A Multi-level Approach to Quantization

Yair Koren, Irad Yavneh, Alon Spira

Department of Computer Science

Technion, Haifa 32000

Israel

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 .

1

2 2

221

( ) [ ] .i

i

dn

ii d

D q E X q X x r p x dx

1[ , )i ix d d ir

Example (Lena, 512x512)

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

Gray Level

Lena, 8 gray levels

Gray Level

Representing Lena with less levels

128 gray levels 64 gray levels

Lena, 4 gray levels

Gray Level

Naïve vs. Optimal Quantization

Lena, 8 levels, Left – optimal, Right - Naive

The Lloyd Max Iterative Process

1

2 2

221

( ) [ ] .i

i

dn

ii d

D q E X q X x r p x dx

We wish to minimize

Differentiating w.r.t. r and d yields the Lloyd-Max equations:

1 1

1

( )

, , 1, , 12

( )

i

i i

i

i

d

d ii id

d

xp x dxr r

r d i n

p x dx

Max and Lloyd proposed a simple iterative process:

The Lloyd Max Iterative Process

1

11 1

1

11

( )

, .2

( )

ki

ki iki

ki

d

k kd ik k

i id

d

xp x dxr r

r d

p x dx

Given some initial guess, , iterate for until some convergence criterion is satisfied:

0d 1, 2,k

The Lloyd Max Iterative Process

1

1

1

1

1, 1, , 1.

2

i i

i i

i i

i i

d d

d di d d

d d

xp x dx xp x dx

d i n

p x dx p x dx

We can rewrite the Lloyd-Max equations in terms of d alone:

This is a generally a nonlinear system.

The Lloyd Max Iterative Process

1

1 1

12 , 1, , 1.

4kk

i i i id d d d i n

However, for the simple case, p = 1, L-M reduces to

This is nothing but a damped Jacobi relaxation with damping factor 1/2 for 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.

Numerical Tests

We compare three algorithms:

1. Lloyd-Max, starting with a uniform representation

2. Our multigrid algorithm, starting similarly

3. LBG (Linze et al., 1980): Sequential refinement (coarse-to-fine).

In all the algorithms, the basic iteration is Lloyd-Max.

P(x)=x

Step Function and local minima

Local Minima

Convergence Criterion

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.

Decision regions (Voronoi cells) and representation levels (centers of mass) for

P(x,y)≡1

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

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

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.

R P G

Discrete Vector Quantization

Sketch of algorithm (V Cycle):

Sketch of Relaxation algorithm:

For 0,1, , log 1, log , log 1, ,1,0,

Relax

j R R R

j

logPartition the variables into, say, 2 aggregates

of 2 representation levels each. Then sweep over the

aggregates, changing each in turn so as to (approximately)

minimize the fine-level function

R j

j

R

al (Gauss-Seidel style).

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.

A Multigrid Approach to Binarization

Ron Kimmel and Irad Yavneh

Image Binarization

Original Image

Nonuniform Illumination

ilted Spherical

Naïve (threshold) binarization

ilted

Naïve (threshold) binarization

Spherical

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.

, :s x y I T

0.u

Edges

ilted Spherical

Results

ilted Spherical