correlation and convolution

There are two closely related concepts for performing linear spatial

filtering. One is the correlation and the other is the convolution. Correlation is the

process of moving a filter mask over the image and computing the sum of

products at each location. The mechanics of the convolution are the same, except

that the filter is rotated by 180 degrees.

4.2.3 Generating spatial filter masks

Generating an m x n linear spatial filter requires specifying mn mask

coefficients. For example, to replace the pixels in an image by the average

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intensity of 3 x 3 neighborhood centered on those pixels, the average value at any

location (x, y) in the image is determined by the sum of the nine intensity values

in the 3 x 3 neighborhood centered on (x, y) divided by 9.

Letting zi = 1,2,…….9, denote these intensities, the average is

9

19

1

iiZR (4.2)

This linear filtering operation with 3 x 3 mask whose coefficients are 1/9

implements the desired averaging. This operation results in image smoothing /1/

4.3 Smoothing spatial filters

The smoothing filters are used for blurring and for noise reduction.

Blurring is used in preprocessing steps, such as removal of small details from an

image prior to (large) object extraction, and bridging of small gaps in lines or

curves. Noise reduction can be accomplished by blurring with a linear and also by

non-linear filtering.

4.3.1 Smoothing Linear Filters

The output of a smoothing, linear spatial filter is simply the average of the

pixels contained in the neighborhood of the filter mask. These filters sometimes

are called averaging filters and they are also referred as low-pass filters. The idea

behind smoothing filters is straightforward. By replacing the value of every pixel

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in an image by the average of the fray levels in the neighborhood defined by the

filter mask. This process results in an image with reduced ‘sharp’ transitions in

gray levels. The most obvious application of smoothing is noise reduction.

However, edges (which almost always are desirable features of an image) also are

characterized by sharp transitions in gray levels, so averaging filters have the

undesirable side effect that they blur edges.

The another application of this type of process includes the smoothing of

false contours that result from using an in sufficient number of gray levels. A

major use of averaging filters is in the reduction of ‘irrelevant’ detail in an image.

By ‘irrelevant’ pixel means pixel regions that are small with respect to the size of

the filter mask/1/.

4.3.2 Order-static (non-linear) filters

The order-static filters are nonlinear spatial filters whose response is based

on ordering (ranking) the pixels contained in the image area encompassed by the

filter, and then replacing the value of the center pixel with the value determined

by the ranking result. The best-known example in this category is the median

filter, which as its name implies, replaces the value of a pixel by the median of the

gray levels in the neighborhood of that pixel ( the original value of the pixel is

included in the computation of the median ).

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4.3.3 Median Filter

The median filters are quite popular because, for certain types of random

noise, it has excellent noise-reduction capabilities, with considerably less blurring

than linear smoothing filters of similar size. These filters are particularly effective

in the presence of impulse noise, also called salt-and-pepper noise because of its

appearance as white and black dots superimposed on an image.

The median M, of a set of values is such that half the values in the set are

less than or equal to M, and half are greater than or equal to M. in order to perform

median filtering at a point in an image. First sort the values of the pixel in

question and its neighbors, determine their median, and assign this value to that

pixel. For example, in a 3 x 3 neighborhood the median is the 5th largest value, in

a 5 x 5 neighborhood the 13th largest value, and so on. When several values in a

neighborhood are the same, all equal values are grouped.

For example, a 3 x 3 neighborhood has values (10, 20, 20, 20, 15, 20, and

25,100).These values are sorted as (10,15,20,20,20,20,20,25,100), which results in

a median of 20. Thus, the principal function of median filters is to force points

with distinct gray levels to be more like their neighbors. In fact, isolated clusters

of pixels that are light or dark with respect to their neighbors, and whose area is

less than n2/2 (one-half the filter area), are eliminated by an n x n median filter. In

this case ‘eliminated’ means forced to the median intensity of the neighbors/1/.

The median is calculated by first sorting all the pixel values from the

surrounding neighborhood into numerical order and then replacing the pixel being

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considered with the middle pixel value. The median filter, especially with larger

window size destroys the fine image details due to its rank ordering process.

Figure 1.Illustrates an example calculation of median value. The central pixel

value of 150 is rather unrepresentative of the surrounding pixels and is replaced

with the median value: 124

Neighborhood values: 115,119,120,123,124,125,126,127,150

Median value: 124

Figure 4.2: Calculating the median value of a 3x3 pixel neighborhood.

4.3.4 Max and min filters

Although the median filter is by far the order-static filter used in image

processing, it is by no means the only one. The median represents the 50th

percentile of a ranked set of numbers, but from basics of statistics the ranking

lends itself to many other possibilities. For example, using the 100th percentile

results in the so called max filter, this filter is useful for finding the brightest

110 125 125 130 140

123 124 126 127 136

114 120 150 125 134

118 115 119 123 134

111 116 111 120 131

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points in an image. Also because the pepper noise has very low values, it is

reduced by this filter as a result of the max selection process in the sub image

areaThe 0th percentile filter is known as the min filter. This filter is useful in

finding the darkest points in an image and also, it reduces salt noise as a result of

the min operation.

4.3.5 Midpoint filter

The midpoint filter simply computes the midpoint between the maximum

and minimum values in the area encompassed by the filter. The filter combines

the order statics and averaging. It works best for randomly distributed noise, like

Gaussian noise or uniform noise.

4.3.6 Mean Filtering

The mean filtering is a simple and easy to implement method for

smoothing images, i.e. it reduces the amount of intensity variation between one

pixel and the next. It is often used to reduce noise in images The idea of mean

filtering is simply to replace each pixel value in an image with the mean value of

its neighbors, including itself.

A 3×3 mean filter is used for smoothing the image containing the salt and

pepper noise. Since the shot noise pixel values are often very different from the

surrounding values, they tend to significantly distort the pixel average calculated

by the mean filter /1/.

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4.4 Modern image denoising methods

Consider an image that has been blurred and contaminated by salt and

pepper noise. Typical sources of blur are defocus and motion of camera/3/. Salt

and pepper noise is a common model for the effects of bit errors in transmission,

malfunctioning pixels and faulty memory locations/4/.

Significant attention has been given to image denoising in the presence of

Gaussian noise /3/. We focus on various methods that have an important role in

modern image denoising research /5-8/. Most methods rely on the standard model

nfhg * , that is applicable to a large variety of image degradation processes

that are encountered in practice. Here ‘h’ represents a known space-invariant blur

kernel (point spread function), ‘f’ is an ideal version of the observed image ‘g’

and ‘n’ is (usually Gaussian) noise. In this research,the focus is on the case of salt

and pepper noise.

The assumption of Gaussian noise is a fundamental element of common

image denoising algorithms. It is therefore not surprising that those algorithms

produce inadequate results in the presence of salt and pepper noise. This fact is

illustrated in figure4.3. The top-left image in figure 4.3 is the 256x256 Lena

image, blurred by a pill-box kernel of radius 3 and contaminated by Gaussian

noise.

Excellent restoration is obtained using the state of the art denoising method

of /7/ (top-right). The bottom-left image in figure 4.3 is the same blurred Lena

image, is contaminated by salt and pepper noise of density 0.01. In this case

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restoration using the method of /7/ is clearly inadequate (bottom-right). Note that

due to the inadequacy of the noise model, the algorithm of /7/ yields poor results

even at lower salt and pepper noise density.

A variety of filtering techniques have been proposed for enhancing images

degraded by salt and pepper noise. The classical linear digital image filters, such

as averaging low pass filters, tend to blur edges and other fine image details.

Therefore, nonlinear filters /15, 16/ are most preferred over linear filters due to

their improved filtering performance in terms of noise suppression and edge

preservation.

Figure 4.3: Top-left: Blurred image with Gaussian noise. Top-right: Restoration using the method of /23/. Bottom-left: Blurred image with salt and pepper noise. Bottom-right: Restoration using the method of /23/.

123

The standard median (SM) filter /17/ is the one of the most robust

nonlinear filters, which exploits the rank-order information of pixel intensities

within filtering window. This filter is very popular due to its edge preserving

characteristics and its simplicity in implementation. Various modifications of the

SM filter have been introduced, such as the weighted median (WM) /18/ filter. By

incorporating noise detection mechanism into the conventional median filtering

approach, the filters like switching median filters /19,20/ had shown significant

performance improvement. Salt and pepper noise removal is considered in the

literature by itself. It is commonly approached using median-type filters /9-11/.

Recently, a promising variational method for impulse denoising was proposed by

/12-14/. At high noise density, image denoising using median type filtering

creates distortion that depends on the neighborhood size.

The median filter, as well as its modifications and generalizations /21/ is

typically implemented invariably across an image. Examples include the mimum-

maximumum exclusive mean filter (MMEM) /22/, Florencio’s /23/, Adaptive

median filter(AMF) /24/. These filters have demonstrated excellent performance

but the main drawbacks of all these filters are, they are prone to edge jitters in the

cases where noise density is high and their large widow size results in blurred

images and significant computational complexity. To overcome this problem, a

modified median filter algorithm called Interpolate Median filter that employs

Interpolated search in determining the desired central pixel value is proposed.

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4.5 Interpolate median filter

In this section a new approach to image denoising called Interpolate

Median Filter (IMF) is proposed. The Interpolate Median filter method considers

each pixel in the image in turn and looks at its neighbors to decide whether or not

it is representative of its surroundings. Instead of replacing the pixel value with

the median of neighboring pixel values, it replaces it with the interpolation of

those values. The interpolation is calculated by first sorting all pixel values from

surrounding neighborhood into numerical order and then replacing the pixel being

considered with the interpolation pixel value. The calculation of interpolation

value is derived from the Interpolation search technique used for searching the

elements. We can also call it a Non- linear filter or order-static filter because there

response is based on the ordering or ranking of the pixels contained within the

mask.

The advantages of this filter over mean and median filter are, it gives more

robust average than both the methods, for some pixels in the neighborhood; it

creates new pixel values like mean filter and for some it will not create new pixel

value like median filter, It has the characteristics of both filters. The algorithm

uses the following relations:

2/])[][( halaK , (4.3)

where K is the ‘key’, Here we make an intelligent guess about ‘key’ which is the

mid value of the array ‘a’, and ][],[ hala are values of bottom and top elements in

the sorted array.

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The value of the ‘key’ K is used in the relation 4.4 to determine the interpolated

value

]))[][((*)( lahaklhlMid . (4.4)

Here value ‘Mid’ gives the optimal mid-point of the array and a[mid] gives the

interpolated value. This interpolated value is the new value of the pixel.

4.5.1 Implementation of Interpolate median filter

The steps of implementing Interpolate Median filter is given followed by

the corresponding flowchart in figure 4.4.

Step 1:

Read the original standard image (Lena.png, Barbara.png,Goldhill.png).

Step 2:

The preprocessing of image is done which resizes the loaded image to a

standard size of 512 x 512, suitable for further processing.

Step 3:

The noise is added to the standard test images. In this work Salt & pepper

noise is used as it is the simplest type of noise among all. According to a given

density D more or less pixels are flipped randomly to black (0) or white (1). D is

just measure of the amount of noise to be added, not a value! This type of noise is

also independent of the image it is added to.

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Step 4:

The noisy image is then subjected to Interpolate Median Filtering. The

resultant image is the denoised image.

Step 5:

The peak signal to noise ratio (PSNR) is calculated for all the standard

images with their noisy and denoised counterparts, respectively. The PSNR values

are compared with some traditional standard methods.

4.5.2 Flowchart for the Interpolate median filter

Figure 4.4 Flow chart for Interpolate median filter

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4.5.3 Matlab code for the Interpolate median filter

% READ THE IMAGE

img1=imread('lena.png');figure,imshow(img1)img2 = imnoise(img1,'salt & pepper',0.5);figure,imshow(img2)img2=double(img2);[Height,width]=size(img2);img = zeros(width,Height);for ycoord=1:1:Height; topbound=ycoord-1; bottombound=ycoord+1; if topbound<1 topbound=1;end if bottombound>Height; bottombound=Height; endfor xcoord=1:1:width; leftbound=xcoord-1; rightbound=xcoord+1; if leftbound<1 leftbound=1; end if rightbound>width rightbound=width; endarrsize = (rightbound-leftbound+1)*(bottombound-topbound+1); arr2 = zeros(arrsize,1); xindex=leftbound; yindex=topbound; index=1;while index<=arrsize arr2(index)=img2(xindex,yindex); xindex=xindex+1; if xindex >rightbound; xindex=leftbound; yindex=yindex+1;end if yindex>bottombound;

128

break;end index=index+1;end

% SORTING OF ARRARY TO FIND INTERPOLATED MEDIANsortedarr = SelectionSort(arr2);a=sortedarr;low=1;l=length(a);high=l;m=ceil((low+high)/2);key=a(m); if (a(high)-a(low)==0) w=0.1+a(high)-a(low); else w=a(high)-a(low); end z=ceil(u*v/w);if ( z==0) z=ceil(low+high)/2);end img(xcoord,ycoord,1)= a(z);%mid; %img(xcoord,ycoord,1)=sortedarr(ceil((arrsize)/2)); endend

img=uint8(img);figure,imshow(img)img=double(img);%img2=uint8(img2);%snr=20*log10(norm(img2))/norm(img2-img)% FINDING SIGNAL TO NOISE RATION = prod(size(img1));img1= double(img1(:)); img = double(img(:));t1 = sum((img1-img).^2); t2 = sum(img1.^2);MSE = t1/NPSNR = 10*log10(255*255/MSE)

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4.7 Simulation results

The proposed method is tested on some natural grayscale test images like

Lena, Barbara and Goldhill of size 512*512 pixel, at different noise levels. Table

4.1, illustrates the comparison of PSNRs of the six denoising methods. The PSNR

(in dB) is calculated using the formula defined by

)(255

log102

dBMSE

PSNR (3.23)

with 21

0

1

0

),(),(1

m

i

n

j

jiKjiImn

MSE , (3.24)

where I and K being the original image and de-noised image, respectively.

The original image and the noisy image of Lena is shown in th fig. 4.5

Figure 4.5 (a): The original test image (Lena) with 512x512 pixels

Figure 4.5 (b): Lena image corrupted by salt & pepper noise(dB) (20%)

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The performance analysis of the method for Lena and Barbara image is given in

table 4.2, the noise density is varied fron 10% to 50% in steps of 10 and PSNR

values are tabulated.

TABLE 4. 1.

PSNR Performance of Different Algorithms for Lena image corrupted with salt and pepper noise

AlgorithmNoise Density in dB

10% 20% 30%

MF(3x3) 31.19 28.48 25.45

MF(5x5) 29.45 28.91 28.43

MMEM [8] 30.28 29.63 29.05

Florencio’s [9] 33.69 32.20 30.95

AMF(5x5) [10] 30.11 28.72 27.84

IMF(Proposed) 33.86 30.59 25.75

TABLE 4.2

Performance analysis

Noise Density in dB (in %)

Lena Barbara

10 33.86 24.83

20 30.59 23.82

30 25.75 21.80

40 20.02 19.00

50 17.23 16.08

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4.7 Conclusions

The proposed algorithm called Interpolate Median filter employs

Interpolated search in determining the desired central pixel value. Interpolation

median filter for image denoising is a simple method and easy to implement.

The simulation results show that the proposed method performs

significantly better than many other existing methods

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