A Multi-level Approach to Quantization. Yair Koren, Irad Yavneh, Alon Spira Department of Computer Science Technion, Haifa 32000 Israel. One dimensional (scalar) quantization.
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Yair Koren, Irad Yavneh, Alon Spira
Department of Computer Science
Technion, Haifa 32000
Left – Lena (gray level image), Right – Lena’s histogram, p(x).
128 gray levels
64 gray levels
Lena, 8 levels, Left – optimal, Right - Naive
We wish to minimize
Differentiating w.r.t. r and dyields the Lloyd-Max equations:
Max and Lloyd proposed a simple iterative process:
Given some initial guess, , iterate for until some convergence criterion is satisfied:
We can rewrite the Lloyd-Max equations in terms of d alone:
This is a generally a nonlinear system.
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.
We compare three algorithms:
In all the algorithms, the basic iteration is Lloyd-Max.
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.
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.
Sketch of algorithm (V Cycle):
Sketch of Relaxation algorithm:
The multi-level approach is very promising for the problem of quantization. In 1D and (semi-) continuous p we get
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.
Ron Kimmel and Irad Yavneh
for some threshold T.
For this we use our version of a multigrid algorithm with matrix-dependent prolongations.