Chapter 5 Image Restoration

Image enhancement◦ Subjective process

Image restoration◦ Objective process to recover the original

image. Involving formula or criteria that will yield an

optimal result.

5. 1 Model of Image degradation and restoration5. 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.

Spatial and frequency property of noise◦ White noise (random noise)

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. All the frequencies are corrupted by an additional

constant frequency.

◦ Periodic noise

Gaussian noise

◦ : mean; : variance

◦ 70% [(-), (+)]◦ 95 % [(-2),

(+2)]

22 2/)(

2

1)(

zezp

•Because of its tractability, Gaussian (normal) noise model is often applicable at best.

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.

Rayleigh noise

= a+(b/4)1/2

= b(4-)/4

azfor

azforeazbzp

baz

0

)(2

)(/)( 2

Erlang (gamma) noise

=b/a; =b/a2

00

0)!1()(

1

zfor

zforeb

zazp

azbb

Exponential noise

=1/a; =1/a2

• A special case Erlang noise model when b = 1.

00

0)(

zfor

zforaezp

az

Uniform noise

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

otherwise

bzaforabzp

0

1)(

Impulse noise (salt and pepper noise)

otherwise

bzfor

azfor

P

P

zp b

a

0

)(

ExampleExample

Results of adding noiseResults of adding noise

Results of adding noiseResults of adding noise

Gaussian noise: ◦ electronic circuit noise and sensor noise due to

poor illumination or high temperature. Rayleigh noise:

◦ Noise in range imaging. Erlang noise:

◦ Noise in laser imaging. Impulse noise:

◦ Quick transients take place during imaging. Uniform noise:

◦ Used in simulations.

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.

Gaussian n: Find the mean and standard deviation of

the histogram (Gaussian noise). Rayleigh, Erlang, and uniform noise:

Calculate the a and b from and . Impulse noise:

Compute the height of peaks at gray levels 0 and 255 to find Pa and Pb.

Estimation of Noise ParametersEstimation of Noise Parameters

Used when only additive noise is present.

Mean filters◦Arithmetic mean filter:

The new pixel value is resulted from averaging the pixels in the area by Sxy.

◦Geometric mean filter:

Achieves smoothing comparable to arithmetic approach, but tends to lose less image detail.

xySts

tsgmn

yxf),(

),(1

),(ˆ

mn

Sts xy

tsgyxf

/1

),(

),(),(ˆ

Median filter

Max filter: find the brightest points to reduce the pepper noise

Min filter: find the darkest point to reduce the salt noise

Midpoint filter: combining statistics and averaging.

),(),(ˆ),(

tsgmedianyxfxySts

),(max),(ˆ),(

tsgyxfxySts

),(min),(ˆ),(

tsgyxfxySts

),(min),(max

2

1),(ˆ

),(),(tsgtsgyxf

xyxy StsSts

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

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 Sxy and average the remaining mn-d pixel, denoted by gr(s, t).

where d=0 ~ mn-1

xyStsr tsg

dmnyxf

),(

),(1

),(ˆ

Filters whose behavior changes based on statistical characteristics of the image.

Two adaptive filters are considered:(1) Adaptive, local noise reduction filter.(2) Adaptive median filter.

Two parameters are considered:◦ Mean: measure of average gray level.◦ Variance: measure of average contrast.

Four measurements:(1) noisy image at (x, y ): g(x, y )(2) The variance of noise 2

(3) The local mean mL in Sxy

(4) The local variance 2L

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 2L = 2

– when the local area has similar properties with the overall image.

– Return arithmetic mean value of the pixels in Sxy.

General expression:

]),([),(),(ˆ2

2

LL

myxgyxgyxf

Adaptive, Local Noise Reduction FilterAdaptive, Local Noise Reduction Filter

Adaptive median filter can handle impulse noise with larger probability (Pa and Pb are large).

This approach changes window size during operation (according to certain criteria).

First, define the following notations:zmin=minimum gray-level value in Sxy

zmax=maximum gray-level value in Sxy

zmed=median gray-level value in Sxy

zxy= gray-level at (x,y)Smax=maximum allowed size of Sxy

The adaptive median filter algorithm works in two levels: A and B

Level A: A1=zmed-zmin A2=zmed-zmax

If A1>0 and A2<0 goto level Belse increase the window sizeIf window size Smax repeat level

Aelse output zxy

Level B: B1=zxy-zmin B2=zxy-zmax

If B1>0 AND B2<0, output zxy

Else output zmed.

Example 5.5Example 5.5