digitalimageprocessing11-imagerestoration1

Course Website: http://www.comp.dit.ie/bmacnamee

Digital Image Processing

Image Restoration:

Noise Removal

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Contents

In this lecture we will look at image restoration techniques used for noise removal

– What is image restoration?– Noise and images– Noise models– Noise removal using spatial domain filtering

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What is Image Restoration?

Image restoration attempts to restore images that have been degraded

– Identify the degradation process and attempt to reverse it

– Similar to image enhancement, but more objective

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“Better” visual representation

Subjective

No quantitative measures

Remove effects of sensing environment

Objective

Mathematical, model dependent quantitative measures

What is Image Restoration?Enhancement vs. Restoration

Enhancement Restoration

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Enhancement Restoration

SpatialDomain

FrequencyDomain

Point Processing• >>imadjust• >>histeq

Spatial filtering• >>filter2

Filtering• >>fft2/ifft2• >>fftshift

• Inverse filtering• Wiener filtering

Image Preprocessing

What is Image Restoration?

Spatial filtering• >>filter2

SpatialDomain

FrequencyDomain

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Image Restoration General

One has to have some a priori knowledge about the degragation process.

Usually we need to know:The noise in the original imageModel for degragationSome information from original image

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A Model of Image Degradation and Restoration

),( ),(*),(),( yxyxfyxhyxg

),( ),(),(),(G vuNvuFvuHvu

The degradation process is modeled as a degradation function that together with an additive noise term.

In frequency domain representation:

Where: f(x,y) is the input image, g(x,y) is the degraded image, h(x,y) is the degradation function, and is the additive noise.),( yx

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Degradation Model

f(x,y) h(x,y) g(x,y)

n(x,y)

Degradation Model: g = h*f + n

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Restoration Model

f(x,y) DegradationModel

f(x,y)RestorationFilter

Unconstrained Constrained• Inverse Filter• Pseudo-inverse Filter

• Wiener Filter

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Restoration Model

The objective of restoration is to obtain an estimate of the original image f(x,y).

Generally, the more we know about H, and , the closer will be to f(x,y).

The approach used throughout most of this chapter is based on various types of image restoration filters.

),( yxf

),( yxf

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Noise and Images

The sources of noise in digital images arise during image acquisition (digitization) and transmission

– Imaging sensors can be affected by ambient conditions

– Interference can be added to an image during transmission

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Noise Models

There are many different models for the image noise term η(x, y):

– Gaussian• Most common model

– Rayleigh– Erlang– Exponential– Uniform– Impulse

• Salt and pepper noise

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Noise Models

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Rayleigh noise:

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Noise Models

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Exponential noise:

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Noise Models

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Uniform noise:

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Impulse noise:

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For pa=pb=0.05 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128

Input image

128 128 255 0 128 128 128 128 128 128 128 128 128 128 0 128 128 128 128 0 128 128 128 128 128 128 128 128 128 128 128 128 0 128 128 128 128 128 128 128 128 128 128 255 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 0 128 128 128 128 255 128 128 128 128 128 128 128 128 128 128 128 128 128 255 128 128 128 128 128 128 128 255 128 128

Degraded image

Noise level p=0.05 means that approximately 5% of pixels are contaminated by salt or pepper noise (highlighted by red color)

Noise Models

Impulse noise is also called salt-and-pepper noise.

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MATLAB Command

>Y = IMNOISE(X,'salt & pepper',p)

Notes:

The intensity of input images is assumed to be normalized to [0,1]. If X is not double, you need to do normalization first, i.e., X=X/255; If X is uint8, MATLAB would do the normalization automatically

The default value of p is 0.05 (i.e., 5 percent of pixels are contaminated)

Imnoise function can produce other types of noise as well (you need to change the noise type ‘salt & pepper’)

Example: impulsenoise.m

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Noise Models

Remarks: The preceding PDFs provide useful tools for modeling a

broad range of noise corruption situations in practice. Gaussian noise arises due to factors such as electronic

circuit noise and sensor noise caused by poor illumination and/or high temperature.

Rayleigh is helpful in characterizing noise phenomena in range imaging.

Exponential and gamma find application in laser imaging. Impuse noise is found in situations where quick transients,

such as faulty switching. Uniform is the least descriptive of practical situations.

However, it is quite useful in simulations.

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Noise Example

The test pattern to the right is ideal for demonstrating the addition of noise

The following slides will show the result of adding noise based on various models to this image

Histogram to go here

Image

Histogram

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Noise Example (cont…)

Gaussian Rayleigh Erlang

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Noise Example (cont…)

Exponential Uniform Impulse

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Periodic Noise

Periodic noise in an image arises typically from electrical or electromechanical interference during image acquisition.

Characteristics– Spatially dependent– Periodic – easy to observe in frequency

domain

Processing method– Suppressing noise component in frequency

domain

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Periodic Noise

spatial Frequency (note the four pairs of bright dots)

See Table 4-1:

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Estimation of Noise Parameters

Periodic noise typically are estimated by inspection of the Fourier spectrum of the image.

The parameters of noise PDFs may be known from sensor specifications.

It is often necessary to estimate them for a particular imaging device.

When images generated by the sensor are available, frequently it is possible to estimate the parameters of the PDF from small patches of constant gray level.

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Gaussian noise Rayleigh noise Uniform noise

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2 2

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i iz S

z p z

z p z

The simplest way to use the data from the image strip is for calculating the mean and variance of the gray levels.

Where zi is the gray level values of the pixels in S, and p(zi) are the corresponding normalized histogram values.

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If the shape is approximately Gaussian, then the mean and variance is all we need. Because Gaussian PDF is completely specified by these two parameters.

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For the other shapes discussed before, we can use the mean and variance to calculate the parameters a and b. For example, Rayleigh noise:

For impulse noise we can estimate the probability of occurrence of white and black pixels. So, a midgray, relatively constant area is needed.

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Restoration of Noise Only Degradation

We can consider a noisy image to be modelled as follows:

where f(x, y) is the original image pixel, η(x, y) is the noise term and g(x, y) is the resulting noisy pixel

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If the noise is known, we can just subtract them from g(x,y) or G(u,v) to restore f(x,y) or F(u,v).

Usually, the noise is unknown, so subtracting them from g(x,y) or G(u,v) is not a realistic option.

In the case of periodic noise, it is possible to estimate N(u,v) from the spectrum of G(u,v), then N(u,v) can be subtracted from G(u,v) to obtain an estimate of the original image.

However, this type of knowledge is the exception, rather than the rule.

When only additive noise is present, spatial filtering is a choice. In this case, enhancement and restoration become almost the same.

Restoration of Noise Only Degradation

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Mean filters

We can use different kinds of spatial filters to remove different kinds of noise

The arithmetic mean filter is a very simple one and is calculated as follows:

This is implemented as the simple smoothing filter

Blurs the image to remove noise

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Other Means

There are different kinds of mean filters all of which exhibit slightly different behaviour:

– Geometric Mean– Harmonic Mean– Contraharmonic Mean

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Other Means (cont…)

There are other variants on the mean which can give different performance

Geometric Mean:

Achieves similar smoothing to the arithmetic mean, but tends to lose less image detail

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Mean Filters

OriginalImage

ImageCorrupted By Gaussian Noise

After A 3*3Geometric Mean Filter

After A 3*3Arithmetic

Mean Filter

Example: meanfilter.m

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Harmonic Mean:

Works well for salt noise, but fails for pepper noise

Also does well for other kinds of noise such as Gaussian noise

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Contraharmonic Mean:

Q is the order of the filter and adjusting its value changes the filter’s behaviourPositive values of Q eliminate pepper noiseNegative values of Q eliminate salt noiseQ=0 Reduced to arithmetic mean filterQ=-1Reduced to the harmonic mean filter

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Mean Filters(cont…)

ImageCorruptedBy Pepper

Noise

Result of Filtering Above

With 3*3 Contraharmonic

Q=1.5

Blurred the dark area, and cleared the background

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ImageCorruptedBy SaltNoise

Result of Filtering Above With 3*3 ContraharmonicQ=-1.5

Blurred the background, and cleared the dark area

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Choosing the wrong value for Q when using contraharmonic mean filter can have drastic results

Q=-1.5 Q=1.5

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Arithmetic and Geometric mean filters Random noise: Gaussian or uniform noise

Harmonic mean filter Salt noise, not good for pepper noise Gaussian noise

Contraharmonic mean filter Impulse noise Prior knowledge about the noise is dark or

light

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Order Statistics Filters

Spatial filters that are based on ordering the pixel values that make up the neighbourhood operated on by the filter

Useful spatial filters include– Median filter– Max and min filter– Midpoint filter– Alpha trimmed mean filter

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Median Filter

Median Filter:

Excellent at noise removal, without the smoothing effects that can occur with other smoothing filters

Particularly good when salt and pepper noise is present

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ImageCorrupted

By Salt AndPepper Noise

Pa=Pb=0.1

Result of 1 Pass With A 3*3 MedianFilter

Result of 2Passes With

A 3*3 MedianFilter

Result of 3 Passes WithA 3*3 MedianFilter

Example: medianfilter1.m

Median Filter

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Note:It is a good example of the power of median filtering in

handling impulse additive noise.Repeat passes of a median filter tend to blur the image.It is desirable to keep the number of passes as low as

possible.

Median Filter

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Max and Min Filter

Max Filter:

Min Filter:

Max filter is good for pepper noise and min is good for salt noise

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Noise Removal Examples (cont…)

ImageCorruptedBy Pepper

Noise

ImageCorruptedBy SaltNoise

Result Of Filtering Above With A 3*3 Min Filter

Result OfFiltering

AboveWith A 3*3Max Filter

Example: maxminfilter.m

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– For max filter, it will remove pepper noise, but it also removed some dark pixels from the borders of dark objects.

– The same as min filter, it will remove some white points around the border of light objects.


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Midpoint Filter

Midpoint Filter:

Good for Gaussian and random uniform noise

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Alpha-Trimmed Mean Filter

Alpha-Trimmed Mean Filter:

We can delete the d/2 lowest and d/2 highest grey levels. So gr(s, t) represents the remaining mn – d pixels

d=0: the arithmetic mean filterd=(mn-1)/2: Median filterOther d: Multiple types of noise: Good for the combination of salt-and-pepper and Gaussian noise

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ImageCorrupted

By UniformNoise

Image FurtherCorruptedBy Salt andPepper Noise

Filtered By5*5 Arithmetic

Mean Filter

Filtered By5*5 Median

Filter

Filtered By5*5 GeometricMean Filter

Filtered By5*5 Alpha-TrimmedMean Filter

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Adaptive Filters

The filters discussed so far are applied to an entire image without any regard for how image characteristics vary from one point to anotherThe behaviour of adaptive filters changes depending on the characteristics of the image inside the filter region

An increase in filter complexityWe will take a look at the adaptive median filter

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Adaptive Median Filtering

The median filter performs relatively well on impulse noise as long as the spatial density of the impulse noise is not large (Pa and Pb are less than 0.2)

The adaptive median filter can handle much more spatially dense impulse noise, and also performs some smoothing for non-impulse noise

An additional benefit of the adaptive median filter is that it seeks to preserve detail while smoothing nonimpulse noise.

The key insight in the adaptive median filter is that the filter size changes depending on the characteristics of the image

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Adaptive Median Filtering (cont…)

Remember that filtering looks at each original pixel image in turn and generates a new filtered pixel

First examine the following notation:

– zmin = minimum grey level in Sxy

– zmax = maximum grey level in Sxy

– zmed = median of grey levels in Sxy

– zxy = grey level at coordinates (x, y)

– Smax =maximum allowed size of Sxy

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Level A: A1 = zmed – zmin

A2 = zmed – zmax

If A1 > 0 and A2 < 0, Go to level BElse increase the window size

If window size ≤ repeat Smax level A

Else output zmed

Level B: B1 = zxy – zmin

B2 = zxy – zmax

If B1 > 0 and B2 < 0, output zxy

Else output zmed

Zmed is an impulse?

Zxy is an impulse?

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The key to understanding the algorithm is to remember that the adaptive median filter has three purposes:

– Remove impulse noise– Provide smoothing of other noise– Reduce distortion, by not changing these

intermediate-level points. (excessive thinning or thickening of object boundaries)

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Adaptive Filtering Example

Image corrupted by salt and pepper noise with

probabilities Pa = Pb=0.25

Result of filtering with a 7 * 7 median filter

Result of adaptive median filtering with smax

= 7

Choice of maximum allowed window size depends on the application. It should be increased as the density of the impulse increases.

A reasonable value can be estimated by experiment with various sizes of standard median filter.

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Summary

In this lecture we looked at image restoration for noise removal

– Restoration is slightly more objective than enhancement

– Spatial domain techniques are particularly useful for removing random noise

digitalimageprocessing11-imagerestoration1

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