A multi level approach to quantization
Download
1 / 31

A Multi-level Approach to Quantization - PowerPoint PPT Presentation


  • 120 Views
  • Uploaded on

A Multi-level Approach to Quantization. Yair Koren, Irad Yavneh, Alon Spira Department of Computer Science Technion, Haifa 32000 Israel. One dimensional (scalar) quantization.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'A Multi-level Approach to Quantization' - claral


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
A multi level approach to quantization
A Multi-level Approach to Quantization

Yair Koren, Irad Yavneh, Alon Spira

Department of Computer Science

Technion, Haifa 32000

Israel


A multi level approach to quantization

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 .


Example lena 512x512
Example (Lena, 512x512)

Gray Level

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



Representing lena with less levels
Representing Lena with less levels

128 gray levels

64 gray levels



Na ve vs optimal quantization
Naïve vs. Optimal Quantization

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


The lloyd max iterative process
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:


The lloyd max iterative process1
The Lloyd Max Iterative Process

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


The lloyd max iterative process2
The Lloyd Max Iterative Process

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

This is a generally a nonlinear system.


The lloyd max iterative process3
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.


Numerical tests
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.






Discrete vector quantization
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.



Equal height contours of p x y x y

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


Discrete vector quantization1
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 quantization2
Discrete Vector Quantization (centers of mass) for P(x,y)≡1

Sketch of algorithm (V Cycle):

Sketch of Relaxation algorithm:


Conclusions
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

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

Ron Kimmel and Irad Yavneh


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

Original Image


A multi level approach to quantization

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

Tilted

Spherical


A multi level approach to quantization

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

Tilted


A multi level approach to quantization

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

Spherical


Yanowitz bruckstein binarization
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.


A multi level approach to quantization

Edges (centers of mass) for P(x,y)≡1

Tilted

Spherical


A multi level approach to quantization

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

Tilted

Spherical