Inversion of the divisive normalization: Algorithms and new possibilities Jesús Malo Dpt. Optics at Fac. Physics, Universitat de València (Spain) + NASA Ames Research Center, Moffett Field, CA

2 Outline Background The model Inversion is needed The model is not analytically invertible The differential method The method Existence and uniqueness of the solution This is not a gradient-based search Inversion Results Applications Image Coding and Vision Reseach Final Remarks

3 Background The model: Transform T A Image a R r Response aiai riri

4 T a A Background The inversion would be useful (stimuli design) A aa AA

5 R a r Background The inversion would be useful (stimuli design) T A r

6 a T Background The inversion would be useful (image coding) A R r

7 Background However, the normalization is not invertible! Without kernelWith kernel ?

8 Locally: The differential method Given the pair: (eq.1) For any response, r (given some initial conditions (r 0, a 0 ) ): (eq.2)

9 The differential method

10 The differential method Existence and Uniqueness of the solution is integrable if Lipschitz: i.e is bounded

11 The differential method aiai riri aiai riri

12 The differential method Summary of the differential method: * The inverse is obtained solving from (r 0, a 0 ) * The differential equation is solved using a 4th order Runge-Kutta algorithm * In each step of the integral, the jacobian is computed using, * The convergence to the appropriate solution is theoretically guaranteed

13 The differential method The conventional gradient-based search Local minima The result highly depends on the initial guess No apriori information about the number of iterations Initial guess Estimated solution Actual solution

14 Inversion Results Convergence Block-DCT 1 step2 steps4 steps6 steps8 steps

15 Inversion Results Convergence Wavelets 3 steps6 steps

16 Applications I: Vision Science Basis functions of the response domain (given a masking pattern A 0, a 0, r 0 ): In the transform domain ( ) In the spatial domain ( ) If the jacobian is not diagonal, the function is not a basis function of T!

17 Applications I: Vision Science Masking Pattern ( A 0 ) Basis functions (space) Basis functions (transform) Basis functions (response)

18 Applications II: Image coding O.27 bpp e=1.69 e=1.74 e=1

19 Final remarks The divisive normalization model is not invertible due to the interaction between transform coefficents The inversion method allows to explore a number of new possibilities both in vision science and image coding The differential method presented here is a non conventional method to solve this nonlinear problem The differential method performs better than comparable gradient based seach algorithms

20 Appendix I:Parameters block-DCT ParametersResponses Response (contrast masking) Jacobian

21 Appendix II:Parameters wavelet ImageTransform Kernel Inhibition Response

22 Conclusions W as complement of  Expression for the computation of W New tools to be used in the transform coding problem: R-1R-1 W The inversion method reconstructs the image from the response in few Runge-Kutta iterations The response representation achieves good decorrelation results in terms of  and W Promising compression results are obtained with a uniform quantization of the response representation