Hidden Markov Chain Models and recent Extensions

Introduction

This web site gives access to a collection of programs, written in C++, I have implemented during the last years regarding the so-called Hidden Markov Chain (HMC) model for time-series analysis. Programs are concerned with the simulation, restoration and estimation of recent extensions of HMC models, e.g. Noise-independent HMC (the classical model as described by L. R. Rabiner [A1]), Pairwise Markov Chain (including HMC with correlated noise [B1, B2, B3])... See section Bibliography for a set of references about these models.

Most of programs have been developed from the work and with the collaboration of Prof. Wojciech Pieczynski. Bibliographical references regarding the underlying models (and many more!) can be found in its web pages. You can also check for draft papers on my personal pages.

This site is only intended to make programs available to everyone, in order to ensure a wide dissemination of research results in the scientific community. The code is made freely available for non-commercial use only, especially for research and teaching purposes, provided that suitable citation is made in published work using the functions and programs (please cite these pages). As a consequence, no responsibility is taken for any errors in the code (but bug report and request are always welcome).

The menu offers several options

Download

For users who have downloaded a previous release and wish to obtain the new functions, it is strongly recommended that the old version be deleted and the entire set downloaded again, to ensure compatibility.

All versions of programs can be accessed here.

What's new?

  >> ./simulHMC-IN

  >> ./simulHMC-IN -i InitHMC-IN.txt

  >> ./simulHMC-IN -i InitHMC-IN.txt -v 2

Installation

The programs are almost self-contained and only require the installation of

Programs have been developed in C++, and compile successfully with gcc version 4.3.2 (under Linux / Debian 4.3.2-1.1. You also need make for compilation). I cannot (and I do not want to!) check for correct installation in other distributions or systems. But programs are expected to work under any environment providing a recent C++ compiler. If you successfully install programs on specific platforms, please tell me.

Download and install

The last tarred and zipped source code can be found in Download section. For users who have downloaded a previous release and wish to obtain the new functions, it is strongly recommended that the old version be deleted and the entire set downloaded again, to ensure compatibility. After downloading, decompress and untar the functions:

  >> tar -xvzf HMC_Ext_???????.tgz

A directory called HMC_Ext will be created, in which you can find a Makefile and 6 directories:

For short, programs installation gives, first

  >> make lib
      ar: creating ./Bin/libHMC_Ext.a
      Library ./Bin/libHMC_Ext.a updated.

  >> make programs
      ...
      restoreHMC-IN done.
      simulHMC-IN done.
      ...

Test installation

To test correctness of your installation, execute command

  >> ./simulHMC-IN

in the Bin directory (no text output). This program generates a 2-classes Markov chain file named X_HMC-IN.txt and its noisy version named Y_HMC-IN.txt.

It is important to note that dafault values can be changed either by providing an initialisation file using ./simulHMC-IN -i InitFile.txt (see Tcl diretory for an example) or manually by specifying values for arguments such as ./simulHMC-IN -v 2 -Y essai.txt. See Help section for a complete explanation on argument values. All options not specified will take default values.

To restore data using both MPM and MAP Bayesian criterion, type command

  >> ./restoreHMC-IN -Y Y_HMC-IN.txt -F X_HMC-IN.txt
    5.3
    5.5

Argument for option -Y provides the file to be restored, whereas argument for option -F provides the original Markov chain (generated by the simulation program) for error rate computing after restoration (this second argument is not required ; in that case no error rate will be computed). The restored files are respectively saved under names Y_HMC-IN_Rest_MPM_HMC-IN.txt and Y_HMC-IN_Rest_MAP_HMC-IN.txt .

The two values show the MPM and MAP classification error rates. You can expect to get different values than those printed due to different stochastic simulation. Default parameters for restoration are the same that default parameters for simulation which justify these low error rates.

For more versatile programs, add option -v 1

  >> ./simulHMC-IN -v 1

   Simulation of a noisy Markov chain with independent noise

   *********************************
   ./simulHMC-IN [arguments] :
      -Chain length : 1000
      -Markov matrix:
   0.45	0.05
   0.05	0.45

      -Markov chain file name (X): X_HMC-IN.txt
      -Observations file name (Y): Y_HMC-IN.txt
      -Laws:
         - Gaussian    (8); mu1->mu4= +0.00,+1.00,+0.00,+3.00, beta1,2= +0.00 & +3.00
         - Gaussian    (8); mu1->mu4= +2.00,+1.00,+0.00,+3.00, beta1,2= +0.00 & +3.00
      -Verbose level   : 1
   *********************************

   Number of data with class 0 in the simulated chain : 464, i.e. 46.4%

   Time spend during following functions (in sec.):
     Simulation time......................................0

   End of - Simulation of a noisy Markov chain with independent noise

and

  >> ./restoreHMC-IN -Y Y_HMC-IN.txt -F X_HMC-IN.txt -v 1

   Restoration of a noisy Markov chain with independent noise

   *********************************
   ./restoreHMC-IN [arguments] :
      -Markov matrix:
   0.45	0.05
   0.05	0.45

      -Observation file name (Y) : Y_HMC-IN.txt
      -Supplied Markov chain file: X_HMC-IN.txt
      -Laws:
         - Gaussian    (8); mu1->mu4= +0.00,+1.00,+0.00,+3.00, beta1,2= +0.00 & +3.00
         - Gaussian    (8); mu1->mu4= +2.00,+1.00,+0.00,+3.00, beta1,2= +0.00 & +3.00
      -Verbose level   : 1
   *********************************

   The error rate for MPM is 5.6
   The error rate for MAP is 7

   Time spend during following functions (in sec.):
     Restoration time for MPM.............................0
     Restoration time for MAP.............................0

   End of - Restoration of a noisy Markov chain with independent noise

Spent time only counts for scientific computation, not for file reading and/or saving.

Also, you can use -h options, to get a list of available parameters for a given program. For example:

   ./simulHMC-IN [options]
     -n Chain lenght [>50]; Actual value=1000
     -Y Filename for simulated observations Y; Actual value=Y_HMC-IN.txt
     -X Filename for simulated Markov chain X; Actual value=X_HMC-IN.txt
     -m Joint Markov probabilities; Actual value=0.45:0.05:0.05:0.45
     -T Parameters for law #1: mean:var>0:beta1>=0:beta2>=1:sign in (+1,-1):law in [0=Gaussian; 1=Gamma; 2=8_2_7 in Pearson system; 3= case 2 + case 1; 4=all Pearson]; Actual value=0:1:0:3:1:0
     -U Parameters for law #1: mean:var>0:beta1>=0:beta2>=1:sign in (+1,-1):law in [0=Gaussian; 1=Gamma; 2=8_2_7 in Pearson system; 3= case 2 + case 1; 4=all Pearson]; Actual value=2:1:0:3:1:0
     -x Save graphic files to a specified directory
     -F Provide the 0/1 text file to be noisied => -m not required
     -v Verbose [0/1/2]; Actual value=0
     -h This help!

To get explanation on argument values for each program, see the lists reported in Help menu.

Help

Remark: Here, I only give some basic instructions regarding use of programs, and do not provide detailed documentation or a comprehensive software support service! News of any errors found however will be gratefully received.

This page provides explanations on all arguments you can provide to programs, e.g. -e or -Y. Arguments keep their meaning in all programs, so you should be able to quickly learn common argument options. It is always possible to pass argument -h to a program to get a comprehensive text list of possible arguments for that program. Each argument has a default value, which should be enough to make basic simulations.

There is 5 kinds of programs:

• Simulation programs, to generate a noise-free and a noisy sample from a Markov model,
• Restoration programs, to restore a noisy signal from model parameter values provided by user,
• Estimation+restoration programs, to restore a noisy signal from model parameter values estimated by using iterative routine (most of the time ICE Iterated Conditional Estimation, EM and SEM are available for estimation),
• Tool programs, to compute an error rate between two binary-valued files, or to get a graphic of a density from the Pearson' system of distributions or from a specific copula..., and
• Tcl-Tk GUI for command generation using simple graphical interfaces (but assuming you have installed Tcl and Tk).

Remark 1: Click on an argument to get information on it.

Remark 2: All simulation and restoration programs are limited to 2 classes (binary Markov chain). The only reason is the ease of use. Estimation programs have no such limitation. They can be used whatever the number of classes.

Remark 3: Program options are presented with their default value (hard-coded), which have been chosen to be the simplest as possible (e.g. always Gaussian margins and Gaussian copulas are chosen by default). Do not hesitate to modify the default choices by changing init value files or by providing specific values manually.

Simulation programs

Simulations programs refer to routines that simulate data according to a given Markov model. Generated data can be use with restoration and estimation programs to evaluate their capacity to deal with data following exactly the underlying model.

Simulation of data following an HMC-IN model

Three other options are available: -F, -x and -h.

Simulation of data following a PMC model

Simulation of PMC model can be degenerated to the simulation of HMC-CN (Correlated Noise) model using option -p (second example). In this case, only options -T and -U are taken into account (i.e. not options -t and -u).

A list of available copulas can be printed using ./simulPMC -z. If selected copulas require parameters (e.g. Gaussian copula require correlations coefficients), then values are asked to the user during program execution. Please note that copula and parameters for (xn=0,xn+1=1) and (xn=1,xn+1=0) should be the same, due to some symmetry condition required by the model. Instead, you can provide the Kendall's tau using option -k (e.g. -k 0.1:0.4:0.4:0.7). Kendall's tau belongs to [-1,1] (such as correlation coefficient) but this range is not valid for all copulas (./simulPMC -z provides the correct range for each copula).

Three other options are available: -F, -x and -h. Option -F is used to specify the binary chain to be noisied (so that only observation are simulated).

Restoration programs

The restoration programs deal with the restoration of a noisy signal according to a Markov model for which user provides parameter values. The programs apply, in most case, bot MPM and MAP criteria, and save the restored chain on the disk. Error rate is computed only if the original data to be compared with the restored signal is provided using -F argument (you can always compute the error rate between text files using the erroRate tool program, see below).

Restoration using an HMC-IN model from provided parameters

Four other options are available: -F, -x, -h and -X. Option -X is used to provide the base name of the MAP and MPM restored chain. If it is not provided, the name of the input file is used for the output files base name. Option -F allows to specify the original binary file in order for the program to compute the error rate between this file and the result of restoration.

Restoration using a PMC model from provided parameters

Restoration of PMC model can be degenerated to the restoration of HMC-CN (Correlated Noise) model using option -p (second example). In this case, only options -T and -U are taken into account (i.e. not options -t and -u).

A list of available copulas can be printed using option -z. If selected copulas require parameters (e.g. Gaussian copula require correlations coefficients), then values are asked to the user during program execution. Please note that copula and parameters for (xn=0,xn+1=1) and (xn=1,xn+1=0) should be the same, due to some symmetry condition required by the model.

Four other options are available: -F, -x, -h and -X. Option -X is used to provide the base name of the MAP and MPM restored chain. If it is not provided, the name of the input file is used for the output files base name. Option -F allows to specify the original binary file in order for the program to compute the error rate between this file and the result of restoration.

Estimation and Restoration programs

The estimation and restoration programs deal with the restoration of a noisy signal according to a Markov model for which parameters are estimated using an iterative procedure (most of the time, the ICE procedure is used, but sometimes EM and SEM are also proposed as alternatives or for comparison purposes). The program applied, in most case, bot MPM and MAP criteria, and save the restored chain on the disk. Error rate is computed only if the original data to be compared with the restored signal is provided using -F argument (you can always compute the error rate between text files using the erroRate tool program, see below).

Estimation and restoration using an HMC-IN model

Four other options are available: -F, -x, -h and -X. Option -X is used to provide the base name of the MAP and MPM restored chain. If it is not provided, the name of the input file is used for the output files base name. Option -F allows to specify the original binary file in order for the program to compute the error rate between this file and the result of restoration.

Estimation and restoration using an HMC-IN model (sub-sampling estimation)

Option -s is used to define the width of the 1D window around the pixel of interest. Option -b defines the way sub-sampling is performed. Option -P provides the percent of data to be used when option b is set to 1 or 2 (for option 3, the percent is computed automatically). Note that the default estimation method is EM here. Indeed, SEM and ICE require the simulation of a posteriori chain which can only be achieved by sub-optimal realization, called ``marginal realization'' (according to the method described in [A2]).

Four other options are available: -F, -x, -h and -X. Option -X is used to provide the base name of the MAP and MPM restored chain. If it is not provided, the name of the input file is used for the output files base name. Option -F allows to specify the original binary file in order for the program to compute the error rate between this file and the result of restoration.

Estimation and restoration using a PMC model

Estimation of PMC model can be degenerated to the estimation of HMC-CN (Correlated Noise) model using option -p (second example). In this case, options -l and -c always take a list of K integers (K in the number of classes set with option -K), whereas the true PMC model requires K^2 integers.

A list of available copulas can be printed using option -z

Four other options are available: -F, -x, -h and -X. Option -X is used to provide the base name of the MAP and MPM restored chain. If it is not provided, the name of the input file is used for the output files base name. Option -F allows to specify the original binary file in order for the program to compute the error rate between this file and the result of restoration.

Tool programs

Here is a collection of programs not directly related to Markov modeling but of some interest for display, error rate computation, use of images...

How to convert a pgm image into a text file for latter use?

The image is decomposed by using the Peano scan [A15]. The name of the generated text file is image.pgm.txt.

One other option is available: -h.

Conversely, how to convert a txt file into a pgm image (reconstruction after segmentation)?

The Tutorial #3 shows how to use this two programs. The image is reconstructed by using the inverse Peano scan. The name of the generated pgm image is image.pgm.txt.pgm.

One other option is available: -h.

Error rate between two binary files

One other option is available: -h.

Draw one or two densities from the Pearson system of distributions

One other option is available: -h. By default, option -x is set to ./.

Draw a 2D pdf from the specification of its 2 margins (within Pearson' system) and its copula

This program draws a 2D pdf built with a Student copula (correlation coeff is specified during program execution) with 2 degrees of freedom (fixed in the program), with a Gaussian margin (as specified by -T option) and a second kind Beta margin (as specified by -U option). It takes a few seconds to generate the figures. By default, option -x is set to ./. To avoid question about parameters during execution time, you can specify Kendall's Tau using option -z.

The example above gives the following display:

   Draw of a density with a specified copula and margins

   Please set the parameter for the Gumbel-Hougaard copula in [1;+infinity[ (but preferably in [1;20]): 4

   *********************************
   ./drawCopula2D [arguments] :
      ->Copula : Gumbel-Hougaard - Theta=4
      ->Margins
         - Beta 2      (6); mu1->mu4= +11.00,+6.00,+12.30,+154.80, beta1,2= +0.70 & +4.30
         - Type V      (5); mu1->mu4= +10.00,+5.00,+19.36,+237.50, beta1,2= +3.00 & +9.50
      -Verbose level    : 1
      -Graphic files to : .
   *********************************
   End of - Draw of a density with a specified copula and margins

and generates the following figures (using Gnuplot. Plplot provides similar black and white draws):

Two other options are available: -h and -z.

Tcl-Tk

To generate command lines to call programs, we have developped, we have developped 4 GUIs for the 4 programs regarding the HMC-IN model. Special thanks to Dhouha Saadealloui, ENSI, Tunis. GUIs for programs about the PMC model is scheduled for next release.

To execute the Tcl script simulHMC-IN.tcl, you can type in the Tcl directory

The default value are the hard-coded ones. If you want to set the default values using an initialisation file you can type

  >> ./simulHMC-IN.tcl InitHMC-IN.txt
  >> ./restoreHMC-IN.tcl InitHMC-IN.txt
  >> ./estimHMC-IN.tcl InitHMC-IN.txt
  >> ./estimHMC-IN_BL.tcl InitHMC-IN.txt

for programs related to the HMXC-IN model and

  >> ./simulPMC.tcl InitPMC.txt
  >> ./restorePMC.tcl InitPMC.txt
  >> ./estimPMC.tcl InitPMC.txt

The GUI for this programs looks like this

Choose the desired parameters and then Validate. The corresponding command line appears in the text box. Finally, "copy and paste" the command line in the Bin directory to run the program with parameters specified in the GUI.

Table of program arguments

Tutorial

Here is a collection of, simple or more sophisticated, examples to illustrate possible uses of programs. This list will growth each time a new model (or variant) is integrated.

Some reported plots are generated using Octave functionalities, from the saved text files. Octave scripts are provided in the Octave directory. Calls to scripts are also reported for ease of reproduction.

Other plots are directly generated by Gnuplot during execution using -x option (assuming that programs have been compiled with Gnuplot enable, see Installation section).

List of tutorials:

• Tutorial #1: Classical HMC model with independent noise but non-Gaussian laws.
• Tutorial #2: Illustration of "PMC model is strictly more general than HMC-CN, which is itself strictly more general than HMC-IN".
• Tutorial #3: How to segment a huge image with bootstrap estimation of the HMC model?

Tutorial #1: Classical HMC model with independent noise and non-Gaussian laws

Simulation of an HMC-IN mixture of two gamma laws

and estimation using 100 ICE iterations

This gives MPM and MAP error rates of 11.25% and 11.89% respectively. Next figure shows the simulated Markov chain (up), the simulated observed data (middle) and the ICE estimation and MPM restoration (down).

Here is the call to script Example1.m used to generate this figure (from Octave directory):

The first two arguments define the extent of the window to be plotted (samples index between 200 and 450).

The -x option can be used to set the repository where graphic files (with .eps extension) will be saved. For program ./estimHMC-IN, we get the one figure which superposes the histogram of classes and estimated densities after classification.

After 100 ICE iterations, estimated parameters for the two laws are

  - Gamma (3); mu1,mu2,mu3,mu4: +0.02,+1.05,+1.05,+4.88,  beta1,2 = +0.96 & +4.43 (Sample size: 4958)
  - Gamma (3); mu1,mu2,mu3,mu4: +2.01,+2.06,+4.00,+24.41, beta1,2 = +1.82 & +5.74 (Sample size: 5042)

whereas Markov joint a priori matrix is given by

	0.401937  0.0938484
	0.09378	  0.410434

with a priori probabilities:

	0.495786
	0.504214

and transition matrix:

	0.810708  0.189292
	0.185992  0.814008

Estimated parameters are close to the parameters used for simulation.

Tutorial #2: "PMC model is strictly more general than HMC-CN, which is itself strictly more general than HMC-IN"

Here is a set of experiments that illustrates the sentence, by first generating a PMC signal and then restoring it with PMC, HMC-CN and HMC-IN models.

We first generate a signal of 100000 samples following a PMC model with default parameters for margins (i.e. Gaussians) and copula (i.e. Gaussians):

Correlation parameters for the 4 copulas are respectively set to 0.2 for (xn=0,xn+1=0), 0.5 for (xn=0,xn+1=1), 0.5 for (xn=1,xn+1=0) and 0.7 for (xn=1,xn+1=1). Supplementary option -v 2 gives the following display:

 - Markov parameters :
  0.4 0.1
  0.1 0.4
 - Laws
   (0,0)
   ->Copula : Gaussian -
     1    0.2
     0.2  1
   ->Margins
   - Gaussian; mu1,mu2,mu3,mu4 : +0.00,+1.00,+0.00,+3.00, beta1,2 = +0.00 +3.00
   - Gaussian; mu1,mu2,mu3,mu4 : +0.00,+1.00,+0.00,+3.00, beta1,2 = +0.00 +3.00
   (0,1)
   ->Copula : Gaussian -
     1    0.5
     0.5  1
   ->Margins
   - Gaussian; mu1,mu2,mu3,mu4 : +0.30,+1.00,+0.00,+3.00, beta1,2 = +0.00 +3.00
   - Gaussian; mu1,mu2,mu3,mu4 : +1.70,+1.00,+0.00,+3.00, beta1,2 = +0.00 +3.00
   (1,0)
   ->Copula : Gaussian -
     1    0.5
     0.5  1
   ->Margins
   - Gaussian; mu1,mu2,mu3,mu4 : +1.70,+1.00,+0.00,+3.00, beta1,2 = +0.00 +3.00
   - Gaussian; mu1,mu2,mu3,mu4 : +0.30,+1.00,+0.00,+3.00, beta1,2 = +0.00 +3.00
   (1,1)
   ->Copula : Gaussian -
     1    0.7
     0.7  1
   ->Margins
    - Gaussian; mu1,mu2,mu3,mu4 : +2.00,+1.00,+0.00,+3.00, beta1,2 = +0.00 +3.00
    - Gaussian; mu1,mu2,mu3,mu4 : +2.00,+1.00,+0.00,+3.00, beta1,2 = +0.00 +3.00

The two generated files X_PMC.txt and Y_PMC.txt contains the simulated PMC.

We next restore the signal with the PMC, the HMC-CN and the HMC-IN models (when required we set the correlation parameters with the values above):

which gives a MPM error rate of 10.33% and a MAP-one of 10.88%.

which gives a MPM error rate of 10.64% and a MAP-one of 11.47%.

which gives a MPM error rate of 13.23% and a MAP-one of 13.49% (we must specify option -Y since, by default, the program looks for a file named Y_HMC-IN.txt).

Difference between restoration rates can be accentuated by playing with parameters (e.g. Markov matrix, correlation value for copulas and non Gaussian laws). To confirm the higher generality of the PMC model vs HMC-IN ones, we can do the inverse experiment: simulating an HMC-IN signal and restoring it with the three models (setting all correlation parameters to 0): error rates are very similar.

Using option -x ./ in restorePMC program generates postscript figures of marge densities and copula densities (of course, if Gnuplot is installed). For (xn=1,xn+1=1), we get the following draws:

Signals can be observed with a call to octave script Example1.m :

we get

Tutorial #3: "How to segment a huge image with bootstrap estimation of the HMC model?"

Experiments are conducted with the free image here. The image is sized 2000x3008. Here is a snapshot of the original image:

Original image (256 gray-levels)

The fist step is to convert the pgm image into a text file using the pgm2Txt program:

This way you get a file named img.pgm.txt. It takes about one minute for my PC to convert the image.

Then you can segment the image according to this call (4 classes):

  Time spend during following functions (in sec.):
    Estimation time (EM)...............................476
    Restoration time for EM-MPM.........................42
    Restoration time for EM-MAP..........................10

It is worth noting that trying to use ./estimHMC-IN or ./estimPMC will conduct to memory overflow. According to the technique used for sub-sampling, the number of pixels that participates to parameters estimation is 1609 (i.e. 0.027%). After 50 EM iterations, you get two text files which contain the MAP and MPM segmentations: img.pgm_EM_MPM_HMC-IN_BL.txt and img.pgm_EM_MAP_HMC-IN_BL.txt.

It is then required to reconstruct the two segmentations:

Please note that images can appear black since each pixel contains the class number (0,1,2,3,...). To stretch the class numbers to visible values (between O and 255), you can use the Color/Auto menu in GIMP software for example.

Here is the results I get for MAP segmentation (dimensions of images have been reduced):

HMC (local EM estimation and MAP) segmentation (4 classes) - 50 iterations, 8 minutes

HMC (EM estimation and MAP) segmentation (4 classes) - 50 iterations, 47 minutes

Blind (EM/MAP) segmentation (4 classes) - 200 iterations, XX minutes

Kmeans segmentation (4 classes) - 12 iterations, a few seconds

Bibliography

Here is a collection of key bibliographical references cited inside these pages. It should not be considered as an exhaustive bibliography! These papers are of particular interest to readers interested in theoretical justifications. When available, a link is provided to download preprint.

HMC and variants, Pearson, copulas & generalized mixtures, ICE principle, Peano scan...

• [A1] L. R. Rabiner, A tutorial on hidden Markov model and selected applications in speech recognition, Proc. of the IEEE, 77(2), 257-286, Feb. 1989. Link
• [A2] B. Benmiloud, W. Pieczynski, Estimation des paramètres dans les chaînes de Markov cachées et segmentation d'images, Traitement du Signal, 12(5), 433-454, 1995. Link
• [A3] N. Giordana, W. Pieczynski, Estimation of generalized multisensor hidden Markov chains and unsupervised image segmentation, IEEE Trans. on Pattern Analysis and Machine Intelligence, 19(5), 465-475, 1997. Link
• [A4] W. Pieczynski, Champs de Markov cachés et estimation conditionnelle itérative, Traitement du Signal, 11(2), 141-153, 1994. Link
• [A5] W. Pieczynski, Sur la convergence de l'estimation conditionnelle itérative, Comptes Rendu de l'académie des Sciences-Mathématique, 346(7-8), 457-460, Avril 2008. Link
• [A6] J.-P. Delmas, An equivalence of the EM and ICE algorithm for exponential family, IEEE Trans. Signal Processing, 45(10), 2613-2615, 1997.Link
• [A7] W. Pieczynski, J. Bouvrais, C. Michel, Estimation of generalized mixture in the case of correlated sensors, IEEE Trans. on Image Processing, 9(2), 308-311, 2000. Link
• [A8] Y. Delignon, A. Marzouki, W. Pieczynski, Estimation of generalized mixture and its application in image segmentation, IEEE Trans. on Image Processing, 6(10), 1364-1375, 1997. Link
• [A9] S. Derrode, G. Mercier, Multiscale oil slick segmentation from SAR image using a vector HMC model, Pattern Recognition, 40(3), 1135-1147, March 2007. Link
• [A10] N. L. Johnson, S. Kotz, Distribution in statistics: Continuous univariate distributions, Vol. 1 and 2. New York. John Wiley and Sons, 1994.
• [A11] S. Derrode, Description du système de Pearson pour son implémentation en langage C à l'aide de la librairie GSL. Rapport interne (in French), Ecole Centrale Marseille & Institut Fresnel, 2009. Link
• [A12] T. Roncalli, Gestion des Risques Multiples ou Copules et Aspects Multidimensionnels du Risque. Cours ENSAI de 3ième année (in French), Groupe de Recherche Operationnelle - Crédit Lyonnais, 233 pages. Link
• [A13] G. Mercier, Mesures de dépendance entre images RSO. Rapport de recherche (in French), RR-2005003-ITI, Octobre 2005. Télécom Bretagne & TAMCIC (CNRS UMR 2872), Equipe TIME, 2009. Link
• [A14] L. Benyoussef, C. Carincotte, S. Derrode, Extension of higher-order HMC modeling with application to image segmentation, Digital Signal Processing, Vol. 18(5), pp. 849-860, 2008, doi: 10.1016/J.dsp.2007.10.010. Link
• [A15] W. Skarbek, Generalized Hilbert scan in image printing, Theoretical Foundations of Computer Vision, p . 45-57, R. Klette et W. G. Kropetsh Ed., Akademik Verlag, 1992.
• [A16] S. Derrode, L. Benyoussef and W. Pieczynski, Sub-sample-based HMC estimation with application to large data sets segmentation, submitted, 2010.

Pairwise Markov chain and HMC with correlated noise

• [B1] N. Brunel, W. Pieczynski, Unsupervised signal restoration using hidden Markov chains with copulas, Signal Processing, 85(12), 2304-2315, 2005. Link
• [B2] S. Derrode, W. Pieczynski, Signal and image segmentation using pairwise Markov chains, IEEE Trans. on Signal Processing, 52(9), 2477-2489, 2004. Link
• [B3] W. Pieczynski, Pairwise Markov chains, IEEE Trans. on Pattern Analysis and Machine Intelligence, 25(5), 634-639, 2003. Link
• [B4] N. Brunel, Sur quelques extensions des chaînes de Markov cachées et couples. Application à la segmentation non supervisée des signaux radar., thèse de l'Université Paris VI (in French), soutenue le 5 décembre 2005. Encadrants: Frédéric Barbaresco (Thales Air Defense), Paul Deheuvels (Paris VI, LSTA), et Wojciech Pieczynski CITI, INT. Link

To come in a short (and not so) future!

To do list (short term) - before publishing next release (version: 0.5)

• Develop a set of programs (simulation, restoration and estimation-restoration) to deal with i.i.d. (i.e. non-Markovian) data.

To do list (middle term) - after the next release (for version: 0.6)

• M-stationary triplet Markov chain

To do list (long term)

• Fuzzy HMC model
• Higher-order HMC model
• Long Memory HMC model