1. Detection theory 1. Definition of the problematic
2. Criterion for evaluating detection quality
3. Optimal detector : the likelihood ratio
4. Nuisance parameters : the GLRT
5. Application to edge detection in images

2. Lagrange multipliers
One searches points x* in which a scalar function of a vectorial parameter reaches an extremum
under the constraint that x verifies :
The solution of this problem is found in the following way :
One defines the « Lagrange function» :
One determines the extrema of this function, for example by annulling it gradient
One obtains solutions that depend on l :
The values of l that correspond to the solutions of the problem are those which verify the constraint:
The solutions are : 4 4 4 4 2 4

3. Lagrange multipliers Example : in the plane, maximize
under the constraint 4 4 4 4 2 4

4. 4 4 4 4 4 2

5. IV. Detection theory 1. Definition of the problematic
2. Criterion for evaluating detection quality
3. Optimal detector : the likelihood ratio
4. Nuisance parameters : the GLRT
5. Application to edge detection in images

6. Problematic Binary decision Multiple decision
Hypothesis 1 :
Object present
Hypothesis 0 :
Object absent
Binary decision
Multiple decision
Detection consists in choosing (making a decision) between two hypotheses -> hypothesis testing

7. In this course, one will only address the problem of detection
Problematic
In practice, one often encounters detection / localization problems
1) One sweeps the image with the « window » F
2) For each position (i,j) of the window, one performs a hypothesis test
H1: object present
in F
(i,j)
H0: no object
in F
In this course, one will only address the problem of detection

8. Decision theory
The measured signal is a vector s that belongs to a P-dimensional space W.
One assumes that it can belong to two classes go ou g1.
The goal of every detection algorithm is to partition the space W into 2 regions w0 et w1 so that the decision rule is: W w1 w0 x s 4 4 4 4 2 4

9. Decision theory
Equivalently, determining a detection algorithm consists in defining a « discriminant surface » between regions w1 et w0.
In practice, one obtains a discriminant function : function g(s) such that :
Discriminant surface W w1 w0
Question : what is the optimal discriminant function ? x s 4 4 4 4 4 2

10. Statistical detection theory
One models the signal s as a random vector -> statistical decision theory.
One knows the following information about the problem :
: a priori probability of class K
NB: « a priori » means « before performing the measure »
Ex : Radar detection :
Property :
: « Likelihood »
NB: probability of obtaining a given value of the vector if one knows its class 4 4 4 4 4 2

11. Example
Detection of a signal r perturbed by an additive white Gaussian noise with zero mean and variance s2.
g0 : signal absent / g1 : signal present
If the signal belongs to class g1 :
If the signal belongs to class g0 : 4 4 4 2 4 4

12. a posteriori probability
Bayes relations:
Joint probability density
a posteriori probability:
It is the probability that signal s belongs to class gk.
Useful relations: 4 4 4 4 2 4

13. Two types of detection errors
Probability of non-detection : W w0 w1
Probability of false alarm :
Can we minimize simultaneously Pnd et Pfa ? 4 4 4 4 4 2

14. Two types of detection errors
One can normalize these probabilities with respect to the a priori probability of each class :
One can also define the detection probability :
Objective : maximize Pd for a given value of Pfa 4 4 4 2 4 4

15. Lagrange multipliers
One searches points x* in which a scalar function of a vectorial parameter reaches an extremum
under the constraint that x verifies :
The solution of this problem is found in the following way :
One defines the « Lagrange function» :
One determines the extrema of this function, for example by annulling it gradient
One obtains solutions that depend on l :
To determine the values of l that correspond to the true solutions, one applies the constraint:
The solutions are : 4 4 4 4 2 4

16. Optimal detector Lagrange function: Optimal decision rule ? w1 w0 4 42 4 4

17. Optimal detector Optimal decision rule ? Likelihood ratio (LR) : avec4 4 4 2 4 4

18. Optimal detector
Detection of a signal r perturbed by an additive white Gaussian noise with zero mean and variance s2.
One compares the scalar product between the reference signal and the measured signal to a threshold. 4 4 4 4 2 4

19. Statistical distribution of the likelihood ratio
The likelihood ratio is a random variable.
Its probability density depends on whether the signal s belongs to class g0 or to class g1: 4 4 4 4 2 4

20. Statistical distribution of the likelihood ratio4 4 4 4 2 4

21. It is easier to work with
Example
Detection of a signal r perturbed by an additive white Gaussian noise with zero mean and variance s2.
It is easier to work with

22. Example
Detection of a signal r perturbed by an additive white Gaussian noise with zero mean and variance s2.
One obtains :

23. Performance of a detectorPd 1
Receiver Operating Characteristic
(ROC).
Pfa 1

24. Performance of a detector
Optimality of the LR : its ROC is above the ROC of any other detector
The ROC depends on two parameters : Pd LR 1
- The nature of the detector (the algorithm)
Other detector
- The intrinsic difficulty of the detection problem (RSB, …)
“Tossing a coin”
Pfa 1

25. Nuisance parameters
In each class, the signal may depend on unknown parameters pk. The expression of the likelihood is thus :
One estimates their values in the ML sense :
One defines the Generalized Likelihood Ratio (GLR) 4 4 4 2 4 4

26. Nuisance parameters
Example : detection in the presence of additive Gaussian noise
Nuisance parameter
ML estimate of the nuisance parameter :
Expression of the GLR : 4 4 4 2 4 4

27. Performance of the GLR Pd LR 1
GLR
“Tossing a coin”
Pfa 1

28. The article is to be handed in, PDF format, on October 28th.
Project
The goal of the project is
To study the resolution that can be reached by PALM imagery, and the optimal parameters of the imaging system parameter. To simplify, this study is to be done on 1 dimensional signals.
To implement the algorithm for reconstruction a higher resolution image from a sequence of images of isolated fluorophores, and apply it to a test image.
This work is to be reported under the form of a scientific journal paper. To this end, a template of the journal “Optics Express” is on Libres Savoirs (in MS Word and Latex).
The article is to be handed in, PDF format, on October 28th.

29. Project The features that must be reported in this work are:
Precision analysis (1D):
Analysis of the CRLB when parameter a is unknown, especially its variation with q.
Expression of the ML estimator, and estimation of its bias and efficiency
Optimal value of the PSF width w
Implementation of the algorithm (2D)
Algorithm for detection and rough estimation of the position of single fluorophores in an image.
Expression and implementation of ML position estimator in 2D.
Application of the method to the proposed test image.
The previous points are only guidelines. You are free order them as you want. The only requirement is that the “flow” of the paper is logical.

30. Project
Low resolution image

31. Project
PALM sequence

32. Project Data (available on Libres Savoirs)
ImageTest.mat: reference image for testing your algorithm. The PSF is an isotropic Gaussian pulse with the same FWHM w=2 in the two directions and maximal value a=1.
CoordinatesTest.mat: exact positions of the fluorophores in image ImageTest.mat
ImagesPALM.mat: sequence of 784 PALM images.
BlurredImage.png: low resolution image of the pattern to reconstruct from ImagesPALM.mat sequence.

33. Structure of a scientific paper
Title, authors, institutions, abstract
1. Introduction
Context, bibliography (literature review)
Description the addressed problem (why is this article worth publishing ?)
Outline of the paper
2. Derivation of the estimator -> ML estimator
3. Implementation of the algorithm -> optimization algorithm
4. Validation of the algorithm on simulated images
Influence of data and algorithm parameters on convergence and reconstructed image quality
5. Application of the algorithm to real data
6. Conclusion
Summary of main results
Perspectives (improvements, other application domains, …)
Appendices
Detail of calculations, technical details of an experiment, …
References : publications cited in the paper (numbered)

34. Reference (publication date !)
Title
Authors and institutions
Abstract
Context and bibliography
Introduction

35. Context and bibliography
Original contribution and content of the paper
Derivation of the estimator

36. Derivation of the estimator
Implementation of the algorithm

37. Validation of the algorithm on simulated images
Parameter study

38. Convergence of the algorithm
Quality of reconstruction

39. Quality of reconstruction
Results on experimental data
CONCLUSION

40. CONCLUSION
Results on experimental data
APPENDIX

41. APPENDIX
REFERENCES

42. Structure of a scientific paper
Figures
Always numbered and cited in the paper
Caption understandable without reading the main text

44. Structure of a scientific paper
Figures
Always numbered and cited in the paper
Caption understandable without reading the main text
Equations
Numbered, and all notations are explained

46. Structure of a scientific paper
Figures
Always numbered and cited in the paper
Caption understandable without reading the main text
Equations
Numbered, and all notations are explained
Bibliographical references
Always cited in the text
Always numbered
Respect the format of the review

48. These advices are also useful for your 3rd year internship report !
Tips for writing
Decide the outline of the paper before beginning to write :
What the important results I want to report ?
What are the figures I want to show ?
Start writing the core of the paper, then the introduction, the conclusion, and finally the abstract.
Length of the text is not proportional to the time spent doing the work !
The outline of the paper is chosen for clarity : it does not always follow the chronology of the work !
These advices are also useful for your 3rd year internship report !