Chapter 3: Image Restoration Introduction. Image restoration methods are used to improve the...
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Transcript of Chapter 3: Image Restoration Introduction. Image restoration methods are used to improve the...
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:
12
)(var
2
0
1)(HISTOGRAM
2
Uniform
abiance
bamean
elsewhere
bgaforabgP
Noise
Noise
• Salt-and-pepper distribution:
)"("
)"(")(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:
4
)4(var
4
2)(HISTOGRAM
2
Rayleigh
iance
mean
eg
gPg
Noise
Noise
• Negative exponential distribution:
2
lExponentia Negative
var
)(HISTOGRAM
iance
egP
g
Noise
Noise
• Gamma distribution:
2
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.