NONLINEAR PRIMARY CORTICAL IMAGE

REPRESENTATION FOR JPEG 2000

Applying natural image statistics and visual perception to image compression

Roberto Valerio

UNIC, CNRS, 1 Avenue de la Terrasse, 91190 Gif-sur-Yvette, France

Rafael Navarro

ICMA, CSIC-Universidad de Zaragoza, Plaza San Francisco s/n, 50009 Zaragoza, Spain

Keywords: Nonlinear models of V1 neurons, Divisive normalization, Natural image statistics, Perceptual quality

metrics, JPEG 2000.

Abstract: In this paper, we present a nonlinear image representation scheme based on a statistically-derived divisive

normalization model of the information processing in the visual cortex. The input image is first decomposed

into a set of subbands at multiple scales and orientations using the Daubechies (9, 7) floating point filter

bank. This is followed by a nonlinear “divisive normalization” stage, in which each linear coefficient is

squared and then divided by a value computed from a small set of neighboring coefficients in space,

orientation and scale. This neighborhood is chosen to allow this nonlinear operation to be efficiently

inverted. The parameters of the normalization operation are optimized in order to maximize the statistical

independence of the normalized responses for natural images. Divisive normalization not only can be used

to describe the nonlinear response properties of neurons in visual cortex, but also yields image descriptors

more independent and relevant from a perceptual point of view. The resulting multiscale nonlinear image

representation permits an efficient coding of natural images and can be easily implemented in a lossy JPEG

2000 codec. In fact, the nonlinear image representation implements in an automatic way a more general

version of the point-wise extended masking approach proposed as an extension for visual optimisation in

JPEG 2000 Part 2. Compression results show that the nonlinear image representation yields a better rate-

distortion performance than the wavelet transform alone.

1 INTRODUCTION

The human visual system (HVS) plays a key role in

the final perceived quality of compressed images.

Therefore, it is desirable to take advantage of the

current knowledge of visual perception in a

compression system. The JPEG 2000 standard

includes various tools that permit to exploit some

properties of the HVS such as spatial frequency

sensitivity, color sensitivity, and visual masking

effects (Zeng et al., 2002). The visual tools sets in

JPEG 2000 are much richer than those in JPEG,

where only spatially-invariant frequency weighting

is used. As a result, visually optimized JPEG 2000

images usually have much better visual quality than

visually optimized JPEG images at the same bit

rates. Nevertheless, the visual optimization tools in

JPEG 2000 are still simplified versions of the latest

models of human visual processing.

In recent years, various authors have shown that

the nonlinear behavior of V1 neurons in primate

visual cortex can be modeled by including a gain

control stage, known as “divisive normalization”

(e.g. Heeger, 1992), after a linear filtering step. In

this nonlinear stage, the linear inputs are squared and

then divided by a weighted sum of squared

neighboring responses in space, orientation, and

scale, plus a regularizing constant. Divisive

normalization not only can be used to describe the

nonlinear response properties of neurons in visual

cortex, but also yields image descriptors more

relevant from a perceptual point of view (Foley,

1994). More recently, Simoncelli and co-workers

519

Valerio R. and Navarro R. (2006).

NONLINEAR PRIMARY CORTICAL IMAGE REPRESENTATION FOR JPEG 2000 - Applying natural image statistics and visual perception to image

compression.

In Proceedings of the First International Conference on Computer Vision Theory and Applications, pages 519-522

DOI: 10.5220/0001377205190522

Copyright

c

SciTePress

(e.g. Schwartz and Simoncelli, 2001) presented a

statistically-derived divisive normalization model.

They demonstrated its utility to characterize the

nonlinear response properties of neurons in sensory

systems, and thus that early neural processing is well

matched to the statistical properties of the stimuli. In

addition, they showed empirically that the divisive

normalization model strongly reduces pairwise

statistical dependences between responses.

In this paper, we describe a nonlinear image

representation scheme (similar to Valerio et al.,

2003) based on a statistically-derived divisive

normalization model of V1 neurons. This scheme

could be useful in a lossy JPEG 2000 codec. Starting

with a 9/7 Daubechies wavelet decomposition, we

normalize each coefficient by a value computed

from a neighborhood. This neighborhood is

suboptimal for dependency reduction, but allows the

transform to be easily inverted. We describe the

empirical optimization of the transform parameters,

and demonstrate that the redundancy in the resulting

coefficients is substantially less than that of the

original linear ones. Compression results show that

the nonlinear representation can improve the

perceptual quality of compressed images.

2 NONLINEAR IMAGE

REPRESENTATION SCHEME

The scheme used here consists of a linear wavelet

decomposition followed by a nonlinear divisive

normalization stage.

2.1 Linear Stage

The linear stage is an approximately orthogonal

four-level wavelet decomposition based on the

Daubechies (9, 7) floating point filter bank. The 9/7

transform is nonreversible and real-to-real, and is

one of the two specific wavelet transforms supported

by the baseline JPEG 2000 codec (the other one is

the 5/3 transform, which is reversible, integer-to-

integer and nonlinear). Lacking the reversible

property, the 9/7 transform can only be used for

lossy coding.

2.2 Nonlinear Stage

The nonlinear stage consists basically of a divisive

normalization. In this stage, the responses of the

previous linear filtering stage, c

i

, are squared and

then divided by a weighted sum of squared

neighboring responses in space, orientation, and

scale,

}{

2

j

c

, plus a positive constant,

2

i

d

:

∑

+

⋅

=

j

jiji

ii

i

ced

ccsign

r

22

2

)(

(1)

Eq. 1 is similar to models of cortical neuron

responses but has the advantage that preserves sign

information. The parameters,

2

i

d

and {e

ij

}, of the

divisive normalization are fixed to the following

values (Schwartz and Simoncelli, 2001):

2

i

d

=

2

i

a

,

e

ij

= b

ij

(i ≠ j) and e

ii

= 0, where

2

i

a

and b

ij

(i ≠ j) are

the parameters of a Gaussian model for the

conditional probability . This choice of

parameters yields approximately the minimum

mutual information (MI), or equivalently minimizes

statistical dependence, between normalized

responses for a set of natural images (Valerio and

Navarro, 2003). In practice, we fix the parameters,

2

i

d

and {e

ij

}, of the divisive normalization for each

subband by using maximum-likelihood (ML)

estimation with a set of natural images (“Boats”,

“Elaine”, “Goldhill”, “Lena”, “Peppers”, and

“Sailboat” in our case). Numerical measures of

statistical dependence in terms of MI for the 6

512x512 B&W images with 8 bpp in the “training

set” show that divisive normalization decreases MI,

with most values much closer to zero. So, for

example, the mean value of MI between two

neighboring wavelet coefficients, c

i

and c

j

(c

j

is the

right down neighbor of c

i

), from the lowest scale

vertical subband is 0.10, whereas between the

corresponding normalized coefficients, r

i

and r

j

, is

only 0.04.

A key feature of the nonlinear stage is the

particular neighborhood considered in Eq. 1. We

consider 12 coefficients {c

j

} (j ≠ i) adjacent to c

i

along the four dimensions (9 in a square box in the

2D space, plus 2 neighbors in orientation and 1 in

spatial frequency). All neighbors belong to higher

levels of the linear pyramid. This permits to invert

the nonlinear transform very easily level by level (to

recover one level of the linear pyramid we obtain the

normalizing values from levels already recovered

and multiply them by the corresponding nonlinear

coefficients). Obviously, in order to invert the

nonlinear transform we need the low-pass residue of

the linear decomposition. More details can be found

in Valerio et al. (2003).

}){|(

2

ji

ccp

VISAPP 2006 - IMAGE ANALYSIS

520

3 PERCEPTUAL METRIC

From the nonlinear image representation it is

possible to define a perceptual image distortion

metric similar to that proposed by Teo and Heeger

(1994). For that, we simply add an error pooling

stage. This computes a Minkowski sum with

exponent 2 of the differences

i

rΔ

(multiplied by

constants k

i

that adjust the overall gain) between the

nonlinear outputs from the reference image and

those from the distorted image (Valerio et al., 2004):

∑

Δ⋅=Δ

i

ii

rkr

2

2

(2)

This perceptual metric has two main differences

with respect to that by Teo and Heeger (1994). First,

the divisive normalization considers not only

neighbouring responses in orientation but also in

position, and scale. Second, the parameters of the

divisive normalization are adapted to natural image

statistics instead of being fixed exclusively to fit

psychophysical data.

4 CODING RESULTS

In order to compare the coding efficiency of the 9/7

transform alone and our nonlinear transform (the 9/7

transform plus the divisive normalization), we have

conducted a series of compression experiments with

a simplified JPEG 2000 codec. Basically, the coding

is as follows. First, the input image is preprocessed

(the nominal dynamic range of the samples is

adjusted by subtracting a bias of 2

P-1

, where P is the

number of bits per sample, from each of the samples

values). Then, the intracomponent transform takes

place. This can be the 9/7 transform or our nonlinear

transform. In both cases, we use the implementation

of the 9/7 transform in the JasPer software (Adams

and Kossentini, 2000). After quantization is

performed in the encoder (we fix the quantizer step

size at one, that is, there is no quantization), tier-1

coding takes place.

In the tier-1 coder, each subband is partitioned

into code blocks (the code block size is 64x64), and

each of the code blocks is independently coded. The

coding is performed using a bit-plane coder. There is

only one coding pass per bit plane and the samples

are scanned in a fixed order as follows. The code

block is partitioned into horizontal stripes, each

having a nominal height of four samples. The stripes

are scanned from top to bottom. Within a stripe,

columns are scanned from left to right. Within a

column, samples are scanned from top to bottom.

The sign of each sample is coded with a single

binary symbol right before its most significant bit.

The bit-plane encoding process generates a sequence

of symbols that are entropy coded. For the purposes

of entropy coding, a simple adaptive binary

arithmetic coder is used. All of the coding passes of

a code block form a single codeword (per-segment

termination).

Tier-1 coding is followed by tier-2 coding, in

which the coding pass information is packaged.

Each packet consists of two parts: header and body.

The header indicates which coding passes are

included in the packet, while the body contains the

actual coding pass data. The coding passes included

in the packet are always the most significant ones

and we use a fixed-point representation with 13 bits

after the decimal point, so that we only need to store

the maximum number of bit planes of each code

block.

In tier-2 coding, rate control is achieved through

the selection of the subset of coding passes to

include in the code stream. The encoder knows the

contribution that each coding pass makes to the rate,

and can also calculate the distortion reduction

associated with each coding pass. Using this

information, the encoder can then include the coding

passes in order of decreasing distortion reduction per

unit rate until the bit budget has been exhausted.

This approach is very flexible and permits the use of

different distortion metrics.

Figs. 1 and 2 show some compression results

with the codec described above. The input image is

in both figures a 128x128 patch (this is for

simplicity, since if we use this image size there is

only one code block per subband) of the 8 bpp

“Baboon” image, and we consider only the lowest

scale vertical subband. The results are very different

depending on the distortion metric used. As we can

see in Fig. 1, if we use the classical mean squared

error (MSE) as distortion metric (note that the MSE

is not very well matched to perceived visual quality)

the 9/7 transform yields better results than the

nonlinear transform. However, the nonlinear

transform yields better perceptual quality than the

9/7 transform (see Fig. 2).

In Fig. 3 we can see that the MSE, or

equivalently the peak signal-to-noise ratio (PSNR),

is not very well matched to perceived visual quality.

So, despite their very different MSE (the PSNR

corresponding to the 9/7 transform is more than 10

dB greater than that of the nonlinear transform), the

two decoded images showed in the figure are almost

visually indistinguishable.

NONLINEAR PRIMARY CORTICAL IMAGE REPRESENTATION FOR JPEG 2000 - Applying natural image statistics

and visual perception to image compression

521

0 2000 4000 6000 8000 10000

0

0.2

0.4

0.6

0.8

1

Size (bytes)

Relative MSE

Figure 1: Relative MSE (1 denotes the MSE when any bit

plane of the considered subband is coded) as a function of

the number of bytes at the output of the encoder, for the

9/7 transform (‘x’) and the nonlinear transform (‘o’).

0 2000 4000 6000 8000 10000

0

0.2

0.4

0.6

0.8

1

Size (bytes)

Relative perceptual error

Figure 2: Relative perceptual error (1 denotes the

perceptual error when any bit plane of the considered

subband is coded) as a function of the number of bytes at

the output of the encoder, for the 9/7 transform (‘x’) and

the nonlinear transform (‘o’).

Figure 3: Decoded images corresponding to the 9/7

transform (left) and the nonlinear transform (right), when

using 7 and 10 bit planes (3395 and 3401 bytes)

respectively to code the considered subband.

5 SUMMARY AND

CONCLUSIONS

We have presented a nonlinear image representation

scheme based on a statistically-derived model of

information processing in the visual cortex. The key

feature of this image representation scheme is that

the resulting coefficients are almost statistically

independent, much more than those of the

orthogonal linear transforms (these cannot eliminate

higher-order dependencies). Such representation has

been also shown relevant to human perception.

This nonlinear image representation could be

very useful in a lossy JPEG 2000 codec. A similar

approach has been proposed in JPEG 2000 Part 2 as

an extension for visual optimisation and also similar

schemes have already been used successfully in

image compression applications. Compression

results with a simplified JPEG 2000 codec show that

the nonlinear image representation yields better

perceptual quality than the 9/7 wavelet transform

alone.

REFERENCES

Adams, M. D., and Kossentini, F., 2000. JasPer: A

software-based JPEG-2000 codec implementation.

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model. Journal of the Optical Society of America A,

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Heeger, D. J., 1992. Normalization of cell responses in cat

striate cortex. Visual Neuroscience, 9: 181-198.

Schwartz, O., and Simoncelli, E. P., 2001. Natural signal

statistics and sensory gain control. Nature

neuroscience, 4(8): 819-825.

Teo, P., and Heeger, D., 1994. Perceptual image

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Valerio, R., Simoncelli, E. P., and Navarro, R., 2003.

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