Chapter 3: Image Restoration

Introduction

Introduction

• Image restoration methods are used to improve the appearance of an image by applying a restoration process that uses a mathematical model for image degradation.

Introduction

• Examples of the types of degradation:– Blurring caused by motion or atmospheric disturbance.– Geometric distortion caused by imperfect lenses. – Superimposed interference patterns caused by

mechanical systems.– Noise from electronic sources.

• It is assumed that the degradation model is known or can be estimated.

• The idea is to model the degradation process and then apply the inverse process to restore the original image.

Introduction

• Let us look at an over-simplified example. Assume that during the modelling stage, it is determined that the image is corrupted by additive noise defined as follows:

d(r,c) = I(r,c) + nd(r,c) : Degraded imageI(r,c) : Original imagen : Constant Noise

Introduction

• Based on this model, the degraded image could be restored back to original image by applying the inverse of the degradation process.– This is done by subtracting n from each

of the pixel in d(r,c). – I(r,c) = d(r,c) - n

Introduction

• However, in practice, the degradation model is often not known and must be experimentally determined and estimated.

• In this chapter, we will consider the various types of degradation that can be modeled and discuss the various techniques available to restore the image.

System Model

• The degradation process model consists of two parts:– The degradation function.– The noise function.

System Model

• The general model in spatial domain:d(r,c) = h(r,c) * I(r,c) + n(r,c)

– d(r,c) = degraded image– h(r,c) = degradation function – I(r,c) = original image– n(r,c) = additive noise function– * denotes convolution

System Model

• Because convolution in the spatial domain is equivalent to multiplication in the frequency domain, the frequency domain model is:

D(u,v) = H(u,v)I(u,v) + N(u,v)– D(u,v) : Fourier transform of degraded image– H(u,v) : Fourier transform of degradation

function– I(u,v) = Fourier transform of the original image– N(u,v) = Fourier transform of the additive noise

function.

System Model

• Based on the definition of image restoration and the general model, in order to perform image restoration, we need to find:– h(r,c) or H(u,v)– n(r,c) or N(u,v)

Noise

• Noise can be defined as any undesired information that contaminates an image.

• Noise appears in images from a variety of sources.

• The digital image acquisition process, which converts an optical image into continuous electrical signal that is then sampled, is the primary process by which noise appears in digital image.

Noise

• At every step in the process, there are fluctuations caused by natural phenomena that add a random value to the exact brightness value for a given pixel.

• In typical images, the noise can be modeled using either a gaussian (“normal”), uniform, or salt-and-pepper (“impulse”) distribution.

• The shape of the distribution of these noise types can be modeled as a histogram.

Noise

• Gaussian distribution:

– g = gray level– m = mean (average)– σ = standard deviation (σ2 = variance)

2

2

2

)(

2Gaussian2

1)(HISTOGRAM

mg

egP

Noise

Noise

• Uniform distribution:

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2

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Uniform

abiance

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elsewhere

bgaforabgP

Noise

Noise

• Salt-and-pepper distribution:

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)"(")(HISTOGRAM Pepper &Salt salt

pepper

bg

ag

for

for

B

AgP

Noise

Noise

OriginalImage

Noise

Image with Gaussiannoise added

Noise

Image with uniformnoise added

Noise

Image withsalt-and-peppernoise added

Noise

• Rayleigh distribution:

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)4(var

4

2)(HISTOGRAM

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Rayleigh

iance

mean

eg

gPg

Noise

Noise

• Negative exponential distribution:

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lExponentia Negative

var

)(HISTOGRAM

iance

egP

g

Noise

Noise

• Gamma distribution:

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1

Gamma

var

)!1()(HISTOGRAM

aiance

ea

ggP a

g

Noise

Noise

Noise sources Typical Statistic Model

Natural noise process such as electronic noise in image acquisition system.

Gaussian

Malfunctioning pixels elements in camera sensors, faulty memory locations or timing errors in digitization process.

Salt & Pepper

Radar range and velocity images. Rayleigh

Laser-based images. Negative Exponential

Lowpass filtered image with noise of Negative Exponential distribution.

Gamma

Noise

• There are various approaches that can be used to determine the type of noise that has corrupted as image.

• Ideally, we want to find an image that contains only noise, and then we can use its histogram for the noise model.– If we have a system that generates a noisy

image, try to take picture of a blank wall.

Noise

• If we cannot find "noise-only" images or access to the actual system, we may start from the sample of degraded image.

• A portion of image where we know what to expect in the histogram is selected (a constant-value image or a well-defined line).

• We can then subtract the known values from the histogram and what is left is our noise distribution.

Noise

• We can then compare this noise model to the ones available and select the best match.

• In order to develop a valid model with any of the above approaches, many sample images need to be evaluated.