Image Restoration Filters in Spatial Domain for various ... · Image Restoration Filters in Spatial...

IJSRD - International Journal for Scientific Research & Development| Vol. 5, Issue 07, 2017 | ISSN (online): 2321-0613

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Image Restoration Filters in Spatial Domain for various Noise Density

Ranges Smita Agrawal1 Sunil Kumar2 Anurag Bajpai3 Vivek Kumar4

1,2,3,4Department of Electronics Design & Technology 1,2,3,4NIELIT, Gorakhpur, India

Abstract— Image Restoration is an area of image processing

that deals with the removal of noise by various filtering

techniques. There are various filters that exist in literature for

various types of noise. Most of them are specific to the noise

type and particular noise ranges. These are classified into

linear and non-linear filters. Linear filters are more suitable

for gaussian noise removal even at higher noise densities

(>30%) and at times for speckle noise removal, whereas non-

linear filters are more suitable for impulse noise removal,

especially at low and medium level densities. However, at

higher noise densities, non-linear filtering efficiency

generally gets degraded. In this paper, a comparative

analytical treatment is done for six filtering methods, at

medium to high noise densities, and the choice of the most

suitable filter technique is determined for particular noise

ranges. The noise and filter functions are implemented in

MATLAB.

Key words: SNR, PSNR, MSE, Salt and Pepper Noise,

Speckle Noise, Gaussian Noise, Hybrid Median Filter

I. INTRODUCTION

Noise removal from a corrupted image has been a prominent

field of research and a large number of algorithms have been

implemented, tested and their results are compared [2][3][16][18][20]. The main thrust on all such algorithms is to

remove impulse noise while preserving image details [21].

Non-linear filters are most suitable for performing noise

removal as well as edge preservation, while linear filters

cause blurring of images [17][20][21][22][23]. Linear filtering of an

image is accomplished through an operation

called convolution. Convolution of neighbour-hood

operation in which each output pixel is the weighted sum of

neighbouring input pixels. The matrix of weights is called

the convolution kernel, also known as the filter. A

convolution kernel is a correlation kernel that has been

rotated 180 degrees [22]. The operation called correlation is

closely related to convolution. In correlation, the value of an

output pixel is also computed as a weighted sum of

neighbouring pixels. The difference is that the matrix of

weights, in this case called the correlation kernel, is not

rotated during the computation. Major linear filters include

Mean Filter, Adaptive Wiener Filter, Gaussian Filter. Linear

filters are more suited for frequency domain as these work on

the frequency spectrum [19][20]. These remove speckle noise

and also, gaussian noise to a great extent, thereby reducing

the size of the image, however, in doing that, they cause

blurring of images, with Wiener filter being an exception.

Non-linear filtering methods, on the other hand, give

excellent salt and pepper (impulse) noise removal in spatial

domain. These, however, are not suited for gaussian noise

removal as their statistical analysis is very difficult.

II. NOISE

Noise arises as a result of un-modelled or un-modellable

processes going on in the production and capture of the real

signal [2]. It is not part of the ideal signal and may be caused

by a wide range of sources, e.g. variations in the detector

sensitivity, environmental variations, the discrete nature of

radiation, transmission or quantization errors, etc. It is also

possible to treat irrelevant scene details as if they are image

noise (e.g. surface reflectance textures). The characteristics

of noise depend on its source, as does the operator which best

reduces its effects [10][14][15]. Many image processing packages

contain operators to artificially add noise to an image.

Deliberately corrupting an image with noise allows us to test

the resistance of an image processing operator to noise and

assess the performance of various noise filters. Noise can

generally be grouped into two classes:

Independent noise.

Noise which is dependent on the image data.

Image independent noise can often be described by

an additive noise model, where the recorded image f(x,y) is

the sum of the true image s(x,y) and the noise n(x,y):

f(x,y) = s(x,y) + n(x,y) (1)

The noise n(x,y) is often zero-mean and described

by its variance σn2 . The impact of the noise on the image is

often described by the signal to noise ratio (SNR), which is

given by:

(1)

Where σs2 and σf

2 are the variances of the true image

and the recorded image, respectively. In the second case of

data-dependent noise (e.g. arising when monochromatic

radiation is scattered from a surface whose roughness is of the

order of a wavelength, causing wave interference which

results in image speckle), it is possible to model noise with a

multiplicative, or non-linear, model. These models are

mathematically more complicated; hence, if possible, the

noise is assumed to be data independent. The major types of

noise that occur in MRI and ultrasound spectroscopy are

impulse noise and speckle noise. In this work, three types of

noise are filtered; viz., salt and pepper (impulse) noise,

speckle noise, and gaussian noise; and the results are analysed

by three parameters; viz., SNR, PSNR and MSE. Six filters

are used for noise removal separately and their performance

is observed for different noise densities.

A. Impulse Noise

The impulse noise may be classified into salt-and-pepper

noise and random valued noise. The salt and pepper noise

pixels can take only the maximum gray and the minimum

gray values. But in random valued noise pixels can take any

random value between the maximum and minimum gray

values. Thus, it could severely degrade the image quality and

Image Restoration Filters in Spatial Domain for various Noise Density Ranges

(IJSRD/Vol. 5/Issue 07/2017/062)


cause some loss of information. An image containing impulse

noise can be represented as

g(x,y) = {η(x, y)with probability P

f(x, y)with probability 1 − P (3)

The detection of random valued impulse noise is

much difficult than the detection of salt-and-pepper. The

mean square error (MSE) is used to determine the peak signal

to noise ratio (PSNR).

MSE = ∑ ∑ (𝐨(𝐱,𝐲)𝐟(𝐱,𝐲))𝟐𝐧−𝟏

𝐲=𝟎𝐦−𝟏𝐱−𝟎

𝐌 𝐍 (4)

The PSNR in dB is given,

PSNR = 10log (𝟐𝟓𝟓𝟐

𝐌𝐒𝐄) (5)

O(x, y) is the original image, f (x,y) represents the

denoised image, and M N is the size of the image. Idea

behind in this PSNR is to compute a single number that

reflects the quality of the restored image. Salt-and-pepper

noise is a form of noise sometimes seen on images. It is also

known as fixed valued impulse noise. This noise can be

caused by sharp and sudden disturbances in the image signal.

It presents itself as sparsely occurring white and black pixels.

Here, the noise is caused by errors in the data transmission [2][8]. The corrupted pixels are either set to the maximum value

(which looks like snow in the image) or have single bits

flipped over. In some cases, single pixels are set alternatively

to zero or to the maximum value, giving the image a `salt and

pepper' like appearance. Unaffected pixels always remain

unchanged. The noise is usually quantified by the percentage

of pixels which are corrupted. Often, salt and pepper noise is

the natural result of creating a binary image via thresholding [11][14]. Salt corresponds to pixels in a dark region that

somehow passed the threshold for bright, and pepper

corresponds to pixels in a bright region that were below

threshold. Salt and pepper might be classification errors

resulting from variation in the surface material or

illumination, or perhaps noise in the analog/digital

conversion process in the frame grabber [5][6]. For this kind of

noise, conventional low pass filtering, e.g. mean filtering or

Gaussian smoothing is relatively unsuccessful because the

corrupted pixel value can vary significantly from the original

and therefore the mean can be significantly different from the

true value [25]. A median filter removes drop-out noise more

efficiently and at the same time preserves the edges and small

details in the image better [3]. Conservative smoothing can be

used to obtain a result which preserves a great deal of high

frequency detail, but is only effective at reducing low levels

of noise. Median filtering techniques and their variations are

quite successful in reducing impulse noise to a great extent.

B. Speckle Noise

Speckle is a granular 'noise' that inherently exists in and

degrades the quality of the active radar, synthetic aperture

radar (SAR), medical ultrasound and optical coherence

tomography images. The vast majority of surfaces, synthetic

or natural, are extremely rough on the scale of the

wavelength. Images obtained from these surfaces by coherent

imaging systems such as laser, SAR, and ultrasound suffer

from a common phenomenon called speckle. Speckle, in both

cases, is primarily due to the interference of the returning

wave at the transducer aperture. The origin of this noise is

seen if we model our reflectivity function as an array of

scatterers [13]. Because of the finite resolution, at any time we

are receiving from a distribution of scatterers within the

resolution cell. These scattered signals add coherently; that is,

they add constructively and destructively depending on the

relative phases of each scattered waveform [15][16]. Speckle

noise results from these patterns of constructive and

destructive interference shown as bright and dark dots in the

image. Speckle noise is a multiplicative noise, i.e. it is in

direct proportion to the local gray level in any area. The signal

and the noise are statistically independent of each other. The

sample mean and variance of a single pixel are equal to the

mean and variance of the local area that is centred on that

pixel. Gaussian filters perform best removal of speckle noise

as their impulse response is a gaussian function itself.

C. Gaussian Noise

This kind of noise is due to the discrete nature of radiation,

i.e. the fact that each imaging system is recording an image

by counting photons. Allowing some assumptions (which are

valid for many applications) this noise can be modelled with

an independent, additive model, where the noise n(x,y) has a

zero-mean Gaussian distribution described by its standard

deviation σ, or variance. This means that each pixel in the

noisy image is the sum of the true pixel value and a random,

Gaussian distributed noise value [14]. Gaussian noise is

statistical noise having a probability density function (PDF)

equal to that of the normal distribution, which is also known

as the Gaussian distribution. In other words, the values that

the noise can take on are Gaussian-distributed. Image noise is

the random variation of brightness or colour information in

images produced by the sensor and circuitry of a scanner or

digital camera [18][20]. The probability density function p of a

Gaussian random variable z is given by:

(6)

In digital image processing, Gaussian noise can be

reduced using a spatial filter, though when smoothing an

image, an undesirable outcome may result in the blurring of

fine-scaled image edges and details because they also

correspond to blocked high frequencies. Conventional spatial

filtering techniques for noise removal include: mean

(convolution) filtering, median filtering and Gaussian

smoothing [15].

III. RESULTS

Six different filters of standard window size 5 x 5 are taken

for noise. The efficiency of each filter is tested against

increasing noise density levels. The noise levels induced were

10%, 20%, 30% and 50% for different iterations.

Fig. 1: Original Image




Fig. 2: Average Filtering at 30% Noise

At high noise densities, average (mean) filters

perform marginally better than other filters for speckle noise

removal. They, however, behave poorly for salt and pepper

noise removal.

Fig. 3: Adaptive Wiener Filtering at 30% Noise

Adaptive Wiener Filters perform fairly well for

speckle and gaussian noise removal at medium to high noise

densities but not for salt and pepper noise.

Fig. 4: Gaussian Filtering at 30% Noise

Gaussian filters perform the best noise filtering at

lower noise densities, and they fairly well for higher noise

densities also. However, these are highly inefficient for

gaussian noise and impulse noise removal.

Fig. 5: Standard Median Filtering at 30% Noise

Salt and pepper noise

Speckle noise

Gaussian noise


Speckle noise

Gaussian noise


Speckle noise

Gaussian noise


Speckle noise

Gaussian noise




Fig. 6: Weighted Median Filtering at 30% Noise

Standard Median Filters exhibit quite superior filtering for

salt and pepper noise at even very high noise densities.

Fig. 7: Hybrid Median Filtering at 30% Noise

Hybrid Median Filters perform best noise removal

for salt and pepper noise at low noise densities. They behave,

however, quite poorly, at higher noise density values.

Table 1: Filter performance at 10% noise

Table 2: Filter performance at 30% noise

From Table.1, it is evident that median filters and

their derivations provide best discrete impulse noise removal

at lower noise densities (Figure.8).

Fig. 8: Bar Graphs for salt and pepper noise filtering at 10%

noise density

However, median filters tend to minimize the mean

square error better than the linear filters for this purpose.

Overall, hybrid median filters provide convenient results both

in terms of SNR and MSE (Figure.9). At lower noise

densities, standard median filters exhibit superior

performance for Gaussian noise removal both in terms of

SNR as well as MSE.


Speckle noise

Gaussian noise

NOISE Para-

meter

Average

(Mean)

Filter

Adaptive

Wiener

Filter

Gaussian

Filter

Standard

Median

Filter

Weighted

Median

Filter

Hybrid

Median

Filter

Salt and

Pepper

(Impulse)

Noise

SNR

(dB)

11.069 9.207 7.391 12.111 4.005 13.321

PSNR

(dB)

19.905 19.081 17.265 21.985 13.879 22.157

MSE 0.0102 0.0124 0.0188 0.0063 0.0409 0.0061

Speckle

Noise

SNR

(dB)

12.174 13.016 13.728 11.112 10.492 13.014

PSNR

(dB)

21.010 22.889 23.601 20.986 20.365 21.849

MSE 0.0079 0.0051 0.0044 0.0080 0.0092 0.0065

Gaussian

Noise

SNR

(dB)

8.781 7.565 5.384 9.578 2.524 6.191

PSNR

(dB)

17.616 17.438 15.257 19.451 12.398 15.026

MSE 0.0173 0.0180 0.0290 0.0113 0.0576 0.0314




Fig. 9: Bar Graphs for speckle noise filtering at 10% noise

density

Adaptive Wiener Filters and Mean Filters are also

quite good in removal of gaussian noise for low noise

(Figure.10).

Fig. 10: Bar Graphs for gaussian noise filtering at 10% noise

density

At low noise densities, Gaussian filters are the best

for speckle noise filtering. At higher noise densities, standard

median filter exhibits, superior performance than all others

for salt and pepper noise removal (Figure.11). For speckle

noise filtering at high noise densities, mean filters and

gaussian filters are efficient both in terms of SNR

improvement as well as minimum MSE (Figure.12).

Fig. 11: Bar Graphs for salt and pepper noise filtering at

30% noise density

Adaptive Wiener filters also provide decent results

in terms of minimizing the MSE, however SNR performance

is not that satisfactory. Standard median and hybrid median

filters perform reasonably well for speckle noise removal at

high noise densities.

Fig. 12: Bar Graphs for speckle noise filtering at 30% noise

density

At higher noise densities, linear filters behave quite

poorly in removal of gaussian noise, as they fail to minimize

the MSE.

Fig. 13: Bar Graphs for gaussian noise filtering at 30% noise

density

Standard median filters, however, concede

satisfactory results for gaussian noise filtering as compared to

other filters at higher noise densities both in terms of SNR as

well as MSE (Figure.13).

IV. CONCLUSION

Six different filters have been implemented in this work; with

an injection of lower as well as higher noise densities, in order

to test filter performance for the same. It is observed that for

lower and medium range noise densities, hybrid median

filters exhibit the best filter performance for salt and pepper

noise removal. However, their performance declines sharply

as noise densities are increased. At high noise densities, they

can give fair results only for speckle noise filtering. Weighted

median filters are inefficient for most noise densities as

compared to other filtering methods, hence they need certain

adaptations for enhanced performance [4][24]. Gaussian filters

perform best speckle noise filtering at low as well as higher

noise densities. Adaptive wiener filters are suitable for

gaussian noise removal at low as well as higher noise

densities, though they are not that effective in reducing the

blurring of edges at higher noise. Average filters give

reasonable results at higher noise ranges, but blurring

continues to be a major issue. Standard median filters provide

quite useful results in most cases, even at higher noise

densities. But they are computationally very expensive, time

consuming and quite hard to implement. Further research is

needed for better noise removal at very high noise levels by

suitable adaptive algorithms, especially for non-linear filters.




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