RECONSTRUCTION OF MULTI-SPECTRAL IMAGES USING MAP

1. RECONSTRUCTION OF MULTI-SPECTRAL IMAGES USING MAP Gaurav

2. OUTLINE • OBJECTIVE • SYSTEM DESCRIPTION • MATHEMATICAL DESCRIPTION • MAP • RESULTS & DISCUSSIONS • PERFORMANCE EVALUATION • CONCLUSION

3. OBJECTIVE • What we have been doing till now: • Improving the state-of-art multi-spectral camera technology (eg. CyberEye 2100 capable of capturing 12 bands of the electromagnetic spectrum) • Use of mosaic focal plane array technology for multi-spectral images • Motive: To improve the use of multi-spectral images in fields like defence, medical, etc. • Next Level: • Focus on improvement of the demosaicked output

4. PROCESS DESCRIPTION • Block Diagram: Additive Noise Mosaicking f Blur Actual Scene/ Input Image m + Demosaicking g Image Restoration Using MAP Output Image

5. Mathematical Description The System can be mathematically formulated as : where, m = mosaicked image f = actual scene/ image (multi-spectral image before acquisition process) h = blur kernel η = additive noise (noise added to all bands of the multi-spectral image) s = system non-linearity : image sampling using mosaicking process

6. PROPOSED MODEL FOR RESTORATION Demosaicking g = distorted demosaicked image MAP

7. MAP – Maximum a Posteriori Probability Estimate Basic Equation: MAP estimate:

8. MAP contd… Problem Transformation: Taking log on both sides Prior Model: Assuming the f ‘s are taken out from an ensemble which have a Gaussian Distribution Noise Model: What is p(g/f) : same as noise distribution

9. MAP contd… To Maximize p(f/g) is same as maximizing log(p(f/g) Optimization: using Gradient Descent Lexicographic Ordering:

10. Results • A synthetic multi-spectral image was used for experimentation. • Bilinear Interpolation algorithm was used for demosaicking. • Initial Guess is taken as a Wiener Filter Result. • Gaussian Noise with constant variance was assumed. • Blur was assumed to be Gaussian in nature. • Three cases for Prior Distribution were considered • Gaussian Distribution (constant variance) • Gibbs Distribution • With Laplacian Kernel • With Quadratic Variation • Convergence when • Metric for comparison of results: Mean Square Error

11. Gibbs Distribution • Prior Term : Gibbs distribution • where, • -V(f) is the of potentials, i.e., in our case difference between neighbourhood pixels. Use of Sobel filter to compute the potentials. • U(f) is the sum of potentials represents the Energy term • T is the absolute temperature (constant) • Z is the normalizing factor

12. Inverse Gaussian • where, • βis the strength of an edge • is the variance • r is the laplacian kernel • Quadratic variation

13. Original Image MAP Images Gaussian Prior Density Gibbs Prior Density

14. Mean Square Error Values

15. Performance Evaluation Speed of Convergence: Noise variance : Cannot really predict how the noise variance affects the speed of convergence. At very less noise variance (<0.1) convergence not possible. Prior Distribution Variance: Remains constant after some particular value of prior distribution variance.

16. Reducing Step size : • Convergence time increases (slow convergence) • Doesn’t affect final MSE value • Increasing Step size: • Convergence guaranteed only till a particular value of step size (alpha = 1). Beyond that value no convergence possible. • Optimum Value : alpha = 0.1 since here we obtain least MSE.

17. MSE reduces drastically at lower values of prior distribution variances. As the prior variance value goes up, the MSE values start to settle down.

18. CONCLUSION • The MAP method successfully restores the least mean square error multi-spectral image. Thank You. -Gaurav