Inverse and Pseudoinverse Filtering
The Problem
Given a known image function, f(x,y), and a blurring function, h(x,y),
we need to recover f(x,y) from the convolution
g(x,y)=f(x,y)*h(x,y).
(Note: * is used to represent convolution and x is used to represent
multiplication).
A Possible Solution: Inverse Filtering
In the frequency domain we have: G(u,v)=F(u,v) x
H(u,v)
- F is the FFT2 of the image matrix, f.
- H is the FFT2 of the complex distoring function, h.
- G is the FFT2 of the "observed" image, g
The novice solution to this problem is to filter by dividing the
2-dimensional fft of the blurred function by the reciprocal of H(u,v):
G(u,v)/H(u,v)=F(u,v) x H(u,v)/H(u,v) = F(u,v).
This is commonly reffered to as the inverse filtering method where 1/H(u,v)
is the inverse filter.
Difficulties with Inverse Filtering
The first problem in this formulation is
that 1/H(u,v) does not necessairily exist. If H(u,v)=0 or is close to zero, it
may not be computationally possible to compute 1/H(u,v).
If there are few values
of H(u,v) which are close to zero then the ideal inverse filter can be
approximated with a stabilized version of 1/H(u,v) given by :
Fapprox(u,v)=G(u,v) x Hinv(u,v)
where
Hinv(u,v)= 1/H(u,v) if |H(u,v)| > threshold value
= 0 otherwise
This works well if few elements of H have a magnitude below the threshold
but if two many elements are lost, the frequency content of Fapprox will be
much lower than F(u,v) and the image will appear distorted.
Here are some examples of how varying the threshold value effects the
inverse filtering of an image:
This is the original image:
Image with a 31-Point
horizontal blur.
Pseudoinverse filtered image
with 512 out of 65536 values of Hinv=0. Note the "ripples".
Pseudoinverse filtered image
with 28682 out of 65536 values of Hinv=0. The image
now appears to have horizontal "ghosts" but still provides more
information than the blurred image.
Pseudoinverse filtered image
with 59648 out of 65536 values of Hinv=0.The image is
no longer intelligible.
More Difficulties with Inverse Filtering
Another problem with inverse filtering is that it dosn't perform well
when used on noisy images.
We revise the "observed image" model by including an error term:
gn(x,y)=g(x,y)+n(x,y)
the corresponding 2 dimensional FFT representation would be:
Gn(u,v)=G(u,v) + N(u,v).
Inverse filtering Gn:
Fn(u,v)=Gn(u,v)/H(u,v)=F(u,v)+N(u,v)/H(u,v).
Since N(u,v) can remain large when H(u,v) gets small, it is possible that
the noise will dominate Fn and consequently also fn. In such instances,
it may be necessary to use a different type of filter
implementation.
The following images compare the results of the restoration of a blurred
image with the addition of normally distributed random noise with a mean of 0.0 and a
variance of 1.
Blurred image with added noise
Image of noise was added to the image.
Results of pseudoinverse filtering with threshold
value set to minimize frequency content losses.
Results using a Wiener filter implementation.
More detail of the original image can be determined from the processed
image using the Wiener filter implementation. The Wiener filter attempts to
minimize the error due to noise.
Topics
Refrences
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