This paper derives new closed-form expressions for the masses of negative multinomial distributions. These masses can be maximized to determine the maximum likelihood estimator of its unknown parameters. An application to polarimetric image processing is investigated. We study the maximum likelihood estimators of the polarization degree of polarimetric images using different combinations of images.
The univariate negative binomial distribution is uniquely defined in many statistical textbooks. However, extensions defining multivariate negative multinomial distributions (NMDs) are more controversial. Most definitions are based on the probability generating function (PGF) of these distributions. Doss [
The family of NMDs introduced in [
As a first goal of this paper, we propose a way of computing the masses of multivariate NMDs
This paper studies the maximum likelihood estimators (MLEs) of the square DoP based on two or three polarimetric images. These estimators are computed by maximizing the masses of bivariate or trivariate NMDs derived in the first part of this work.
The paper is organized as follows. Section
An
In this section, we derive new expressions for the coefficients
Denote
In the trivariate case defined by
Let
The masses of NMDs can be directly obtained from this theorem. The particular cases of bivariate and trivariate NMDs are considered in the following subsections since the corresponding masses will be useful in the application considered in the second part of this paper.
Consider the affine polynomial of order 2 with variables
The result
Consider the affine polynomial with the three variables
The state of the polarization of the light can be described by the random behavior of a complex vector
This paper considers practical applications where the intensity level of the reflected light is very low (low flux assumption), which leads to an additional source of fluctuations on the detected signal. Under the low flux assumption, the quantum nature of the light leads to a Poisson-distributed noise which can become very important relatively to the mean value of the signal at a low photon level. As a consequence, the observed pixels of the low flux polarimetric image are discrete random variables contained in the vector
The joint distribution of the intensity vector
The PGF of
The ML estimator of
The PGF of
The MLE of
The performance of the ML estimators of the square DoP based on two or three polarimetric images has been evaluated via several experiments. The first simulations compare the log Mean Square Errors (MSEs) of the square DoP estimators constructed from two or three images. Eleven different covariance matrices of the Jones vector have been considered in order to define typical values of the DoP. The values of
Covariance matrix elements and square DoP values for the Jones vector.
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Figure
Simulation results for the estimation of
2 images MLE | 3 images MLE | |||||||
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Bias | std | MSE | avar | Bias | std | MSE | avar | |
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Low flux
High flux
In order to appreciate the influence of the Poisson noise due to the low flux assumption, experiments have been conducted using the high flux assumption. In this case, the intensity vector
The next set of simulations studies the performance of the different estimators as a function of the sample size
Simulation results for the estimation of
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2 images MLE | 3 images MLE | ||||||
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Bias | std | MSE | avar | Bias | std | MSE | avar | |
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Simulation results for the estimation of
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2 images MLE | 3 images MLE | ||||||
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Bias | std | MSE | avar | Bias | std | MSE | avar | |
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Matrix
Matrix
Figures
Matrix
Matrix
In order to appreciate the estimation performance on polarimetric images, we consider a synthetic polarimetric image of size
Polarimetric properties of elements that compose the scene displayed in Figure
Object | Polarization matrix |
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Remarks |
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Background |
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Very depolarizing and dark background |
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Very bright and weakly depolarizing object (typically steel) |
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Bright object quite depolarizing |
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Dark object whose mean total intensity is the same as the background |
Composition of the scene used to generate synthetic polarimetric low flux images and associated theoretical squared DoP.
Scene
Theoretical squared DoP
Synthetic intensity images (negative colors) for the scene depicted in Figure
Low flux intensity
Low flux intensity
Low flux intensity
Low flux intensity
Estimates of
MLE
MLE
Finally, the ML estimator based on three images is applied on real polarimetric data. These images are acquired by using a laser as a coherent illumination source. The scene consists of two disks. The first one, intended to provide low DoP, is a grey diffuse material (left object in Figure
Real-world polarimetric intensity images of a scene composed of a plastic disk (left) and a steel disk (right).
Low flux intensity
Low flux intensity
Low flux intensity
Low flux intensity
Figure
Estimates of
The set of affine polynomials with real coefficients and variables Substituting Substituting Without loss of generality, if
The relation (
Denote
The relation (
The authors would like to thank Gérard Letac for fruitful discussions regarding multivariate gamma distributions and negative multinomial distributions. The Authors are also grateful to Mehdi Alouini for acquiring and providing them the polarimetric images.