Survey of Image Distortions

Given the enormous variety of ways to form images (Chapter 7), there are many reasons for image degradation. Imperfections of the optical sys- tem, known as lens aberrations, limit the sharpness of images. However, even with a perfect optical system, the sharpness is limited by diff rac- tion of electromagnetic waves at the aperture stop of the lens. While these types of degradation are an inherent property of a given optical system, blurring by defocusing is a common misadjustment that lim- its the sharpness in images. Further reasons for blurring in images are unwanted motions and vibrations of the camera system during the ex- posure time. Especially systems with a narrow fi eld of view (telelenses) are very sensitive to this kind of image degradation. Blurring can also occur when objects move more than a pixel at the image plane during the exposure time.

Defocusing and lens aberrations are discussed together in this section as they are directly related to the optical system. The eff ect of blurring or aberration is expressed by the point spread function h( x ) and the op- tical transfer function (OTF ); see Section 7.6. Thus, the relation between object g( x ) and image g'( x ) is in the spatial and Fourier domain

g'( x ) = (h ∗ g)( x )       ◦ • gˆ ^'( k ) = hˆ ( k )gˆ ( k ).                     (17.81)

17.8 Inverse Filtering                                                                     475

b

a

1

0.8

0.6                                  1

0.4                     2

0.2     8 4

0

-0.2

1

k

0    0.2  0.4  0.6  0.8 ~

 

Figure 17.14: a Transfer functions for disk-shaped blurring. The parameters for the diff erent curves are the radius of the blur disk; b defocused image of the ring test pattern.

 

Lens aberrations are generally more diffi cult to handle. Most aberra- tions increase strongly with distance from the optical axis and are, thus, not shift invariant and cannot be described with a position-independent PSF. However, the aberrations change only slowly and continuously with the position in the image. As long as the resulting blurring is limited to an area in which we can consider the aberration to be constant, we can still treat them with the theory of linear shift-invariant systems. The only diff erence is that the PSF and OTF vary gradually with position.

If defocusing is the dominant blurring eff ect, the PSF has the shape of the aperture stop. As most aperture stops can be approximated by a circle, the function is a disk. The Fourier transform of a disk with radius r is a Bessel function of the form (± R5):

  1  Π  .| x | Σ  ◦ • 2J₁( | k |r)

 

 

(17.82)

π r 2           2r

.

| k |r

This Bessel function, as shown in Fig. 17.14a, has a series of zeroes and, thus, completely eliminates certain wave numbers. This eff ect can be observed in Fig. 17.14b, which shows a defocused image of the ring test pattern.

While blurring by defocusing and lens aberrations tend to be isotropic, blurring eff ects by motion are one-dimensional, as shown in Fig. 17.15b. In the simplest case, motion is constant during the exposure. Then, the PSF of motion blur is a one-dimensional box function. Without loss of generality, we fi rst assume that the direction of motion is along the x

476                                                                     17 Regularization and Modeling

 

A                                                                    b

Figure 17.15: Simulation of blurring by motion using the ring test pattern:

a small and b large velocity blurring in horizontal direction.

 

axis. Then,

2u∆ t

2u∆ t

ku∆ t/2

h_Bl(x) =    1   Π .   x   Σ  ◦      • hˆ Bl(k) = sin(ku ∆ t/2),                  (17.83)

 

=

where u is the magnitude of the velocity and ∆ t the exposure time. The blur length is ∆ x       u∆ t.

If the velocity u is oriented in another direction, Eq. (17.83) can be generalized to

h_Bl( x ) =     1  Π .    xu ¯     Σ  δ ( ux ) ◦ • hˆ Bl( k ) = sin ( ku ∆ t / 2 ),

2| u |∆ t

1| u |∆ t

ku ∆ t/2

(17.84)

where u ¯ = u /| u | is a unit vector in the direction of the motion blur.

Deconvolution

H

H

Common to defocusing, motion blur, and 3-D imaging by such tech- niques as focus series or confocal microscopy (Section 8.2.4) is that the object function g( x ) is convolved by a point spread function. Therefore, the principal procedure for reconstructing or restoring the object func- tion is the same. Essentially, it is a deconvolution or an inverse fi ltering as the eff ect of the convolution by the PSF is to be inverted. Given the simple relations in Eq. (17.81), inverse fi ltering is in principle an easy procedure. The eff ect of the convolution operator is reversed by the application of the inverse operator − 1. In the Fourier space we can write:

ˆ '

Hˆ  '

G ˆ R = G   = H ˆ  − 1 · G ˆ '.                                             (17.85)

The reconstructed image G _R is then given by applying the inverse Fourier transform:

G _R = F− 1 H ˆ  − 1 · F G ˆ '.                                            (17.86)

17.8 Inverse Filtering                                                                     477

 

F

The reconstruction procedure is as follows. The Fourier transformed image,   G ', is multiplied by the inverse of the OTF, H ˆ  − 1, and then trans-

formed back to the spatial domain. The inverse fi ltering can also be per- formed in the spatial domain by convolution with a mask that is given by the inverse Fourier transform of the inverse OTF:

G _R = (F− 1 H ˆ  − 1) ∗ G '.                                           (17.87)

At fi rst glance, inverse fi ltering appears straightforward. In most cases, however, it is useless or even impossible to apply Eqs. (17.86) and (17.87). The reason for the failure is related to the fact that the OTF is often zero in wide ranges. The OTFs for motion blur (Eq. (17.84)) and defocusing (Eq. (17.82)) have extended zero range. In these areas, the inverse OTF becomes infi nite.

Not only the zeroes of the OTF cause problems; already all the ranges in which the OTF becomes small do so. This eff ect is related to the infl uence of noise. For a quantitative analysis, we assume the following simple image formation model:

G ' = H ∗ G + N          ◦ •

G ˆ ' = H ˆ   · G ˆ   + N ˆ

(17.88)

 

Equation (17.88) states that the noise is added to the image after the image is degraded. With this model, according to Eq. (17.85), inverse fi ltering yields

G ˆ R = H ˆ − 1 · G ˆ ' = G ˆ   + H ˆ − 1 · N ˆ

(17.89)

provided that H ˆ   ≠ 0. This equation states that the restored image is the restored original image G ˆ   plus the noise amplifi ed by H ˆ − 1.

If H ˆ   tends to zero, H ˆ − 1 becomes infi nite, and so does the noise level.

Equations (17.88) and (17.89) also state that the signal to noise ratio is not improved at all but remains the same because the noise and the useful image content in the image are multiplied by the same factor.

From this basic fact we can conclude that inverse fi ltering does not improve the image quality at all. More generally, it is clear that no linear technique will do so. All we can do with linear techniques is to amplify the structures attenuated by the degradation up to the point where the noise level still does not reach a critical level.

As an example, we discuss the 3-Dreconstruction from microscopic focus series. A focus series is an image stack of microscopic images in which we scan the focused depth. Because of the limited depth of fi eld (Section 7.4.3), only objects in a thin plane are imaged sharply. There- fore, we obtain a 3-Dimage. However, it is distorted by the point spread function of optical imaging. Certain structures are completely fi ltered out and blurred objects are superimposed over sharply imaged objects. We can now use inverse fi ltering to try to limit these distortions.

478                                                                     17 Regularization and Modeling

 

It is obvious that an exact knowledge of the PSF is essential for a good reconstruction. In Section 7.6.1, we computed the 3-D PSF of optical imaging neglecting lens errors and resolution limitation due to diff rac- tion. However, high magnifi cation microscopy images are diff raction- limited.

The diff raction-limited 3-D PSF was computed by Erhardt et al. [35]. The resolution limit basically changes the double cone of the 3-D PSF (Fig. 7.13) only close to the focal plane. At the focal plane, a point is no longer imaged to a point but to a diff raction disk. As a result, the OTF drops off to higher wave numbers in the k_xk_y plane. To a fi rst approximation, we can regard the diff raction-limited resolution as an additional lowpass fi lter by which the OTF is multiplied for geometrical imaging and by which the PSF is convolved.

The simplest approach to obtain an optimal reconstruction is to limit application of the inverse OTF to the wave number components that are not damped below a critical threshold. This threshold depends on the noise in the images. In this way, the true inverse OTF is replaced by an eff ective inverse OTF which approaches zero again in the wave number regions that cannot be reconstructed.

× ×

The result of such a reconstruction procedure is shown in Fig. 17.16. A 64 64 64 focus series has been taken of the nucleus of a cancer- ous rat liver cell. The resolution in all directions is 0.22 µm. The images clearly verify the theoretical considerations. The reconstruction consid- erably improves the resolution in the xy image plane, while the resolu- tion in the z direction — as expected — is clearly worse. Structures that change in the z direction are completely eliminated in the focus series by convolution with the PSF of optical images and therefore can not be reconstructed.

 

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