Bhattacharyya distance as a contrast parameter for statistical processing of noisy optical images

François Goudail; Philippe Réfrégier; Guillaume Delyon

doi:10.1364/JOSAA.21.001231

Journal of the Optical Society of America A
Vol. 21,
Issue 7,
pp. 1231-1240
(2004)
•https://doi.org/10.1364/JOSAA.21.001231

Bhattacharyya distance as a contrast parameter for statistical processing of noisy optical images

François Goudail, Philippe Réfrégier, and Guillaume Delyon

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François Goudail, Philippe Réfrégier, and Guillaume Delyon, "Bhattacharyya distance as a contrast parameter for statistical processing of noisy optical images," J. Opt. Soc. Am. A 21, 1231-1240 (2004)

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Abstract

In many imaging applications, the measured optical images are perturbed by strong fluctuations or noise. This can be the case, for example, for coherent-active or low-flux imagery. In such cases, the noise is not Gaussian additive and the definition of a contrast parameter between two regions in the image is not always a straightforward task. We show that for noncorrelated noise, the Bhattacharyya distance can be an efficient candidate for contrast definition when one uses statistical algorithms for detection, location, or segmentation. We demonstrate with numerical simulations that different images with the same Bhattacharyya distance lead to equivalent values of the performance criterion for a large number of probability laws. The Bhattacharyya distance can thus be used to compare different noisy situations and to simplify the analysis and the specification of optical imaging systems.

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Law	Pdf	Free Parameters	Fixed Parameters (if any)
Bernoulli	$p δ (x) + (1 - p) δ (1 - x)$	p	None
Poisson	$\sum_{n \in N} δ (x - n) exp (- λ) \frac{λ^{n}}{n!}$	λ	None
Gamma	${(\frac{L}{m})}^{L} \frac{x^{L - 1}}{Γ (L)} exp [- \frac{L}{m} x]$	m	L
Weibull	$\frac{μ x^{μ - 1}}{m^{μ}} exp [- {(\frac{x}{m})}^{μ}]$	m	μ
Gaussian	$\frac{1}{\sqrt{2 π} σ} exp [- \frac{(x - m)^{2}}{2 σ^{2}}]$	m , σ	None
Geometric	$\sum_{n \in N} δ (x - n) p (1 - p)^{n}$	p	None

Law	$F$	$K$	$B$
Bernoulli	$\frac{(p_{a} - p_{b})^{2}}{p_{a} q_{a} + p_{b} q_{b}}$	$(p_{a} - p_{b}) ln [\frac{p_{a} q_{b}}{p_{b} q_{a}}]$	$- ln [\sqrt{p_{a} p_{b}} + \sqrt{q_{a} q_{b}}]$
Poisson	$\frac{(λ_{a} - λ_{b})^{2}}{λ_{a} + λ_{b}}$	$(λ_{a} - λ_{b}) ln [\frac{λ_{a}}{λ_{b}}]$	$\frac{1}{2} (\sqrt{λ_{a}} - \sqrt{λ_{b}})^{2}$
Gamma	$L \frac{(r - 1)^{2}}{r^{2} + 1}$	$L {(\sqrt{r} - \frac{1}{\sqrt{r}})}^{2}$	$L ln [\frac{1}{2} (\sqrt{r} + \frac{1}{\sqrt{r}})]$
Weibull	$A \frac{(r - 1)^{2}}{r^{2} + 1}$	${[(\sqrt{r})^{μ} - {(\frac{1}{\sqrt{r}})}^{μ}]}^{2}$	$ln \{\frac{1}{2} [(\sqrt{r})^{μ} + {(\frac{1}{\sqrt{r}})}^{μ}]\}$
Gaussian	$\frac{β}{1 + α}$	$\frac{1}{2} \frac{β (1 + α)}{α} + \frac{1}{2} {(\sqrt{α} - \frac{1}{\sqrt{α}})}^{2}$	$\frac{1}{4} \frac{β}{1 + α} + \frac{1}{2} ln [\frac{1}{2} (\sqrt{α} + \frac{1}{\sqrt{α}})]$
Geometric	$\frac{(p_{a} - p_{b})^{2}}{q_{a} {p_{b}}^{2} + q_{b} {p_{a}}^{2}}$	$\frac{p_{b} - p_{a}}{p_{a} p_{b}} ln (\frac{q_{a}}{q_{b}})$	$ln [\frac{1 - \sqrt{q_{a} q_{b}}}{\sqrt{p_{a} p_{b}}}]$

Pdf	$J (x, w)$	$f (z)$
Bernoulli	$N_{a} f (\hat{m_{a}}) + N_{b} f (\hat{m_{b}})$	$- z ln z - (1 - z) \times ln (1 - z)$
Poisson	$N_{a} f (\hat{m_{a}}) + N_{b} f (\hat{m_{b}})$	$- z ln z$
Gamma	$N_{a} f (\hat{m_{a}}) + N_{b} f (\hat{m_{b}})$	$L ln z$
Gaussian^b	$N_{a} f (\hat{{σ_{a}}^{2}}) + N_{b} f (\hat{{σ_{b}}^{2}})$	$1 / 2 ln z$
Gaussian with identical variances^c	$Nf (\hat{σ^{2}})$	$1 / 2 ln z$

Law	Pdf	Free Parameters	Fixed Parameters (if any)
Bernoulli	$p δ (x) + (1 - p) δ (1 - x)$	p	None
Poisson	$\sum_{n \in N} δ (x - n) exp (- λ) \frac{λ^{n}}{n!}$	λ	None
Gamma	${(\frac{L}{m})}^{L} \frac{x^{L - 1}}{Γ (L)} exp [- \frac{L}{m} x]$	m	L
Weibull	$\frac{μ x^{μ - 1}}{m^{μ}} exp [- {(\frac{x}{m})}^{μ}]$	m	μ
Gaussian	$\frac{1}{\sqrt{2 π} σ} exp [- \frac{(x - m)^{2}}{2 σ^{2}}]$	m , σ	None
Geometric	$\sum_{n \in N} δ (x - n) p (1 - p)^{n}$	p	None

Law	$F$	$K$	$B$
Bernoulli	$\frac{(p_{a} - p_{b})^{2}}{p_{a} q_{a} + p_{b} q_{b}}$	$(p_{a} - p_{b}) ln [\frac{p_{a} q_{b}}{p_{b} q_{a}}]$	$- ln [\sqrt{p_{a} p_{b}} + \sqrt{q_{a} q_{b}}]$
Poisson	$\frac{(λ_{a} - λ_{b})^{2}}{λ_{a} + λ_{b}}$	$(λ_{a} - λ_{b}) ln [\frac{λ_{a}}{λ_{b}}]$	$\frac{1}{2} (\sqrt{λ_{a}} - \sqrt{λ_{b}})^{2}$
Gamma	$L \frac{(r - 1)^{2}}{r^{2} + 1}$	$L {(\sqrt{r} - \frac{1}{\sqrt{r}})}^{2}$	$L ln [\frac{1}{2} (\sqrt{r} + \frac{1}{\sqrt{r}})]$
Weibull	$A \frac{(r - 1)^{2}}{r^{2} + 1}$	${[(\sqrt{r})^{μ} - {(\frac{1}{\sqrt{r}})}^{μ}]}^{2}$	$ln \{\frac{1}{2} [(\sqrt{r})^{μ} + {(\frac{1}{\sqrt{r}})}^{μ}]\}$
Gaussian	$\frac{β}{1 + α}$	$\frac{1}{2} \frac{β (1 + α)}{α} + \frac{1}{2} {(\sqrt{α} - \frac{1}{\sqrt{α}})}^{2}$	$\frac{1}{4} \frac{β}{1 + α} + \frac{1}{2} ln [\frac{1}{2} (\sqrt{α} + \frac{1}{\sqrt{α}})]$
Geometric	$\frac{(p_{a} - p_{b})^{2}}{q_{a} {p_{b}}^{2} + q_{b} {p_{a}}^{2}}$	$\frac{p_{b} - p_{a}}{p_{a} p_{b}} ln (\frac{q_{a}}{q_{b}})$	$ln [\frac{1 - \sqrt{q_{a} q_{b}}}{\sqrt{p_{a} p_{b}}}]$

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Equations (18)

Journal of the Optical Society of America A