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# A Multi-level Approach to Quantization - PowerPoint PPT Presentation

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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A Multi-level Approach to Quantization

Yair Koren, Irad Yavneh, Alon Spira

Department of Computer Science

Technion, Haifa 32000

Israel

• 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 .

Gray Level

Left – Lena (gray level image), Right – Lena’s histogram, p(x).

Gray Level

128 gray levels

64 gray levels

Gray Level

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:

• 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.

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.

Equal height contours of P(x,y) = x*y (centers of mass) for P(x,y)≡1

Decision regions for P(x,y)=x*y (centers of mass) for P(x,y)≡1

Discrete Vector Quantization (centers of mass) for P(x,y)≡1

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.

Discrete Vector Quantization (centers of mass) for P(x,y)≡1

Sketch of algorithm (V Cycle):

Sketch of Relaxation algorithm:

Conclusions (centers of mass) for P(x,y)≡1

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.

### A Multigrid Approach to Binarization (centers of mass) for P(x,y)≡1

Image Binarization (centers of mass) for P(x,y)≡1

Original Image

Nonuniform (centers of mass) for P(x,y)≡1Illumination

Tilted

Spherical

Naïve (threshold) binarization (centers of mass) for P(x,y)≡1

Tilted

Naïve (threshold) binarization (centers of mass) for P(x,y)≡1

Spherical

Yanowitz-Bruckstein Binarization (centers of mass) for P(x,y)≡1

• 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.

Edges (centers of mass) for P(x,y)≡1

Tilted

Spherical

Results (centers of mass) for P(x,y)≡1

Tilted

Spherical