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

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

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

4 4 4 4 4 2

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

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

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

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

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

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

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

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

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

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

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

Optimal detector Lagrange function: Optimal decision rule ? w1 w0 4 4 2 4 4

Optimal detector Optimal decision rule ? Likelihood ratio (LR) : avec 4 4 4 2 4 4

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

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

Statistical distribution of the likelihood ratio 4 4 4 4 2 4

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

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

Performance of a detector Pd 1 Receiver Operating Characteristic (ROC). Pfa 1

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

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

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

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

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.

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.

Project Low resolution image

Project PALM sequence

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.

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)

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

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

Derivation of the estimator Implementation of the algorithm

Validation of the algorithm on simulated images Parameter study

Convergence of the algorithm Quality of reconstruction

Quality of reconstruction Results on experimental data CONCLUSION

CONCLUSION Results on experimental data APPENDIX

APPENDIX REFERENCES

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

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

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

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 !