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Inverse Filtering

Inverse filtering is a technique used in image restoration to recover an original image from a degraded version by applying a known degradation function. The process involves minimizing an error function to estimate the original image, with the relationship f̂ = H⁻¹g being central to the method. While it offers advantages like requiring only the blur point spread function and achieving perfect reconstruction without noise, it has drawbacks such as difficulties in obtaining an inverse and potential noise amplification.

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207 views2 pages

Inverse Filtering

Inverse filtering is a technique used in image restoration to recover an original image from a degraded version by applying a known degradation function. The process involves minimizing an error function to estimate the original image, with the relationship f̂ = H⁻¹g being central to the method. While it offers advantages like requiring only the blur point spread function and achieving perfect reconstruction without noise, it has drawbacks such as difficulties in obtaining an inverse and potential noise amplification.

Uploaded by

adrijazen
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INVERSE FILTERING

Image restoration is to restore a degraded image back to the original image

A model of the image degradation/restoration process

Where,
f(x,y) - input image
f^(x,y) - estimated original image
g(x,y) - degraded image
h(x,y) - degradation function
h(x,y) - additive noise term

The degraded image is given by

g(x,y)=f(x,y)*h(x,y)+h(x,y) – Spatial domain

OR

g=Hf+h ( coordinates are ignored)

The error function or noise in the image is given by

h=Hf-g

Usually H is a degradation function which is known ( from different mathematical modelling


and knowledge of channels etc). But noise is a parameter of which there is no knowledge. Hence
it is desirable to use 𝑓" to such that 𝐻𝑓" approximates g in the least square sense.

The error function thus becomes

*
𝐽%𝑓" & = ‖𝜂‖* = +(𝑔 − 𝐻𝑓" )+

Where

‖𝜂‖* is the norm and is given by ‖𝜂‖* = 𝜂1 𝜂

* 1
Thus, +(𝑔 − 𝐻𝑓"+ = %𝑔 − 𝐻𝑓" & . (𝑔 − 𝐻𝑓")
*
Hence, 𝐽%𝑓"& = ‖𝜂‖* = +(𝑔 − 𝐻𝑓" )+ = 𝑔* + 𝐻* 𝑓" * − 2𝑔𝐻𝑓"

To find the minimum of 𝐽%𝑓"& , the above equation is differentiated wrt 5𝑓 and equating it to
zero.

𝜕𝐽%𝑓"&
= 0 + 2𝐻* 𝑓" − 2𝑔𝐻 = −2𝐻(𝑔 − 𝐻𝑓")
𝜕𝑓"

Now,

𝜕𝐽%𝑓" &
= 0 ⟹ −2𝐻%𝑔 − 𝐻𝑓" & = 0
𝜕𝑓"

Therefore,

%𝑔 − 𝐻𝑓"& = 0 ⟹ 𝑓" = 𝐻9: 𝑔

The above relation 𝑓" = 𝐻9: 𝑔 is used by an inverse filter for implementing a simple
degradation.

Taking the Fourier Transform of both sides,

𝐺(𝑢, 𝑣)
𝐹< (𝑢, 𝑣) =
𝐻(𝑢, 𝑣)

Usually H(u,v) is known . It is a low pass filter and the inverse filter can be designed by taking
inverse of low pass filter i.e a high pass filter.

But if any element of the LP filter is zero, the inverse may result into infinity.

:
Thus, A(B,C) is designed in such a way that

1
𝑖𝑓 𝑢* + 𝑣 * ≤ 𝜔I *
𝐻(𝑢, 𝑣) = D𝐻(𝑢, 𝑣)
1 𝑓 𝑢* + 𝑣 * > 𝜔I *

Where 𝜔I is the cut off frequency

Advantages:

• It requires only blur PSF


• It gives perfect reconstruction in the absence of noise

Drawbacks:

• It is not always possible to obtain an inverse . For inverse to exist the matrix must be
non-singular .
• If noise is present, inverse filter amplifies noise. (better option is wiener filter)

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