Image Restoration

Image Restoration (Sections 5.1, 5.5, 5.7, 5.9) CS474/674 – Prof. Bebis

Image Restoration • Image Enhancement – A subjective process using mostly heuristics (i.e., takes advantage of the psychophysical aspects of the human visual system)

• Image Restoration – An objective process assuming a-priori knowledge of the degradation process.

blur due to motion

Modeling Image Degradation • Degradation process can be modeled through a degradation function H and additive noise η(x,y).

Goal of Image Restoration • Given some knowledge of H and noise η(x,y), the objective of restoration is to obtain an estimate of the original image.

Assumptions about degradation function H • H is linear:

• H is shift invariant:

(i.e., shifting the input merely shifts the output by the same amount)

Degradation Model (assuming linearity and shift invariance) • Any function f(x,y) can be written as follows:

• Then, H[f(x,y)] is:

f(α,b)H[δ(x-α,y-b)dαdb

Degradation Model - Continuous Case (cont’d) • Since H(x,y) is shift invariant: if impulse response

• Therefore

Degradation Model - Continuous Case (cont’d) • Under the assumptions of linearity and shift invariance:

or

G(u, v)  H (u, v) F (u, v)  N (u, v)

Noise Properties • Noise arises typically during image acquisition and /or transmission. • For simplicity, we will assume that: (1) noise is independent of spatial coordinates (2) there is no correlation between pixel values and values of noise components (* not true for periodic noise)

Noise Models • Most types of noise are modeled using a probability density function. • Noise model is typically chosen based on some understanding of the noise source. salt & pepper

Gaussian noise p( z ) 

1 2 

e

 ( z  z ) 2 / 2 2

• Gaussian noise arises in an image due to factors such as: (1) electronic circuit noise (2) sensor noise due to poor illumination and/or high temperature

Rayleigh noise 2 2  ( z  a )e  ( z  a ) / b z  a p( z )   b 0 za b( 4   ) z  a  b / 4  2  4

• Typically used to characterize noise phenomena in range imaging.

Gamma (Erlang) noise  a b z b 1  az  e z0 p( z )   (b  1)! 0 z0 b b 2 z   2 a a

• Typically used to characterize noise phenomena in laser imaging.

Exponential noise  ae  az z  0 p( z )   z0 0 1 1 z 2  2 a a

• Typically used to characterize noise phenomena in laser imaging.

Uniform noise   p( z )   0 ab z 2

1 a zb ba otherwise (b  a ) 2   12 2

• Least used in practice. • Useful as the basis for random number generators.

Impulse (salt and pepper) noise  Pa z  a  p ( z )   Pb z  b 0 otherwise 

• Pa=Pb, a=0, and b=255, then we salt and pepper noise. •Common in situations where quick transients (e.g., faulty switching), takes place during imaging.

Noise Models - Example noise corrupted images and their histograms

test pattern

Noise Models - Example noise corrupted images and their histograms

test pattern

Estimation of noise parameters • Estimate the parameters of the noise PDFs from small patches of reasonably constant background intensity. • For impulse noise, estimate probability of black/white pixels (i.e., choose a midgray region). (i) Estimate mean and variance (ii) Compute a and b (i.e., parameters of noise distributions)

Restoration in the presence of noise only

Restoration in the presence of noise only (cont’d) • If noise can not be estimated, filtering methods can be used to suppress noise: – Mean filters (arithmetic, geometric, harmonic etc.) – Order statistic filters (median, max and min, midpoint) – Frequency-domain filters (band-reject, band-pass, etc. )

Arithmetic/Geometric mean filters

geometric mean filters tend to preserve more details (i.e., less blurring)

mxn mask

1

1   mn fˆ ( x, y)  g ( s, t ) ˆ  f ( x, y )    g ( s , t )  mn ( s ,t )Sx , y  ( s ,t )S x , y 

Contra-Harmonic filters Q: order of filter fˆ ( x, y ) 



 g ( s, t ) 



 g ( s, t ) 

( s ,t )S x , y

( s ,t )S x , y

Q 1

Q

good for removing salt or pepper noise (but not both simultaneously) Q>0  pepper noise Q