1. Approaches for Retinex and Their Relations Yu Du March 14, 2002

2. 2 Presentation Outline u Introductions to retinex u Approaches for retinex u The variational framework u Relation of these approaches u Conclusions

3. 3 What Is Retinex u Lightness and retinex theory s E. H. Land 1971 u Visual system of human s Retina: the sensory membrane lining the eye that receives the image formed by the lens (Webster) s Reflectance and illumination s Edges and independent color senstion

4. 4 Model of retinex (1) The given image The reflectance part The illumination part

5. 5 Model of retinex (2) Input Image Log Estimate the Illumination Exp +

6. 6 Three Types of Previous Approaches u Random walk algorithms s E. H. Land (1971) u Homomorphic filtering s E. H. Land (1986), D. J. Jobson (1997) u Solving Poisson equation s B. K. P. Horn (1974)

7. 7 Random Walk Algorithms (1) u First retinex algorithm u A series of random paths s Starting pixel s Randomly select a neighbor pixel as next pixel on path u Accumulator and counter

8. 8 Random Walk Algorithms (2) u Adequate number of random paths s Cover the whole image s Small variance u Length of paths s >200 for 10x10 image (D. H. Brainard)

9. 9 Special Smoothness of Random Walk u The value in the accumulator u The illumination part

10. 10 Homomorphic Filtering u Assume illumination part to be smooth u Apply low pass filter

11. 11 Poisson Equation Solution (1) u Derivative of illumination part close to zero u Reflectance part to be piece-wise constant u Get the illumination part s Take the derivative of the image s Clip out the high derivative peaks

12. 12 Poisson Equation Solution (2) u Solve Poisson equation u Iterative method u Apply low-pass filter (invert Laplacian operator)

13. 13 Comments on Above Approaches u Random walk algorithm s Too slow u Homomorphic filtering s Low-pass filtering first or log first? u More work needed to be done on Poisson equation solving

14. 14 Variational Framework u Presented by R. Kimmel etc. u From assumptions to penalty function u From penalty function to algorithm

15. 15 Assumptions On Illumination Image u Spatial smoothness of illumination u Reflectance is not pure white u Illumination close to intensity image u Spatial smoothness of reflectance u Continues smoothly beyond boundaries

16. 16 Penalty Function and Restrictions u Goal to minimize: u Subject to: And on And on

17. 17 Solve the Penalty Function (1) u Euler-Lagrange equations And And

18. 18 Solve the Penalty Function (2) u Projected normalized steepest descent (PNSD) u Iteratively to get illumination part

19. 19 Multi-resolution u Make PNSD algorithm converges faster u Illumination part is smooth u Coarse resolution image first u Upscale coarse illumination as initial of finer resolution layer u Not multi-scale technique

20. 20 Relationship of Different Approaches (1) u Random walk and Homomorphic filtering u R. Kimmel’s words on Homomorphic filtering and remove constraint and remove constraint

21. 21 Relationship of Different Approaches (2) u Apply appropriate scaling on images, Homomorphic filtering satisfies constrain and and u Poisson equation approach:

22. 22 Conclusions u Retinex is trying to simulate human vision process u Different approaches are from same assumptions u Implementation details are important for results

23. Thank You March 14, 2002