Microscope Image Processing

6.3.2 Deconvolution Using a Least-

Squares Approach

The observed image, g, can be expressed in a matrix form as

g ¼ Hf þ n (6:22)

where g, f, and n are N

2

 1 column vectors, H is an N

2

 N

2

matrix, f is the

original image, n is the noise, and H stands for blurring. When the blurring is

shift-invariant, the matrix H becomes a block-circulant matrix. If n ¼ 0, we can

find the approximate solution by minimizing the mean square error

e(

ˆ

f ) ¼kg  H

ˆ

f k

2

¼ g  H

ˆ

f



t

g  H

ˆ

f



(6:23)

by setting the derivative of e

ˆ

f



in respect to f

ˆ

to zero:

@e

ˆ

f



@

ˆ

f

¼2H

t

g  H

ˆ

f



¼ 0(6:24)

The solution for f

ˆ

becomes

ˆ

f ¼ H

t

H

ðÞ

1

H

t

g ¼ H

1

g (6:25)

If n is nonzero, the problem can