Directio-Pyramidal Decomposition

In multidimensional signals a directional decomposition is equally im- portant as a scale decomposition. Directional decompositions require suitable directional fi lters. Ideally, all directional components should add up to the complete image. A combined decomposition of an image into a pyramid and on each pyramid level into directional components is known as a directiopyramidal decomposition [78]. Generally, such a de- composition is a diffi cult fi lter design problem. Therefore, we illustrate a directiopyramidal decomposition here only with a simple and effi cient decomposition scheme with two directional components.

142                                                                              5 Multiscale Representation

 

B

B

The smoothing is performed by separable smoothing fi lters, one fi lter that smoothes only in the x direction ( _x) and one that smoothes only in the y direction ( _y): then the next higher level of the Gaussian pyramid is given as in Eq. (5.47) by

G (q+1) =↓ ₂ B_xB_y G (q).                                            (5.51)

The Laplacian pyramid is

L (q) = G (q)− ↑ ₂ G (q+1).                                           (5.52)

Then, the two directional components are given by

 

L

L

(q) x (q)

y

= 1/2( G (q)− ↑ ₂ G (q+1) − (B_x

= 1/2( G (q)− ↑ ₂ G (q+1) + (B_x

− B_y) G (q)),

− B_y) G (q)).

 

(5.53)

 

= +

From Eq. (5.53) it is evident that the two directional components L _x and L _y add up to the isotropic Laplacian pyramid: L L _x L _y. Example im- ages with the fi rst three levels of a directional decomposition are shown in Fig. 5.7.

 

5.4 Further Readings‡

 

Multiresolutional image processing developed in the early 1980ies. An excellent overview of this early work is given by Rosenfeld [156]. Linear scale spaces are described in detail by the monograph of Lindeberg [113], nonlinear scale spaces including inhomogeneous and anisotropic diff usion by Weickert [196]. Readers interested in the actual development of scale space theory are referred to the proceedings of the international conferences on “Scale-Space”: 1997 [180], 1999 [131], and 2001 [95].

 

 

Part II

Image Formation and Preprocessing

 

 

 

Quantitative Visualization

Introduction

An imaging system collects radiation emitted by objects to make them visible. The radiation consists of a fl ow of particles or electromagnetic or acoustic waves. In classical computer vision scenes and illumination are taken and analyzed as they are given, but visual systems used in scientifi c and industrial applications require a diff erent approach. There, the fi rst task is to establish the quantitative relation between the object feature of interest and the emitted radiation. It is the aim of these eff orts to map the object feature of interest with minimum possible distortion of the collected radiance by other parameters.

Figure 6.1 illustrates that both the incident ray and the ray emit- ted by the object towards the camera may be infl uenced by additional processes. The position of the object can be shifted by refraction of the emitted ray. Scattering and absorption of the incident and emit- ted rays lead to an attenuation of the radiant fl ux that is not caused by the observed object itself but by the environment, which thus falsifi es the observation. In a proper setup it is important to ensure that these additional infl uences are minimized and that the received radiation is directly related to the object feature of interest. In cases where we do not have any infl uence on the illumination or setup, we can still choose radiation of the most appropriate type and wavelength range.

As illustrated in Sections 1.2 and 6.4, a wealth of phenomena is avail- able for imaging objects and object features, including self-emission, induced emission (fl uorescence), refl ection, refraction, absorption, and scattering of radiation. These eff ects depend on the optical properties of the object material and on the surface structure of the object. Basically, we can distinguish between surface-related eff ects caused by disconti- nuity of optical properties at the surface of objects and volume-related eff ects.

It is obvious that the complexity of the procedures for quantitative visualization strongly depends on the image processing task. If our goal is only to make a precise geometrical measurement of the objects, it is suffi cient to set up an illumination in which the objects are uniformly illuminated and clearly distinguished from the background. In this case, it is not required that we establish quantitative relations between the ob- ject features of interest and the radiation emitted towards the camera.

145

B. Jä hne, Digital Image Processing                                                                                                       Copyright © 2002 by Springer-Verlag

ISBN 3–540–67754–2                                                                                                    All rights of reproduction in any form reserved.

146                                                                             6 Quantitative Visualization

 

Object (to be observed)

by reflection, refraction, emission absorption or scattering

 

 

Scattering

 

 

Incident ray

(from light source)

 

Absorption Refraction

 

Refraction Absorption

 

 

Scattering

Emitted ray (towards camera)

 

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