Iterative Inverse Filtering

H

Iterative techniques form an interesting variant of inverse fi ltering as they give control over the degree of reconstruction to be applied. Let

H = I − H

be the blurring operator. We introduce the new operator                         '              .

Then the inverse operator

H =

− 1          I I− H '

can be approximated by the Taylor expansion

 

(17.90)

H − 1 =I + H ' +H '2 +H '3 +...,                   (17.91)

or, written explicitly for the OTF in the continuous Fourier domain,

hˆ − 1( k ) = 1 + hˆ ' + hˆ '2 + hˆ '3 +....                                     (17.92)

17.8 Inverse Filtering                                                                    479

 

A                              b                              c                              d

E                              f                               g                              h

Figure 17.16: 3-D reconstruction of a focus series of a cell nucleus taken with conventional microscopy. Upper row: a – c selected original images; d xz cross section perpendicular to the image plane. Lower row: e – h reconstructions of the images a – d ; courtesy of Dr. Schmitt and Prof. Dr. Komitowski, German Cancer Research Center, Heidelberg.

 

In order to understand how the iteration works, we consider periodic structures. First, we take one that is only slightly attenuated. This means that hˆ  is only slightly less than one.  Thus, hˆ ' is small and the iteration converges rapidly.

The other extreme is when the periodic structure has nearly vanished.

Then, hˆ ' is close to one.  Consequently, the amplitude of the periodic structure increases by the same amount with each iteration step (linear convergence). This procedure has the signifi cant advantage that we can stop the iteration as soon as the noise patterns become noticeable.

A direct application of the iteration makes not much sense because the increasing exponents of the convolution masks become larger and thus the computational eff ort increases from step to step. A more effi - cient scheme known as Van Cittert iteration utilizes Horner’s scheme for polynomial computation:

G ₀ = G ',     G _k+1 = G ' + ( I − H ) ∗ G _k.                             (17.93)

In Fourier space, it is easy to examine the convergence of this iteration. From Eq. (17.93)

.

k

gˆ _k( k ) = gˆ ^'( k )       (1 − hˆ ( k ))i.                             (17.94)

i=0

480                                                                     17 Regularization and Modeling

 

= −                                              | |= | − |

=

This equation constitutes a geometric series with the start value a₀ gˆ ' and the factor q 1 hˆ. The series converges only if q  1 hˆ < 1. Then the sum is given by

 

gˆ ( k ) = a

1 − qk = gˆ '( k ) 1 − |1 − hˆ ( k )|k

 

                                              

 

(17.95)

k                   0 1 − q

hˆ ( k )

and converges to the correct value gˆ '/hˆ.  Unfortunately, this condition for convergence is not met for all transfer functions that have negative values. Therefore the Van Cittert iteration cannot be applied to motion blurring and to defocusing.

A slight modifi cation of the iteration process, however, makes it pos- sible to use it also for degradations with partially negative transfer func- tions. The simple trick is to apply the transfer function twice. The trans-

∗

fer function hˆ 2 of the cascaded fi lter H                      H is positive.

The modifi ed iteration scheme is

G ₀ = H ∗ G ',     G _k+1 = H ∗ G ' + ( I − H ∗ H ) ∗ G _k.                    (17.96)

=            = −

With a₀ hˆ gˆ ' and q 1 hˆ 2 the iteration again converges to the correct value

 

lim gˆ ( k )

lim hˆ gˆ ' 1 − |1 − hˆ 2|k

 

gˆ ' ,   if

 

 

1 hˆ 2

 

< 1    (17.97)

k

k→ ∞

= k→ ∞

hˆ 2                =  hˆ

|  − |

 

17.9 Further Readings‡

 

This subject of this chapter relies heavily on matrix algebra. Golub and van Loan

[54] give an excellent survey on matrix computations. Variational methods (Sec- tion 17.3) are expounded by Jä hne et al. [83, Vol. 2, Chapter 16] and Schnö rr and Weickert [164]. The usage of the membran model (Section 17.3.5) was fi rst re- ported by Broit [12], who applied it in computer tomography. Later it was used and extended by Dengler [29] for image sequence processing. Nowadays, elas- ticity models are a widely used tool in quite diff erent areas of image processing such as modeling and tracking of edges [91], reconstruction of 3-Dobjects [183] and reconstruction of surfaces [182]. Anisotropic diff usion (Section 17.5) and nonlinear scale spaces are an ongoing research topic. An excellent account is given by Weickert [195] and Jä hne et al. [83, Vol. 2, Chapter 15]. Optimal fi lters for fast anisotropic diff usion are discussed by Scharr and Weickert [162] and Scharr and Uttenweiler [161].

 

 

Morphology

Introduction

In Chapters 16 and 17 we discussed the segmentation process that ex- tracts objects from images, i. e., identifi es which pixels belong to which objects. Now we can perform the next step and analyze the shape of the objects. In this chapter, we discuss a class of neighborhood operations on binary images, the morphological operators that modify and analyze the form of objects.

 

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