Bayesian Fusion of Ensemble of Multifocused Noisy Images

Abstract

This paper addresses the problem of simultaneous fusion and denoising of an ensemble of multifocused noisy source images using statistical approach. The central theme of the paper is to develop a novel generalized Bayesian framework based on maximum a posteriori (MAP) estimation technique to obtain the fused image from the noisy observations using a multiscale wavelet transform. A mathematically tractable multivariate a priori function is used in the MAP estimator to derive the closed-form expression of the fusion rule for the wavelet coefficients of noisy images. Experiments are carried out on a number of test-sets having an ensemble of multifocused source images with varying noise strengths to evaluate the performance of the proposed MAP-based fusion method as compared to the existing methods. Results show that the performance of the proposed method is better than that of the other wavelet or principal component analysis-based methods in terms of various metrics such as the structural similarity, peak signal-to-noise ratio and cross-entropy, uses of which are common both in the areas of fusion and denoising. In addition, the proposed method yields excellent results in terms of visual quality even in the case of non-Gaussian noise as well as computational load.

Appendix

The joint PDF of $X_1$, $X_2$, and $X_{\text {f}}$ is given by [33]

$$\begin{aligned}&p_{X_1,X_2,X_{\text {f}}}\big (x_1,x_2,x_{\text {f}}\big ) =\frac{1}{\sqrt{8\pi ^3\sigma _{11}\sigma _{22}\sigma _{\text {ff}}\gamma }}\nonumber \\&\quad \exp \Bigg [-\frac{1}{2\gamma }\bigg \{\alpha _1 x_1^2+ \alpha _2 x_2^2 +\alpha _3 x^2_{\text {f}}-2\beta _1x_1x_2-2\beta _2x_1x_{\text {f}}- 2\beta _3x_2x_{\text {f}}\bigg \}\Bigg ] \end{aligned}$$

(21)

Since the noise is uncorrelated with the reference images as well as the fused image

$$\begin{aligned} p_{X_1,X_2,N,X_{\text {f}}}\left( x_1,x_2,n,x_{\text {f}}\right)&=p_{X_1,X_2,X_{\text {f}}}\left( x_1,x_2,x_{\text {f}}\right) p_{N}\big (n\big )\nonumber \\&=\frac{1}{\sqrt{8\pi ^3\sigma _{11}\sigma _{22}\sigma _{\text {ff}}\gamma }} \exp \Big [-\frac{1}{2\gamma }\Big \{\alpha _1 x_1^2+ \alpha _2 x_2^2 + \alpha _3 x^2_{\text {f}}\nonumber \\&\qquad -2\beta _1x_1x_2-2\beta _2x_1x_{\text {f}}- 2\beta _3x_2x_{\text {f}}\Big \}\Big ]\nonumber \\&\quad \times \frac{1}{\sqrt{2\pi \sigma _{nn}}}\exp \left[ -\frac{n^2}{2\sigma _{nn}}\right] \end{aligned}$$

(22)

Consider the transformation of variables $Y_1=X_1+N$, $Y_2=X_2+N$ and $Y_3=N$. The transformation is one-to-one, so there exists a unique inverse $X_1=Y_1-Y_3$, $X_2=Y_2-Y_3$ and $N=Y_3$. The Jacobian of the transformation is

$$\begin{aligned} |J|=\bigg |\frac{\partial (X_1, X_2, N)}{\partial (Y_1,Y_2,Y_3)}\bigg |=1 \end{aligned}$$

(23)

Then the quadvariate joint PDF can be obtained as

$$\begin{aligned}&p_{Y_1,Y_2,Y_3,X_{\text {f}}}\left( y_1,y_2,y_3,x_{\text {f}}\right) \nonumber \\&\quad =\frac{1}{\sqrt{8\pi ^3\sigma _{11}\sigma _{22}\sigma _{\text {ff}}\gamma }} \exp \Big [-\frac{1}{2\gamma }\Big \{\alpha _1 (y_1-y_3)^2+ \alpha _2 (y_2-y_3)^2 + \alpha _3 x^2_{\text {f}}\nonumber \\&\qquad -2\beta _1(y_1-y_3)(y_2-y_3)-2\beta _2(y_1-y_3)x_{\text {f}}- 2\beta _3(y_2-y_3)x_{\text {f}}\Big \}\Big ]\nonumber \\&\qquad \times \frac{1}{\sqrt{2\pi \sigma _{nn}}}\exp \left[ -\frac{y_3^2}{2\sigma _{nn}}\right] \end{aligned}$$

(24)

Expanding the squared terms in the first exponent and collecting the terms that contain $y_3^2$ and $y_3$ gives

$$\begin{aligned}&p_{Y_1,Y_2,Y_3,X_{\text {f}}}\left( y_1,y_2,y_3,x_{\text {f}}\right) \nonumber \\&\quad =\frac{1}{\sqrt{8\pi ^3\sigma _{11}\sigma _{22}\sigma _{\text {ff}}\gamma }}\frac{1}{\sqrt{2\pi \sigma _{nn}}} \exp \Big [-\frac{1}{2\gamma }\Big \{\alpha _1 y_1^2+\alpha _2 y_2^2 + \alpha _3 x^2_{\text {f}}\nonumber \\&\qquad -2\beta _1y_1y_2-2\beta _2y_1x_{\text {f}}- 2\beta _3y_2x_{\text {f}}+(\alpha _1+\alpha _2-2\beta _1+\frac{\gamma }{\sigma _{nn}})y_3^2\nonumber \\&\qquad -2\Big ((\alpha _1-\beta _1)y_1+(\alpha _2-\beta _1)y_2- (\beta _2+\beta _3)x_{\text {f}}\Big )y_3\Big \}\Big ]\nonumber \\&\quad =\frac{1}{\sqrt{8\pi ^3\sigma _{11}\sigma _{22}\sigma _{\text {ff}}\gamma }}\frac{1}{\sqrt{2\pi \sigma _{nn}}} \exp \Big [-\frac{1}{2\gamma }\Big \{\alpha _1 y_1^2+ \alpha _2 y_2^2 + \alpha _3 x^2_{\text {f}}\nonumber \\&\qquad -2\beta _1y_1y_2-2\beta _2y_1x_{\text {f}}-2\beta _3y_2x_{\text {f}} \Big \}\Big ]\exp \left( -\frac{1}{2}\left( \lambda _3y_3^2-2\lambda _2y_3\right) \right) \end{aligned}$$

(25)

The required PDF $p_{Y_1,Y_2,X_{\text {f}}}\big (y_1,y_2,x_{\text {f}}\big )$ is obtained after integrating out $y_3$ and noting that $\lambda _1=\sqrt{\frac{1}{\lambda _3\sigma _{nn}}}$