This document summarizes digital image processing techniques including algebraic approaches to image restoration and inverse filtering. It discusses: 1) Unconstrained and constrained restoration, with unconstrained having no knowledge of noise and constrained using knowledge of noise. 2) Inverse filtering which is a direct method that minimizes error between degraded and original images using matrix operations, but can be unstable due to noise or near-zero filter values. 3) Pseudo-inverse filtering which adds a threshold to the inverse filter to avoid instability, working better for noisy images by not amplifying high frequency noise.

1. Digital Image Processing
Algebraic Approach to Image
Restoration
&
Inverse Filtering
Submitted by,
M. Kavitha,
II – M.Sc(CS&IT),
Nadar Saraswathi College of
Arts & Science, Theni.

2. Algebraic Approach to Restoration
1. Unconstrained Restoration :
It is an absence of any knowledge about n
noise term in degradation model.
n = g – Hf
Euclidean Norm
|| || = || f || f
by definition square norm,

4. 2. Constrained Restoration :

6. Inverse Filtering for Image Restoration
* Inverse filtering is a deterministic and direct method
for image restoration.
* The images involved must be lexicographically
ordered. That means that an image is converted to a column
vector by pasting the rows one by one after converting them to
columns.
* An image of size 256×256 is converted to a column
vector of size 65536×1.
* The degradation model is written in a matrix form,
where the images are vectors and the degradation process is a
huge but sparse matrix 𝐠=𝐇𝐟.
* The above relationship is ideal. What really happens is
𝐠=𝐇𝐟+𝐧.

7. * In this problem we know 𝐇 and 𝐠 and we are looking
for a descent 𝐟. The problem is formulated as follows:
To minimize the Euclidian norm of the error.
* The first derivative of the minimization function must
be set to zero.
* If 𝐇 is a square matrix and its inverse exists then
𝐟= 𝐇−𝟏 𝐠

8. * We have that
𝐇 𝑻 𝐇𝐟=𝐇 𝑻 𝐠
* If we take the DFT of the above relationship in both
sides we have
* Note that the most popular types of degradations are
low pass filters (out-of-focus blur, motion blur).

9. Inverse Filtering for noise - free scenarios :
* We have that 𝐹𝑢,
𝑣= 𝐺(𝑢,𝑣)/𝐻(𝑢,𝑣)
* Problem: It is very likely that 𝐻(𝑢,𝑣) is 0 or very small
at certain frequency pairs.
* For example,
𝐻(𝑢,𝑣) could be a 𝑠𝑖𝑛𝑐 function.
* In general, since 𝐻(𝑢,𝑣) is a low pass filter, it is very
likely that its values drop off rapidly as the distance of (𝑢,𝑣) from
the origin (0,0) increases.

10. Pseudo – Inverse Filtering :
* Instead of the conventional inverse filter, we
implement one of the following :
* The parameter 𝜖 (called threshold in the figures in
the next slides) is a small number chosen by the user.
* This filter is called pseudo-inverse or generalized
inverse filter.

11. Pseudo – Inverse Filtering with different Thresholds :

12. Pseudo-inverse filtering in the case of noise :

15. Thank You