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STOCHASTIC SAMPLING FROM IMAGE CODER

INDUCED PROBABILITY DISTRIBUTIONS

regu@isis.poly.edu

presenting author:

memon@poly.edu

viresh@google.comGoogle Inc., Mt. View, CA

Polytechnic University, Brooklyn, NY

oguleryuz@erd.epson.comEpson Palo Alto Laboratory

Palo Alto, CA

Regunathan Radhakrishnan, and Nasir Memon

Onur G. Guleryuz, Viresh Ratnakar,

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Overview•We determine the probability distributions that today’s popular,

state-of-the-art coders induce on image outcomes.

•We consider the set of images that are well-coded by today’s

popular state-of-the-art coders.

•We use stochastic sampling techniques to obtain typical samples

from these sets.

•We compare typical samples to well-known images like Lena

and Barbara.•Are today’s coders giving us the best possible performance on natural images?

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Image Compression Systems

Encoder Decoder

Original Image

Decoded Image

Encoded bitstream

< format bits > < data bits > < data bits >< format bits > < … >

E.g.:Image dimensions,Quantizer information,Marker information,…

Compressed data that specifies the imagepixel values.

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Image Compression Systems

Encoder Decoder

Original Image

Decoded Image

Encoded bitstream

< format bits > < data bits > < data bits >< format bits > < … >

For a good image coder these bits should be random

(coin tosses - i.i.d., prob(0)=prob(1)=1/2)

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This Paper

Decoder

Decoded ImageEncoded bitstream

< format bits > < data bits > < … >

512x512, grayscaleimage, encoded

at ~1 bit/pixel, … Random bits!

Decoded Images shown below.

(The decoding syntax will decode each data bit using a certain probability distribution. We provided random bits drawn from the appropriate distributions.)

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Background

Encoder

Original Image

“Quantizer” Entropy Coder

Introduces loss (if desired)

(Same as the decoded image)

image i il bits

U

The set of all (512x512) grayscale images

il2~prob(image i)

UAn image coder induces a probability distribution on the image space,

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BackgroundU

,1C UE( )1CEfficient set of coder

U

,2C UE( )2CEfficient set of coder

…

(contains most of the induced probability)

•We would like to find out what the typical elements of these sets look like.

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Why?

•Are highly probable outcomes close to “the set of natural images”, ?Ni.e., is ?NE( ~)iC

If not, there is room to improve.

When we are encoding Lena with coder we are spending precious bits to distinguish Lena from all the other images in .

,iC)iCE(

Image compression is dead.< Insert coder here > is the final word in image compression.

…

•What kind of images is coder good for?iC

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How do we know …

Image coders utilize data structures that allow us to talk about:

• JPEG: typical random blocks,• SPIHT, JPEG2000: typical random trees of wavelet coefficients,• JPEGLS: typical random sequences of pixels.…

)iCE(1. How do we know exists?

in the sense of typical sequences and Asymptotic Equipartition Theorem [1].

Coders induce typical sets that contain most of the probability.

[1] T. M. Cover and J. A. Thomas, ``Elements of Information Theory.'‘ New York: Wiley, 1991.

2. How do we know we are sampling from ?)iCE(The probability of not sampling from is very, …, very small.)iCE(

10[3] Emmanuel Bacry, LastWave software: http://www.cmap.polytechnique.fr/~bacry/LastWave

[2] S. Mallat and S. Zhong, ``Characterization of signals from multiscale edges,'' IEEE Trans. Pattern Anal. Machine Intell., vol. 14, pp. 710-732, July 1992.

Conclusion•Are today’s coders giving us the best possible performance on natural images?You decide.

Qualitatively: Typical images are provided in this presentation. Please examine them.

Quantitatively: We provide a metric that shows how important an image’s edges are in representing the image with the help of [2,3].Please ask the presenter for details and examine the results.(For natural images, edges are very important).

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SPIHT – 1 (simulations by Onur G. Guleryuz)

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SPIHT - 2

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SPIHT - 3

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SPIHT

length of wavelet maxima chains starting from the finest scale

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JPEG2000 - 1 (simulations by Regunathan Radhakrishnan and Nasir Memon)

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JPEG2000 - 2

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JPEG2000 - 3

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JPEG2000

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JPEG – 1 (simulations by Viresh Ratnakar)

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JPEG - 2

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JPEG - 3

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JPEG

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JPEGLS - 1 (simulations by Regunathan Radhakrishnan and Nasir Memon)

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JPEGLS - 2

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JPEGLS - 3

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JPEGLS