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Thèse de doctorat/ PhD Thesis Citation APA:
Schockaert, C. (2007). Three dimensional object analysis and tracking by digital holography microscopy (Unpublished doctoral dissertation). Université libre de Bruxelles, Faculté des sciences appliquées – Chimie, Bruxelles.
Disponible à / Available at permalink : https://dipot.ulb.ac.be/dspace/bitstream/2013/210718/4/29df4207-a530-4460-a53c-11cee828fc96.txt
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D 03565
UNIVERSITE LIBRE DE BRUXELLES Faculté des Sciences Appliquées
THREE DIMENSIONAL OBJECT
ANALYSIS AND TRACKING BY DIGITAL HOLOGRAPHY MICROSCOPY
Kalman Tracking software DHM Oject Détection & Analysis sofb//are
Ir. Schockaert Cédric - 2006-2007
Promoteur : Prof. Frank Dubois
Service de Chimie-Physique EP CP 165/62 Avenue F. Roosevelt, 50
1050 Bruxelles (Belgium)
Dissertation originale présentée en vue
de l’obtention du grade de Docteur en
Sciences Appliquées
1
UNIVERSITE LIBRE DE BRUXELLES Faculté des Sciences Appliquées
THREE DIMENSIONAL OBJECT
ANALYSIS AND TRACKING BY DIGITAL HOLOGRAPHY MICROSCOPY
Kalman Tracking software DHM Oject Détection & Analysis software
Ir. Schockaert Cédric - 2006-2007
Promoteur : Prof. Frank Dubois Dissertation originale présentée en vue
de l’obtention du grade de Docteur en Service de Chimie-Physique EP CP 165/62 Sciences Appliquées
Avenue F. Roosevelt, 50
1050 Bruxelles (Belgium)
Acknowledgements
This adventure closed by this thesis reflects only a part of what I hâve lived everyday. I would like first to acknowledge my advisor Prof. Frank Dubois who gave me a real passion for the scientific research in vision technology and more precisely digital holography, who really motivated me everyday, who was always professionally critic with the different directions I took and that lead to the présentation of this work. I would like to thank the director of the research laboratory Microgravity Research Center, Prof. Jean-Claude Legros, for the trust he showed me, which constitutes a source of high motivation.
My gratitude goes particularly to ail the member of the optical team of the Microgravity Research Center, Christophe Minetti for ail the conversations we had about a lot of different scientific subjects and other, and for his valuable help that has increased my programming skills in C++, Natacha Callens who encouraged me every day, for her help, for her enthusiasm with respect to my research work and results, Catherine Yourassowsky who gave me a real attraction for biological applications, who allowed me to work on high quality holograms by designing the microscopes, Patrick Queeckers for his high availability and professional collaboration on space project together with Christophe Minetti, Olivier Monnom for the technical conversations we had, for his support when I began my thesis, Andrei Vedemikov for the motivation he showed me about my tracking research, for the discussions we had on scientific experiments, for introducing me Prof. Jurgen Blum from TU Braunschweig in Germany, Patrick Grosfils for the theoretical travel where he guided me to discover his scientific research world.
My gratitude goes to ail the staff of the Microgravity Research Center, André, Jean-Charles, Rachid, Pierre, Philippe, Patrick, Hakon, Denis, Carlo, Benoit, Marcel, Séverine, Stefan, Oleg, Valentina, Igor, Ilia, Sasha, Sergei...
I would like to deeply acknowledge my mother and my grandmother for their support and for everything.
Finally, I would be pleased to thank Pierre Coquay from the Politique
Scientifique Federale for the subvention of my research.
Abstract
Digital Holography Microscopy (DHM) is a new 3D measurement technique that exists since Charge Coupled Devices (or CCD caméras) allow to record numerically high resolution images. That opens a new door to the theory of holography discovered in 1949 by Gabor; the door that masked the world of digital hologram Processing. A hologram is a usual image but that contains the complex amplitude of the light coded into intensities recorded by the caméra. The complex amplitude of the light can be seen as the combination of the energy information (squared amplitude modulus) with the information of the propagation angle of the light (phase of the amplitude) for each point of the image. When the hologram is digital, this dual information associated with a diffractive model of the light propagation permits to numerically investigate back and front planes to the recorded plane of the imaging System. We understand that 3D information can be recorded by a CCD caméra and the acquisition rate of this volume information is only limited by the acquisition rate of the unique caméra. For each digital hologram, the numerical investigation of front and back régions to the recorded plane is a tool to numerically refocus objects appearing unfocused in the original plane acquired by the CCD.
This thesis aims to develop general and robust algorithms that are devoted to automate the analysis process in the 3D space and in time of objects présent in a volume studied by a spécifie imaging System that permits to record holograms.
Indeed, the manual processing of a huge amount of holograms is not realistic and
has to be automated by software implementing précisé algorithms. In this thesis, the
imaging System that records holograms is a Mach-Zehnder interferometer working
in transmission and studied objects are either of biological nature (crystals, vesicles,
cancer cells) or latex particles. We propose and test focus criteria, based on an
identical focus metric, for both amplitude and phase objects. These criteria allow the
détermination of the best focus plane of an object when the numerical investigation
is performed. The précision of the best focus plane is lower than the depth of field of
the microscope. From this refocus theory, we develop object détection algorithms
that build a synthetic image where objects are bright on a dark background. This
détection map of objects is the first step to a fully automatic analysis of objects
présent in one hologram. The combination of the détection algorithm and the focus criteria allow the précisé measurement of the 3D position of the objects, and of other relevant characteristics like the object surface in its focus plane, or its convexity or whatever. These extra relevant measures are carried out with a segmentation algorithm adapted to the studied objects of this thesis (opaque objects, and transparent objects in a uniform refractive index environment). The last algorithm investigated in this research work is the data association in time of objects from hologram to hologram in order to extract 3D trajectories by using the prédictive Kalman filtering theory.
These algorithms are the abstract bricks of two software: DHM Object Détection
and Analysis software, and Kalman Tracking software. The first software is
designed for both opaque and transparent objects. The term object is not defined by
one other characteristic in this work, and as a conséquence, the developed
algorithms are very general and can be applied on varions objects studied in
transmission by DHM. The tracking software is adapted to the dynamic applications
of the thesis, which are flows of objects. Performance and results are exposed in a
spécifie chapter.
Contents
ACKNOWLEDGEMENTS 2
ABSTRACT 3
CONTENTS 5
CHARTER 1 - INTRODUCTION 9
1.1. Digital holography OverView 9
1.2. Related works and applications 12
1.3. Contribution of the thesis 12
1.4. Studied applications of the thesis 17
1.4.1. phase objects 1 : vesicles 18
1.4.2. phase objects 2: biological cells 21
1.4.3. Latex particles 23
1.4.4. Protein crystals 25
1.5. Convention 27
CHARTER 2 - FOCUS RLANE DETECTION CRITERIA IN DIGITAL HOLOGRARHY MICROSCORY BY AMRLITUDE
ANALYSIS 30
2.1. Introduction 30
2.2. Theoretical aspects 32
2.2.1. Conservation of the integrated amplitude in digital holography 32 2.2.2. Focus criterion for pure amplitude objects 34
2.2.3. Focus criterion for pure phase objects 37
2.2.4. Focus criteria: a graphical approach 39
2.2.5. Précision of the focus criteria 42
2.3. Experimental results 43
2.3.1. Amplitude objects 44
2.3.2. Phase objects 47
2.4. Extension: Local focus metric 49
2.4.1. Définition of the local focus metric 50
2.4.2. Analysis of the local focus metric 53
2.4.3. Experimental results 61
2.4.3.1. pure amplitude objects: particles 61
2.4.3.2. pure phase objects: vesicles 63
2.5. Conclusion 65
CHAPTER 3 - OBJECT DETECTION IN DIGITAL
HOLOGRAPHY MICROSCOPY 66
3.1. Introduction 66
3.2. Amplitude object détection by local amplitude analysis 68
3.2.1. Algorithm 68
3.2.2. Experimental results 70
3.2.3. Latéral précision of the détection algorithm 74 3.2.4. Other applications of the amplitude object détection algorithm 77 3.2.4.1. synthetic image with focus depth coded in color 77
3.3. Undefined object détection by local energy analysis 823.3.1. Algorithm 83
3.3.2. Experimental results 84
3.4. Conclusion 88
CHAPTER 4 - ACTIVE CONTOURS FOR OBJECT
SEGMENTATION IN DIGITAL HOLOGRAPHY MICROSCOPY 89
4.1. Introduction 89
4.2.
OverView of the segmentation algorithms
924.2.1. Homogeneity criterion 92
4.2.2. Discontinuity criterion 93
4.2.3. Active contour 93
4.2.3.1. parametric active contour 94
4.2.3.2. Géométrie active contour 96
4.3. Mumford-Shah model of Active contour 101
4.3.1. Theoretical aspects 101
4.3.2. Numerical implémentation 110
4.4. Segmentation of amplitude objects in DHM 113
4.5. Segmentation of pure phase objects in DHM 119
4.6. Conclusion 123
CHAPTER 5 - PREDICTIVE KALMAN FILTERING FOR OBJECT TRACKING IN A 3-DIMENSIONAL ENVIRONMENT
125
5.1. Introduction 125
5.2. Tracking algorithm based on the prédictive Kalman fdter 127
5.2.1. Dynamic and measure Models 128
5.2.2. Kalman prédiction équations 134
5.2.3. Multiple objects tracking algorithm and initialization 136
5.3. Experimental results 139
5.3.1. Tracking of amplitude objects: particles 140
5.3.1.1. noises modelization 140
5.3.1.2. initialization of the tracking algorithm 141
5.3.1.3. results 141
5.3.2. Tracking of phase objects: vesicles 145
5.3.2.1. noises modelization 145
5.3.2.2. initialization of the tracking algorithm 146
5.3.2.3. results 147
5.4. Conclusion 151
CHAPTER 6 - SOFTWARE DEVELOPMENT FOR DIGITAL HOLOGRAPHY MICROSCOPY: BIOLOGICAL
APPLICATIONS 153
6.1. DHM Object Détection and Analysis software 153
6.1.1. Performance assessment for particles 160
6.1.2. Performance assessment for vesicles 162
6.2. Kalman Tracking software 169
CHAPTER 7 - CONCLUSION 176
BIBLIOGRAPHY 178
ANNEX I - OPTICAL SCHEMA OF THE DHM WITH
PARTIAL SPATIAL COHERENCE 186 ANNEX II - THE FOURIER METHOD TO EXTRACT
COMPLEX AMPLITUDE OF LIGHT FROM HOLOGRAM 188
ANNEX III - 4-PHASE STEPPING METHOD TO EXTRACT
COMPLEX AMPLITUDE OF LIGHT FROM HOLOGRAM 192
ANNEX IV - IMAGE SELECTION ALGORITHM FOR THE
PROMISS MISSIONS 194
Chapter 1
Introduction
1.1. Digital holography overview
In standard holography, a holograra is recorded on a photographie film. The word
hologram is originating from Greece and means the whole or the entire (holos)
writing (gramma). Indeed, holography allows the recording not only of the intensity
of the light like in photography, but also of its phase which gives the direction of the
propagating light. As photographie films record only the light intensity, the intensity
and phase information of the light has to be coded in an auxiliary intensity which
constitutes the recording stage of the hologram. The codification of the complex
amplitude of light into intensity is of interferometric nature. The knowledge of the
intensity and the phase permits to reproduce the complété 3D wave front of the
captured object. In 1949 Gabor [1] invented holography to improve the resolution of
electronic microscopy in order to visualize atoms. The principle of holography is
based on the interférence of a cohérent beam diffracted by objects either with the
background beam {in-line holography) or, as proposed by Leith and Upatnieks [2] in
1962, with a reference beam similar to the beam diffracted by objects {off-axis
holography) but the reference beam is combined to the object beam with an offset
angle. This interferometric information is recorded on a photographie film and
defines the recording stage of the hologram. In the Gabor experiment, the cohérent
beam was of électrons and the measurement procedure was in-line holography. The
holography with cohérent wave of visible light will begin ten years later with the
invention of a cohérent beam device known as laser [3]. When a hologram is
recorded on a photographie film, the second stage consists in the 3D visualization of the captured objects in the hologram by illuminating the photographie film with the same cohérent beam. This second stage constitutes the holographie reconstruction.
The photographie film can be illuminated with white light, but a spécial recording configuration has to be performed [4], By changing the position of the viewer with respect to the illuminated hologram, the perspective of the objects changes and the objects recorded by the hologram are seen like real objects in a 3D space and not as a projection in a 2D area like in a usual image.
The évolution of high resolution Charge Coupled Devices (CCD caméras) allows the recording of the hologram of an object directly on it. As in this case the hologram is in a digitalized form, it can be easily numerically processed and there is no need of Chemical development like in analog recording of hologram on a photographie film. The digital hologram is numerically processed to extract the complex information of the object: its intensity and phase. From this complex information, the optical diffractive laws are emulated to numerically investigate the whole object in 3D. We note that in in-line holography the conjugate object can not be separated from the object information. The complex information of the object itself can only be computed in off-axis holography where the conjugate and non conjugate objects are separated thanks to the offset angle between the two beams in the recording stage [5].
In this thesis, the off-axis holography is implemented in microscopy and holograms are recorded by a CCD caméra. Fig. 1.1 represents schematically this imaging System used to record holograms and called Mach-Zehnder interferometer.
The laser beam / is separated by a beam splitter (55/) into the object beam O that
goes thru the experimental cell, and the reference beam R. Two mirrors are
represented respect!vely by Ml and M2. Both beams O and R are combined by a
second beam splitter (552) on the CCD captor and their interférence pattern is
recorded.
I ceii waiiSj O Ln\J "<
Cell wallsi BSl
i_\
\ \
R experimental cell
\ \
Ml BS2
Fig, 1.1. Off-axis digital holography configuration: Mach-Zehnder interferometer
A 3D experimental cell is studied by the laser beam of the microscope in transmission. Digital holography microscopy (DHM), allows the numerical investigation of a volume which contains objects that are not necessarily in the focus plane of the imaging System. This numerical investigation is a powerful tool to refocus objects in a non-invasive way. In microscopy, the high numerical aperture of lenses and the high magnification ratios induce small depth of field. Usually a mechanical scanning has to be performed manually by an operator to refocus objects. It is not possible to register in real time 3D information of objects under rapid motions with this procedure. The numerical investigation is a post processing operation that can be automated by algorithms to refocus objects. As a conséquence, recording speed of the 3D experiment is only limited by the acquisition rate of the CCD. Those holograms contain the complex information in the object plane within the experimental cell, which is the focus plane recorded by the imaging System.
Knowing the complex amplitude in the object plane, the complex amplitude in
planes parallel to the object plane and perpendicular to the optical axis are
numerically computed by simulating the optical diffractive laws with the Kirchhoff-
Fresnel équation [6].
1.2. Related works and applications
Digital holographie microscopy has been demonstrated in many applications as in refractometry [7], observation of biological samples [8-12], living cells analysis [13- 17] and velocimetry [18]. Due to its digital nature, this technique allows to implement powerful processing to improve the digital holographie reconstruction [19], to control the image size as a fonction of the refocusing distance and the wavelength [20], to perform 3D pattern récognition [21-23], to process the border artifacts [24,25], to implement twin-image noise élimination techniques [26,27] and to perform quantitative phase contrast imaging [28,14,15,19] and aberration compensation [29,30]. In [31], it is proposed a tomographie imaging System that uses a digital holographie microscope with wavelength-scanning capability and allows both front reconstruction along the optical axis and side reconstruction along a perpendicular to the optical axis. A recent application of digital holography for tomography [32] permits to déterminé the 3D refractive index in an experimental volume. As in DHM the entire information of the objects within the experimental volume is hold in the hologram, non focus objects inside the hologram appear diffracted and can affect the visual information of other focus objects. For this reason DHM is more suitable for small objects. Indeed, a wide object covering a large part of the field of view highly dégradés the visual information of other objects when it appears defocus. An important application is the study of particles in a 3D environment [33,34]. In [35] a digital holographie particle tracking velocimeter is develop to study the 3D flow in in-line holography using a pulsed laser source that allows to compute the 3D velocity of tracer particles in a water flow around obstacle. An optical holographie configuration is proposed in [36] and combines two views of a set of particles in an experimental volume to measure the 3D components of their velocity. For the semiconductor industry, it has been demonstrated the capabilities of DHM for defect détection [37] and microstructures analysis [38].
1.3. Contribution of the thesis
In this thesis we develop numerical tools for DHM that permit to follow in time (tracking) objecta within a 3D experimental cell in order to extract trajectories in the volume of the experimental cell. These numerical tools are not related to one particular object. The keywords of the thesis are Image processing, Digital holography microscopy, Tracking, Object analysis.
Image processing: the numerical tools developed are designed for images, and more precisely for digital holograms which correspond to grayscale interferometric images where constructive and destructive interférences appear respectively bright and dark. The objective of the development of those algorithms is the implémentation of stand-alone software that process a sequence of digital holograms acquired by a digital holographie microscope. Without the automation of the processing of the 3D information inside the hologram, only a weak quantity of the 3D information could be extract manually by a human operator which limits drastically the study of the sequence of holograms. One other important aspect is that the algorithms analyze images always in the same way (not operator dépendant) and are more précisé, in a lot of applications, than the human eyes for the automatic refocusing stage to compute the 3D position of objects.
Digital holography microscopy constitutes the décor of the scene where the thesis plays its own rôle. This décor has been presented in Section /./ and as been clapped thunderously in Section 1.2.
Tracking: by tracking the reader has to understand that each moving object of a hologram sequence is followed in each hologram in order to obtain its trajectory.
The extraction of the 3D trajectories of moving objects from the hologram sequence constitutes the final objective of the thesis. In order to reach this stage, the holograms are processed by a stand-alone software that extracts the 3D information of objects, and a second software, integrating a data association algorithm of objects from hologram to hologram, builds trajectories from the results of the first software.
Both software are built for this thesis and are discussed in the related chapter.
Object analysis: We can classify objects in two classes: amplitude objects and phase objects. Firstly, amplitude objects are defined as connected area in the intensity field of the complex light amplitude at their focus plane. As the digital holographie microscope works in transmission, amplitude objects appear darker than the rest of the intensity field. Amplitude objects absorb energy of the light.
Secondly, phase objects are defined as transparent objects. As a conséquence, they
don’t absorb energy of the light but change the phase of the complex amplitude light by refraction. The shape of a phase object is materialized in the phase field at the focus plane of the object by a connected area. This connected area represents the local change of phase induced by the object refractive index which is different than the refractive index of the solution in which the object is studied. Typical phase objects are biological cells. The analysis of objects of a hologram is defined as the automatic refocusing of each object by developing focus criteria for both object nature, and as the surface analysis of each object in its focus plane. The complément area of objects in the hologram defines the background.
As the définition of object is general in this thesis, ail the associated image Processing algorithms and the tracking theory are general and not developed for one particular object. However, the tracking theory is adapted to the dynamic applications of the thesis which are flows of objects in one main direction.
DHM permits to numerically refocus objects in order to measure their 3D
positions, which constitutes the first stage before tracking objects. However the
focus plane of an object has to be determined by a criterion. This criterion can be
simply the eyes of a user but this is not realistic when the amount of holograms to
process increases. One other problem with this criterion is that the choice of the
focus plane of a particular object is subjective and of course with a précision limited
by the depth of field of the microscope. A focus plane criterion from the numerical
reconstruction of the complex amplitude, computed by emulating the optical
diffractive laws in the experimental volume along the optical axis, must be
developed. Few focus criteria has been proposed for DHM in the literature. Usually
they are valid only for amplitude objects and not for pure phase objects like
biological cells which are an important application in DHM [14,16,42]. Among
those criteria for amplitude objects, we note that they are sometimes adapted only
for a particular object like particles [40,34]. In [41] they hâve developed recently a
general focus criterion for amplitude objects but this method is computationally time
consuming. In this thesis we propose a focus metric that permits to build a focus
criterion for amplitude objects (defined as the minimum of the focus metric) and
also for pure phase objects (defined as the maximum of the focus metric). The focus
criteria are based on the integrated amplitude modulus of which the properties are
unique respectively in the focus plane of an amplitude object and in the focus plane
of a phase object. Those criteria are applied successfully in this thesis on different
object like latex particles, a ruler, protein crystals, biological cells, and vesicles. The theory of these focus criteria is presented in Chapter 2 of this thesis.
Each recorded hologram contains the complex amplitude information of a plane perpendicular to the optical axis and determined inside the experimental volume by the lenses of the imaging System of the digital holographie microscope. This complex amplitude holds the information of ail the objects présent in the experimental volume. Some objects can be focus and other not. Objects that are not focus appear diffracted and their signal is expanded in a spatial région on the image plane that grows with the defocus distance of the object. This is problematic if we want to select a particular object on the recorded hologram to compute its focus plane because its 2D localization on the plane can be noisy by the information of other objects. To automatically compute the focus plane of ail objects within the experimental volume, their ail hâve to be detected in the 2D plane of the complex amplitude decoded from the recorded hologram. The aim of an object détection algorithm is to separate on the recorded hologram, areas related to objects from areas related to the background (for example the solution in which the objects are in suspension in the 3D volume). In this thesis we develop a détection algorithm for amplitude objects. This new tool in DHM builds a synthetic image of the same size than the digital hologram where pixels related to the background are dark, and pixels related to amplitude objects are bright. AU amplitude objects in this synthetic image appear focus, as if the depth of field was not limited by the high numerical aperture of lenses. This synthetic image opens different perspectives in DHM. The first contribution is the automatic computation of the focus plane of ail amplitude objects présent in the hologram because they are ail now automatically detected. There is no need to select manually objects we want to refocus with the focus criterion.
Secondly, as amplitude objects in the synthetic image appear focus, we can directly
apply the well-know theory of segmentation in image processing and analyze the
surface of objects directly on the synthetic image. One other perspective is the
object counting in a 3D experimental cell. The theory of the amplitude object
détection is proposed in Chapter 3 of this thesis. In Chapter 3 we présent also an
object détection algorithm for amplitude and phase objects. With this algorithm one
other synthetic image is created representing again background in dark and objects
in bright, but with less précision than in the synthetic image representing amplitude
objects only. By less précision the reader has to understand that the bright objects in
this synthetic image don’t represent the phase or amplitude objects in its focus plane
with the same latéral précision than the first algorithm dedicated to amplitude objects. However, this algorithm is general and not limited to amplitude objects. It constitutes general object détection method for DHM where each detected object can be studied independently by a numerical refocusing (Chapter 2) to reach its focus plane and, as a conséquence, its focus shape.
The object détection algorithms combined with the focus plane criteria allows the détermination of the coordinate along the optical axis of ail the amplitude or phase objects présent in the experimental volume. To hâve the complété 3D information and, if required, an analysis of the surface or contour of objects, ail focus objects must be segmented. This segmentation is performed for amplitude objects directly in the synthetic image of the amplitude object détection algorithm, and, for phase objects, in the unwrapped and non tilted phase image of the numerically reconstructed complex amplitude at the focus plane of the studied pure phase object along the optical axis. In both cases, i.e. amplitude and phase objects, the segmentation is performed in a image which can be separated in two classes represented by a mean grey-level: one for the objects and one for the background.
For amplitude objects, as we explained earlier in this paragraph, the synthetic image
of the détection algorithm in which the segmentation is performed, represents the
background with dark pixel, whereas the objects appear bright. For pure phase
objects, the phase image at its refocusing distance along the optical axis can be
processed First to remove the background phase induced by lens aberrations and/or
non parallelism between optical éléments or walls of the cell [44] and secondly to
unwrap the phase image in order to remove the phase Jump, induced by the modulo
27
tphase, and translated in the image by a spatial Jump from a high grey-value to a
low grey-value. After this processing, the phase of the solution of the experimental
cell supposed of a constant refractive index is uniform, and the phase of the objects
is higher or lower than the phase of the solution depending on the refractive index of
phase objects. It results two classes: the phase (coded into grey-level in the image)
of the solution and the phase related to objects. In [84] they developed active
contour technique based on the Mumford-Shah functional well adapted to this
common characteristic of two classes for the segmented maps of amplitude or phase
objects. This segmentation theory is presented and applied on several objects studied
by DHM in Chapter 4. We hâve at our disposai the coordinate of each object along
the optical axis with the theory developed in Chapter 2 and 3. This information
focus shape. This processing permits the computation of the object 2D centroid in the focus plane. Combined with the computed refocusing distance, the 3D coordinate of objects in the experimental volume are obtained. Other important characteristics of objects like the 2D orientation and surface in their focus plane, the convexity of the contour, etc. can be computed easily. From these algorithms, a stand-alone software is built and process ail holograms of a sequence in order to extract the surface and 3D information of objects.
For each object of the holograms we hâve built an information label defined as the 3D position and other information that can be relevant depending on the studied objects in the experiment like the surface, the 2D orientation in the focus plane, the convexity of the contour, etc. The Chapter 5 describes a data association algorithm (tracking algorithm) that permits to link in time the different objects in order to croate trajectories of objects. The tracking is performed from hologram to hologram.
The data association algorithm is based on the Kalman prédictive filtering theory [98] that performs association in time of objects described by their information label and with a probability of the association. This tracking theory is applied to moving objects. The first application is performed on amplitude objects: latex particles. The second application is the processing of a sequence of holograms of pure phase objects: vesicles.
The Chapter 6 is dedicated to the évaluation of the performances of two software implementing the algorithms developed in the thesis. This évaluation is performed on both applications quoted above. The first software, DHM Object Détection and Analysis software, implements the focus criteria, the object détection methods and the segmentation technique. This stand-alone software analyzes a sequence of holograms. The resulting information labels of every hologram are processed by a second software in order to extract trajectories in 3D from the sequence of holograms: the Kalman Tracking software.
We close the stage curtain of the thesis in Chapter 7, and we présent there the perspectives.
1.4. Studied applications of the thesis
To demonstrate the theory for each chapter of this thesis we apply it on actual holograms representing amplitude objects or pure phase objects. Those holograms bave been acquired by off-axis digital holographie microscopes working in transmission as described in Fig. 1.1. We summarize now the experiments and the digital holographie microscope for each application.
1.4.1. phase objects 1: vesicles
A vesicle is a phase object of size varying from about 1 to 100 microns. They are déformable objects composed of a bilayer interface of phospholipids separating an internai solution from the solution in which there are suspended. The 3D dynamic of vesicles is studied in order to simulate the dynamic of a unique cell and the rheology of a suspension like blood [43,118].
The digital holographie microscope uses a partially cohérent source created by a laser incident on a rotating ground glass [53]. The optical schéma is provided in Annex I. The detailed operating mode of the microscope is described in [54] where it has been applied to study the motion of particles. The use of a partial cohérent source reduces the cohérent noise which is inhérent to a fully cohérent illumination and highly dégradés the signal. The vesicles are in a 3D circular experimental cell, called shear flow chamber and developed within the BIOMICS project of the European Space Agency (Dynamics of Cells and Biomimetic Systems - ESA AO- 2004-113). The shear flow chamber is studied in transmission by a laser incoming perpendicularly to the circular walls of the cell. A motor applies a rotating motion at constant velocity to one of the circular wall of the cell. This induces a shear flow in the cell. The thickness of the volume in which the vesicles can evolve is 250 pm.
The size of digital holograms acquired by the CCD is 1024 x 1024 pixels and the
size of one pixel is 0.41 pm. The acquisition rate of the caméra is 24 frames by
second. Holograms are recorded in a plane within the experimental volume of the
cell. This plane corresponds to the object plane of the imaging System of the
microscope. From those holograms, the complex amplitude field in the object plane
is extracted by filtering the Fourier transform of the hologram [56]. Details of this
method are presented in Annex II.
To numerically investigate the whole volume of the experimental cell, a numerical holographie reconstruction is applied along the optical axis by using the Kirchhoff-Fresnel équation [6]. As we will see in Section 1.5, this équation is composée! of three parts: firstly, the Fourier transform of the complex amplitude field is computed, then the results is multiplied by a quadratic phase, and finally the inverse Fourier transform is applied. To compute numerically the Fourier transform, the well-known Fast Fourier Transform algorithm (FFT) is used [58]. The FFT supposes the spatial periodicity of the processed signal. If no preprocessing on the complex amplitude field is performed before the holographie reconstruction, a diffraction of the borders appears on the reconstructed complex amplitude. To solve this problem, different solutions are proposed in literature [24, 25]. However in [25], the algorithm is time consuming and in [24] it induces a loss of information on the hologram border. In this thesis, to avoid border diffraction when a numerical holographie reconstruction is performed by emulating the diffractive laws, the entire recorded complex field is centered in a wider area of 2048 x 2048 pixels and the outside régions are simply filled with the border complex amplitude. This doesn’t avoid diffractions of border but they appear at the border of the enlarged 2048 x 2048 pixels image and don’t dégradé the image information of the centered image for not excessive numerical reconstruction distances from the recorded plane.
Figures 1.2(a), 1.2(b) and 1.2(c) show respectively a recorded hologram and the
corresponding intensity and phase images obtained by filtering the Fourier transform
of the hologram. The reader is referred to Annex II for details about the filtering
process of the hologram to extract the intensity and phase information.
a)
c)
Fig. 1,2: (a) recorded digital hologram of vesicles; (b) corresponding intensity image; (c) corresponding phase image.
1.4.2. phase abjects 2: biological cells
The objective of this biological application is to analyze cancer cell migration in a 3D matrix gel [55], They are almost completely transparent and can be considered as phase objects. As the cell migration is a slow phenomenon in comparison with the typical image acquisition time, we used a digital holographie microscope working with the phase stepping method [17] which is explained below. The light source is a 660 nm Epitex LED with a 20 nm spectral width. The microscope objectives are lOx Leica objectives HC PL Fluotar, NA= 0.30. The caméra is a Hamamatsu Orca 1 with 1024 X 1024 pixels for a field of view of 685 pm x 685 pm.
To compute the complex information in the recorded plane, the intensity and the
phase fields, four holograms are recorded at different phase shifts, induced by a
piezo-transducer. The phase and intensity information in the recorded plane are
obtained directly knowing the optical phase shift induced by the piezo-transducer between two consecutive hologram acquisitions. This method is the 4-Frame phase stepping method and is described in Annex III. To avoid border diffraction when holographie reconstruction is performed, the entire complex amplitude of the recorded hologram is centered in a wider area of 2048 x 2048 pixels and the outside régions are filled with the border complex amplitude Unes. Figures 1.3(a) and 1.3(b) show respectively a computed intensity and phase image.
a)
Fig. 1.3: (a) intensity image ofbiological cells; (b) phase image ofbiological cells.
1.4.3. Latex particles
The motion of latex particles is studied in a 3D experimental cell. As the flow of
particles is a rapid phenomenon, it is necessary to record, for each caméra
acquisition, the complété 3D information (holograms) for further processing and to
avoid methods with multiple recording of phase shifted holograms like described in
Section 1.4.2. The digital holographie microscope is the same as the microscope
used for the vesicles in Section 1.4.1 and is presented in [53]. The optical schéma is
provided in Annex I. The detailed operating mode and experiment are described in
référencé [54]. The field of view is 330 pm x 330 pm on a 1024 x 1024 pixels CCD
sensor size. The acquisition rate of the caméra is 24 frames by second. The
extraction of the complex amplitude from the recorded hologram is performed, as in
Section 1.4.1, by filtering the Fourier plane [56] as described in Annex 11. Again, to
avoid border diffraction, the entire digital complex amplitude of the recorded plane
is centered in a wider area of 2048 x 2048 pixels and the outside régions are filled
with the border complex amplitude lines. The numerical holographie reconstruction
is applied on this enlarged image. Figures 1.4(a), 1.4(b) and 1.4(c) show respectively
a recorded hologram and the corresponding intensity and phase images.
c)
Fig. 1.4: (a) recorded digital hologram of latex particles; (b) corresponding intensity image; (c) corresponding phase image.
1.4.4. Protein crystals
This application studies the growth of protein crystals in 3D and in microgravity
condition. This experiment is performed in the International Space Station (ISS) and
named PROMISS (PROteine Microscopy for ISS). The whole experiment and the
microscope are described in [63]. To compute the complex information in the
recorded plane, the 4-Frame phase stepping method of Section 1.4.2 is used and four
holograms are recorded at different phase shifts induced by a piezo-transducer (see
Annex III for details). The successive phase shifts are of 7t/2. The NASA imposed
the use of an analogue caméra. As a conséquence, it is not possible to record a
determined number of four images to implement the 4-Frame phase stepping
method. The images are continuously recorded by the caméra while the phase shifts
are performed and an algorithm has been developed in order to automatically extract
the best 4 images which hâve a successive determined phase shift of nl2. The
algorithm implemented into a software is detailed in Annex IV. As the growing
process of protein crystals is a slow process, the caméra records one complex information (which required 4 images) by hour. The Field of View (FOV) is 2.27 x 1.84 mm, recorded by a caméra and sampled into images of 720 x 480 pixels. Again, in order to alleviate disturbances by border effects, the complex amplitude computed in the recorded plane is centered in a larger area of 1024 x 1024 pixels where the outside régions are filled with the border complex amplitude Unes. Figures 1.5(a), 1.5(b), 1.5(c), 1.5(d), 1.5(e) and 1.5(f) show respectively the four recorded holograms and the corresponding intensity and phase images.
e) 0
Fig. 1.5: (a)-(d) 4 recorded digital holograms of protein crystals with a successive phase shift of n!2\ (e) computed intensity image; (f) computed phase image.
1.5. Convention
In this thesis we follow several conventions for the numerical investigation of the amplitude light in the space (digital holographie reconstruction). First, we assume that the optical beam is propagating in the z-axis direction in the experimental cell where numerical reconstructions are performed. A digital holographie reconstruction consists in the computation of the complex amplitude in a plane P’
perpendicular to the z-axis knowing the complex amplitude in a plane P parallel to P’ and separated from it by a distance d. The spatial coordinates {x,y) and are respect!vely in the plane P and P'. The relationship between
v^(xand M
q(
x,>^) is given by the Kirchhoff-Fresnel équation that we assume paraxial:
v,(x',y) =
exp{jkd)F-\^.y exp ^ I..2 . ..2 ( 1 . 1 )
Where À is the wavelength, k=2n/À, (v„Vy) are the spatial frequencies, 7'=-/, and
^g{CX, P) dénotés the direct or inverse 2D continuons Fonder transformations defmed by:
F(nn4Sioc, P)=Y\ exp{+ 2j7r{arj + P^)}g{a, P)dadp (1.2)
As we can assume that the factor exp(y^i/) at the right hand side of Eq. (1.1) does not play any significant rôle in practice, we define the effective digital holographie reconstructed or propagated amplitude Uj{x',y') in F”by:
uAx\y') = F^
-1exp jkdÀ^ (d+d: (Dr,,!/ “0 0
-3
)The second convention consists in the définition of the plane P as the recorded plane of the hologram, which corresponds to the object plane of the imaging System, inside the experimental cell. The distances for the holographie reconstructions along the optical axis, defined as the z-axis in this convention, are referenced to the position of the plane P along this axis. We will use the name of reference plane to qualify the plane P in the whole thesis. This reference plane represents the origin along the z-axis. Upstream planes to the reference plane are at négative distances from it, and downstream planes at positive distances.
The third convention consists in the définition of 3D Cartesian axes of reference and related unities. We hâve already defined the z-axis as the optical axis and the origin of this axis is the reference plane. Figure 1.6 represents the two other axes, the x-axis and the y-axis, which define the image plane.
Fig. 1.6. Définition of the Cartesian axes of reference and related unities for digital holographie image reconstruction.
We work for the z-axis with unities different than the two other axes in order to
point out the différence of methods used to compute the 3D position coordinate of
objects in the theory of the thesis. Indeed, the coordinate along the z-axis of the 3D
position of an object is measured by holographie reconstruction along the z-axis by
steps expressed in micron in this thesis. In the other hand, when the focus plane of
an object is determined (Chapter 2), the two other coordinates, along the x-axis and y-axis, correspond to the object centroid computed in the image plane (object segmentation Chapter 4) and are expressed in pixel.
The fourth convention is the holographie vocabulary used in this thesis:
■ Digital holographie reconstruction: from the complex amplitude in the reference plane, other planes along the z-axis are obtained numerically.
■ The complex amplitude in a plane perpendicular to the z-axis is represented by an intensity image and a phase image. In the thesis we work with grayscale BMP (Bitmap) format images coded in 8 bits (256 grey-levels). This means that the intensity information of the complex amplitude is quantified in 256 levels. Similarly, the phase information covering the range [0,..,2Tt\ is coded in 256 grey-levels. By convention, the lowest grey-level (black color) corresponds to the phase 0, and a multiplicative factor of 40 is used to stretch the phase information in almost the entire grey-value range. As a conséquence, the phase 2n is coded by the quantified grey-value of 25/ as 2it*40=251.32.
Fifthly, when a sequence of holograms is studied for tracking applications, the
time zéro is related to the first hologram of the sequence. This time is qualified as
the reference time of the hologram sequence.
Chapter 2
Focus plane détection criteria in digital holography microscopy by amplitude analysis
Abstract: We propose and test a focus plane détermination method that
computes the digital refocus distance of an object investigated by digital holography microscopy working in transmission. For this purpose we analyze the integrated amplitude modulus as a function of the digital holographie reconstruction distance. It is shown that when the focus distance is reached, the integrated amplitude modulus is minimum for pure amplitude object and maximum for pure phase object. After a theoretical analysis, the method is demonstrated on actual digital holograms for the refocusing of pure amplitude and of pure phase microscopie samples.
This chapter is based on [61] and partially on [62],
2.1. Introduction
Digital holography provides a tool to numerically refocus an object by holographie
reconstruction. However, it does not provide any criterion when the best focus
distance is reached. Indeed, if digital holographie reconstruction can refocus a
sample slice-by-slice as the focusing stage of a classical imaging System, the
refocusing of an object has to be determined by an extemal criterion. In that way, a
refocus criterion based on a gradient computation has been investigated in [49]. This
approach can be powerful when the objects under investigation give rise to sharp images at the best refocus distance. A sharpness metric based on the self-entropy bas been described in [50]. It bas been also proposed to use a criterion based on the maximization of the intensity local variance in [51]. Ferraro and al. proposed a method where an operator focuses the object under test and the amount of focus change in time is obtained by measuring its phase change [52]. The defocus distance is then introduced to digitally refocus the object. Recently, the theory of Fresnelet has been applied to compute a new sharpness metric related to the sparsity of the wavelet coefficients and their energies [41]. This method is however computationally time consuming. We propose a focus détermination methodology that takes benefit of the invariance of both energy and amplitude intégration on the global hologram plane. Those invariance properties allow to build two focus criteria, respectively for pure amplitude and for pure phase objects, that are based on the score of the integrated amplitude modulus. In Section 2.2, we describe the theoretical aspects and we dérivé the focus criteria for the both types of objects.
Although the general case concems objects that hâve mixed amplitude and phase
modulations, the pure amplitude or phase cases are of considérable practical
importance in microscopy. For example, biological cells are mostly transparent and
can be considered as pure phase objects while, for 3D velocimetry applications, the
objects are very often opaque particles. The Section 2.3 is devoted to the
experimental démonstrations on actual digital holograms. In the Section 2.4, those
criteria are extended for the détermination of the best focus planes of a set of objects
that could occur at different locations in the same digital hologram. The objective of
that section is the local study of one particular object of the hologram.
2.2. Theoretical aspects
2.2.1. Conservation of the integrated amplitude in digital holography
By taking the Fourier transform of Eq. (1.3) and by computing the squared modulus, we obtain that the power spectrum |f/^ )j of Uj{x',y') is equal to the power spectrum
(2-4)
Equation (2.4) means that the information is preserved by the digital holographie reconstruction of Eq. (1.3). Therefore the information content is insensitive to any refocus change by digital holography. This situation is different with respect to the classical imaging Systems using incohérent lighting. Indeed, with such an imaging System, a defocus provides a loss of the image sharpness in a low-pass filtering process that modifies the shape of the power spectrum. As a conséquence, there is no way to fmd a refocusing criterion based on the only digital holographie signal and it is necessary to add a criterion to détermine the refocusing distance of an object along the z-axis.
We consider some invariant properties of the optical field under the propagation defmed by Eq. (1.3). The energy E is invariant:
oo oo oo oo
j ^\uj{x\y''^dx'dÿ - ^ ^\uç^{x,y'f dxdy = E (2.5)
The intégral in the xy directions of the complex amplitude is also invariant:
Consider that we propagate the complex amplitude field equal to M
q(
jc,>’)+/I,
where A is an arbitrary amplitude value constant in the xy plane. The energy
conservation leads to:
( 2 . 6 )
J +\Â\ + A*U
ç,+Aul)dx'dÿ = + A Uj + Au *j)dxdy
where we used the fact that the constant amplitude A is invariant under the propagation defined by Eq. (1.3), and where we omitted, for conciseness, the explicit spatial dependency of and Uj{x,y^. We note that the intégration range from to + °o in the x and y direction is only theoretical because of the space limitation in the xy plane of the hologram captured by the caméra. As a conséquence, the constant amplitude A is defined in the xy plane only where the hologram is defîned, else its value is zéro. From Eq. (2.6), we hâve:
oo oe oc oo
J \(A\ + Aul)dx'dy= I j(Ay, + Auj)dxdy (2.7)
By combining the Eq. (2.7) respectively obtained with pure real and pure imaginary A values, we can conclude that:
J ^Uj{x',ÿ)dx'dy^ ^ ^u^{x,y)dxdy^ B (2.8)
Equation (2.8) means that B, the intégral of the amplitude in the xy directions, is invariant with respect to the digital holographie propagation. It results that the modulus of B is also an invariant. Therefore, using the usual triangle inequality relation, we obtain:
1^1 ^ J \\^d{x,y\dxdy^ Mj (2.9)
Equation (2.9) indicates that the intégral Mj of the amplitude modulus has a global
lower bound which is independent of d.
2.2.2. Focus criterion for pure amplitude abjects
We consider a pure amplitude object located in a plane. We will show that its image is refocused at distance d when the intégral defined in Eq. (2.9) is minimum. The refocus distance is d if:
J \\uAx.y}dxdy is minimum ( 2 . 10 )
Indeed, consider first an object that is a real positive transparency t{x,y) illuminated by a constant real positive amplitude C We hâve in the focus plane:
^*d{x,y) = t{x,yY: (2.11)
As uAx,y) is real positive valued, its phase is uniform on the xy plane and it results that:
00 OQ PO OQ OO OO
1 ^Uj{x,y)dxdy = JJ” Ax,y)dxdy\^ JJI” j{x,y)fixdy (2.12)
We dénoté the complex amplitude in an out of focus plane by Uj.{x',y'\ By invoking the invariance of the integrated amplitude, we obtain:
OO OO OO OO
J \^d{x,y)dxdy = j ^Uj\x\y)dx'dÿ (2.13)
As is not the amplitude in the focus plane, it is a priori complex and the second equality of Eq. (2.12) has to be replaced by:
j J Uj.{x\y')dx'dy ^ J \ \^æ{x',y')\dx'dy' (2.14)
( 2 . 15 )
J \\uA^^y}dxdy < J ^\uj.{x',ÿ\ix'dy'
Equation (2.15) expresses that the integra! of the amplitude modulus is minimized when the focused plane is reached. When we consider the same object illuminated by a tilted plane wave, the amplitude modulus pattern reconstructed by digital holography remains identical except that there is a small shift in its position in the xy plane with respect to the un-tilted illumination case. Therefore, the intégration of the amplitude modulus of the diffraction pattern remains identical to the one of the un-tilted case and the minimum of the modulus intégration occurs also at the refocusing distance d. In most of the practical cases of interest, even in the case of non-plane wave illumination, the objects are small enough to be considered as illuminated by a plane wave.
Consider the example of the real amplitude object Mg(x, jp) shown by the Fig.
2.1 (a). It représenta a clear disk in a screen back-illuminated by a constant phase plane wave. When the object is out of focus by a distance d, diffraction effects are appearing with the typical oscillations shown by the Fig. 2.1 (b).
Figures 2.1(c) and 2.1(d) represent the horizontal amplitude profiles across the
center of the patterns of, respect!vely, Fig. 2.1 (a) and Fig. 2.1 (b). As the diffraction
pattern présents positive and négative régions (Fig. 2.1(d)) while the un-diffracted
one (Fig. 2.1(c)) has only a positive région, it results that the intégration of the
modulus |
mj(
x,>')| will be minimized when d=0.
a) b)
Fig. 2.1: (a) A pure real valued amplitude object. It represents a back-illuminated aperture by constant amplitude field; (b) The same object that is defocused by the digital holographie propagation; (c) Horizontal amplitude profile across the middle of (a); (d) Horizontal amplitude profile across the middle of (b).
This example is an illustration of the optical field behavior that is constrained to
fulfill the invariance of the integrated intensity and amplitude. The refocus criterion
is based on this phenomenon.
2.2.3. Focus criterîon for pure phase abjects
We consider now the case of pure phase objects. We assume that a pure phase object is illuminated in transmission by a plane wave that is propagating along the z axis and that the physical thickness of the object is small in comparison with the depth of field of the imaging System. As in the amplitude case, we assume that the object is located at a distance d with respect to the focus plane of the imaging System. The beam emerging out of the object is expressed by:
the uniform real amplitude of an illuminating plane wave. As we consider a pure phase object, the emerging intensity is constant when the focus image of the object is reached. Indeed the intensity image of the object disappears from the focus image as it is well known in optical microscopy. It results also that the modulus of the amplitude is constant in the focus plane. When the image of the object is defocused, the refraction created by the object’s phase modulâtes the amplitude and the resulting intensity. As a conséquence the defocus is at the origin of fluctuations of the amplitude modulus and the intensity. As in the previous case, the invariant energy and amplitude intégral are fulfilled. We will show that its image is refocused at distance d when the intégral defined in Eq. (2.9) is maximum. The refocus distance is d if:
To understand how this criterion is operating, consider a collection of N sampled amplitudes that are obtained from a pure phase object, not necessarily focus, that is illuminated in transmission by a plane wave of amplitude a. For the purpose of the démonstration, there is no need to consider the explicit 2D nature of the optical field and we can consider only one discrète spatial variable s. We hâve:
(2.16)
Where ç{x,y) is the optical phase change introduced by the phase object and a is
J is is maximum (2.17)
(2.18)
Where s = a, is the amplitude modulus and Ç?, the optical phase at the sampling point s.
The energy conservation is expressed by:
(2.19)
The focus metric defined by Eq. (2.9) becomes with the sampled signal:
A^-l
( 2 . 20 )
.v=0
When the image of the object is focus, ail amplitudes a, are equal to a. To demonstrate that is maximum when ail a, = a, we consider the problem that consists to look at the a, that maximizes Mj under the constraint of a constant energy E. The Lagrange multipliers method solves this problem. We form the fonction/(ûfç):
where P is the Lagrange multiplier.
The partial dérivatives of/ with respect to the a, hâve to be equal to zéro. It results a relationship between F and the a, that hâve ail to be equal to a. This condition on the a, corresponds to the best focus plane of the object. With a simple physical interprétation, we observe that the extremum of Mj is a maximum.
Therefore Mj is maximum when the best focus plane is reached.
( 2 . 21 )
2.2.4. Focus criteria: a graphical approach
From the invariance of the integrated complex amplitude in the xy planes with respect to holographie reconstruction distance along the z-axis, two focus criteria based on the same focus metric hâve been proposed; one for pure amplitude objects and one for pure phase objects. We can represent the integrated complex amplitude by an invariant vector in the complex plane. This vector representing the integrated complex amplitude is a sum of vectors in the complex plane, each corresponding to a pixel of complex amplitude in the xy plane at a distance d along the z-axis. Figure 2.2 represents this invariant vector in the complex plane. The invariant modulus of this vector is defined by Eq. (2.22) and the invariant phase by Eq. (2.23) where the sampled notation Eq. (2.18) of the complex amplitude is used.
( 2 . 22 )
(2.23)
Fig. 2.2. Invariant complex vector representing the invariance of the integrated complex amplitude. Ak is the amplitude modulus and (pk the phase of the integrated complex amplitude.
For every plane perpendicular to the z-axis, if we plot on the complex plane of Fig.
2.2 the sampled complex amplitudes one by one starting from the point (0,0), a path
made of vectors is drawn and reaches the point P = (A^
cossin as a
conséquence of the invariance of the integrated complex amplitude. It results that
each reconstructed plane perpendicular to the z-axis can be represented by a path in
a complex plane starting at the point (0,0) to reach the point P. Figure 2.3 illustrâtes
this property with a green path representing an arbitrary reconstructed plane.
( 0 , 0 ) Real
Fig. 2.3. In red: invariant complex vector representing the invariance of the integrated complex amplitude. Ai is the modulus amplitude and pk the phase of the integrated complex amplitude; in green: path representing a complex amplitude in a reconstructed plane perpendicular to the z-axis.
The discretized expression Eq. (2.20) of the focus metric defined by Eq. (2.9) is the
measure of the length of the path for each reconstructed plane. For a pure amplitude
object, the focus criterion has been built with the property that the phase of the
complex amplitude is uniform when the focus plane of the object is reached and
corresponds to the minimum of the focus metric. Indeed, when the phase is uniform,
each (p^ in Eq. (2.23) are equal to (Pj^. This condition corresponds to the most
direct path from (0,0) to P in Fig. 2.3, and only in this case, the length of the path, so
the focus metric, is minimum. Also, for a pure phase object, we showed that the
focus metric is maximum when the object is almost transparent, which defines its
focus plane. The reconstructed plane along the z-axis which gives the longest path
from (0,0) to P corresponds to the focus plane of the studied pure phase object. As
the energy E is an invariant, the longest path imposes the condition that the £3^ are
equal as demonstrated by the solution of Eq. (2.21).
2.2.5. Précision of the focus criteria
A non focus object can be defined as an object situated in the upstream région of the lenses of the imaging System at a distance which doesn’t compensate the quadratic phase factor of the light induced by lenses [6]. In other words, for an imaging System composed of a unique lens, the imaging condition is not satisfied. The imaging condition is defined by \ja + 1/è — where a, b and/are respectively the upstream distance with respect to the lens, the downstream distance with respect to the lens and the focal distance of the lens as shown in Fig. 2.4. The distance a defines the position of the object plane and b the position of the image plane.
Fig. 2.4. imaging System composed of one lens.
