IMAGE RESTORATION

Image restoration

SUMMARY:

• Summary of frequency domain filtering

• Linear degradation model

• Linear restoration model

• Linear shift-invariant systems

• Inverse filter

• Parametric Wiener filter

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Image restoration

Process of removing or reducing image degradations whichoccur during image formation.

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Image degradation

Degrading factors:

• blurring produced by the optical system or object motion duringacquisition

• noise from electronic and optical devices.

3

Restoration

• Noise only: spatial filtering, frequency filtering

• Linear shift-invariant (LSI) degradations

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Noise models

Gaussian; Rayleigh; Gamma; Exponential; Uniform; Impulse(shot noise).

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Noise only filtering

Degraded image g(x, y) :

g(x, y) = f(x, y) + η(x, y)

• Mean filter: f̂(x, y) = 1mn

∑(s,t)∈Sxy g(s, t)

• Median filter: f̂(x, y) = median(s,t)∈Sxy g(s, t)

• Bandreject frequency domain filters

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Linear shift-invariant systems

A linear shift-invariant (LSI) system O maps an input signal f(t)to an output signal g(t), denoted as f(t) O→ g(t), satisfying:

• Linearity: if f1(t)O→ g1(t) and f2(t)

O→ g2(t), then, for arbitraryconstants a, b,

a f1(t) + b f2(t)O→ a g1(t) + b g2(t)

• Shift invariance: if f(t) O→ g(t), then for each time shift T ,

f(t− T ) O→ g(t− T )

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Convolution

• Output of a LSI system is given by the convolution of the inputwith a kernel h(t):

g(t) = (h ∗ f)(t) =∫ ∞

−∞h(t− s) f(s) ds

• Filter kernel h(t) is called the impulse response function or in2-D point spread function.

• The Fourier transform H(µ) is called the transfer function.

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Linear shift-invariant degradations

• Degraded image g(x, y) is convolution of ideal image f(x, y)by kernel h(x, y) plus noise η(x, y):

g(x, y) = (h ∗ f)(x, y) + η(x, y).

In frequency domain:

G(u, v) = H(u, v)F (u, v) +N(u, v)

• Finite images of size N ×N :

g = Hf + n

where g, f ,n are column vectors of length M × N , and H isan MN ×MN convolution matrix.

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Linear restoration model

Deconvolution: compute an estimate f̂(x, y) of the ideal imageby using a reconstruction kernel r(x, y):

f̂(x, y) = (r ∗ g)(x, y)

h

hg ^

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Constrained least squares restoration

• Solution Estimated solution f̂ should satisfy the model equation:∥∥∥g −Hf̂

∥∥∥2

= ‖n‖2

• Constraints: introducing a matrix Q such that the term∥∥∥Q f̂

∥∥∥2

is minimized.

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Minimization problem and solution

• Minimize

E(f̂) = K∥∥∥Q f̂

∥∥∥2

+

(∥∥∥g −Hf̂∥∥∥2

− ‖n‖2)

where K is a constant, called the regularization parameter,which determines the relative weight of both terms

• Least squares solution:

f̂ =(HtH+KQtQ

)−1Htg

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Inverse filter

• Put K = 0 in formula for least squares solution:

f̂ = (HtH)−1Htg = H−1g

= H−1(Hf + n) = f +H−1n

Expressed in the frequency domain:

F̂ (u, v) = F (u, v) +N(u, v)

H(u, v)

This is the ‘inverse’ filter.

• This filter is useless in practice: small values in H(u, v) amplifythe noise.

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Wiener filter

Assume f and n to be random vectors with zero mean andcovariance matrices Rf = E(f f t) for the signal and Rn = E(nnt)

for the noise. Let Q be the noise-to-signal ratio Q =(RnRf

)12

andlet K = 1.

f̂ =(HtH+R−1f Rn

)−1Htg

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Parametric Wiener filter

Minimum norm least squares solution:

f̂ =(HtH+K I

)−1Htg

or, expressed in the frequency domain:

F̂ (u, v) =[ H∗(u, v)

|H(u, v)|2 +K

]G(u, v)

This is the Parametric Wiener filter or Pseudo-inverse filter.

K is the regularization parameter.

Case K = 0: inverse filter.

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Example

(a): Original. (b): Degraded image. (c): Inverse filtering. (d): Pseudo-inversefiltering (K = 0.01).

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