A Multi-level Approach to Quantization

1. A Multi-level Approach to Quantization Yair Koren, Irad Yavneh, Alon Spira Department of Computer Science Technion, Haifa 32000 Israel

2. One dimensional (scalar) quantization • Consider the image I consisting of G representation (gray) levels. We would like to represent I with n < G representation levels as best as possible. • More formally, given a signal X (image, voice, etc.), with probability density function (histogram) p(x), we would like an approximation q(x)of X, which minimizes the distortion: • Here, all are represented by .

3. Example (Lena, 512x512) Gray Level Left – Lena (gray level image), Right – Lena’s histogram, p(x).

4. Lena, 8 gray levels Gray Level

5. Representing Lena with less levels 128 gray levels 64 gray levels

6. Lena, 4 gray levels Gray Level

7. Naïve vs. Optimal Quantization Lena, 8 levels, Left – optimal, Right - Naive

8. The Lloyd Max Iterative Process We wish to minimize Differentiating w.r.t. r and dyields the Lloyd-Max equations: Max and Lloyd proposed a simple iterative process:

9. The Lloyd Max Iterative Process Given some initial guess, , iterate for until some convergence criterion is satisfied:

10. The Lloyd Max Iterative Process We can rewrite the Lloyd-Max equations in terms of d alone: This is a generally a nonlinear system.

11. The Lloyd Max Iterative Process However, for the simple case, p = 1, L-M reduces to This is nothing but a damped Jacobi relaxation with damping factor 1/2for the discrete Laplace equation. Evidently, multigrid acceleration is likely to help. We employ a nonlinear multigrid algorithm, using the Lloyd Max process for relaxation (with over-relaxation 4/3), and a nonlinear interpolation which retains the order of d.

12. Numerical Tests We compare three algorithms: • Lloyd-Max, starting with a uniform representation • Our multigrid algorithm, starting similarly • LBG (Linze et al., 1980): Sequential refinement (coarse-to-fine). In all the algorithms, the basic iteration is Lloyd-Max.

13. P(x)=x

14. Step Function and local minima

15. Local Minima

16. Convergence Criterion

17. Discrete Vector Quantization The 1D problem is used mainly as a preliminary study towards higher-dimensional problems, viz., vector quantization (e.g., for color images). Also, the p histogram is discrete in practice, and usually quite sparse and patchy and there are many different “solutions” (local minima). “Standard” multigrid methods do not seem appropriate.

18. Decision regions (Voronoi cells) and representation levels (centers of mass) for P(x,y)≡1

19. Equal height contours of P(x,y) = x*y

20. Decision regions for P(x,y)=x*y

21. Discrete Vector Quantization Let G denote the number of possible representation-levels (D-tuples), P the number of such levels for which p does not vanish, and R the number of quantized representation levels. Typically, A Lloyd Max iteration costs at least O(P) operations. As it doesn’t seem possible to usefully coarsen p, coarse–level iterations will be equally expensive, resulting in O(P log(R)) complexity for the multigrid cycle.

22. Discrete Vector Quantization Sketch of algorithm (V Cycle): Sketch of Relaxation algorithm:

23. Conclusions The multi-level approach is very promising for the problem of quantization. In 1D and (semi-) continuous p we get • Much faster convergence. • Often better minima. • Sounder convergence criterion. The real dividends are expected for vector quantization (as in color images). This is a significantly harder and more important problem. Research on this is in progress, led by Yair Koren.

24. A Multigrid Approach to Binarization Ron Kimmel and Irad Yavneh

25. Image Binarization Original Image

26. NonuniformIllumination Tilted Spherical

27. Naïve (threshold) binarization Tilted

28. Naïve (threshold) binarization Spherical

29. Yanowitz-Bruckstein Binarization • Isolate the locations of edge centers, for example, the set of points, for some threshold T. • Use the values I(x,y), for (x,y) in s, as constraints for a threshold surface, u, which elsewhere satisfies the equation For this we use our version of a multigrid algorithm with matrix-dependent prolongations.

30. Edges Tilted Spherical

31. Results Tilted Spherical