Inversion of the divisive normalization:

Algorithms and new possibilities

Jesús Malo

Dpt. Optics at Fac. Physics, Universitat de València (Spain)

+NASA Ames Research Center, Moffett Field, CA

2Outline

• Background• The model• Inversion is needed• The model is not analytically invertible

• The differential method• The method• Existence and uniqueness of the solution• This is not a gradient-based search

• Inversion Results• Applications Image Coding and Vision Reseach

• Final Remarks

3Background

The model:

Transform

TA

Image

a

k

kikii AT a

Rr

Response

j

q

jijq

i

p

ii

hr

a

a

ai

ri

4

TaA

Background

The inversion would be useful (stimuli design)

i

i-ikii

k Ti0

δ )(1

)(A 10

0kii

k Ti )(1

)(A 10

0

A aaAA

5

Ra r

Background

The inversion would be useful (stimuli design)

i

i-ikii

k RTi ) )(1

)(A0

110 (δ

TA r

6

aT

Background

The inversion would be useful (image coding)

a][A' 11 QT

AR

r

])[(A' 111 rQRT

7Background

However, the normalization is not invertible!

Without kernel

22

2

a

a

ii

iir

222 aa iiiii rr

jjiji

ii

hr 22

2

a

a

222 aa ij

jijiii hrr

With kernel

i

iii r

r

1

a2 ?

8

')'()'()''( drrRrRdrrR -1-1-1 Locally:

The differential method

) )'( r' ,a'( aRGiven the pair:

'))'a('('a drrRd -1 (eq.1)

'))'('(aa0

0 drarRr

r

1-

For any response, r (given some initial conditions (r0, a0) ):

(eq.2)

9

-1-1 RrR )a()( aa drrR

r

r

1- b

a

)(aa ab

The differential method

10The differential method

Existence and Uniqueness of the solution

is integrable if Lipschitz: i.e is bounded -1R

ij

j

q

jijq

i

q

j

p

iij

j

q

jijiq

p

i

j

iij h

h

qh

pR

R 2

11

a

aa

a

a

a)a(

j

q

jijq

i

p

ii

hR

a

a)a(

'))'a('('a drrRd -1

11The differential method

ai

ri

ij

j

q

jijq

i

p

i

q

jij

j

q

jijq

i

p

iijij h

h

qh

pR 2

11

a

aa

a

a

1001

)a(

j

q

jijq

i

p

iii

hR

a

aa

1001

)a(

ai

ri

12The differential method

Summary of the differential method:* The inverse is obtained solving from (r0, a0) '))'a('('a drrRd -1

'))'('(aa0

0 drarRr

r

1- * The differential equation is solved using a 4th order Runge-Kutta algorithm

* In each step of the integral, the jacobian is computed using,

ij

j

q

jijq

i

p

i

q

jij

j

q

jijq

i

p

iijij h

h

qh

pR 2

11

a

aa

a

a

1001

)a(

-1-1 RrR )a'()'(

* The convergence to the appropriate solution is theoretically guaranteed

13The differential method

The conventional gradient-based search

•Local minima

•The result highly depends on the initial guess

•No apriori information about the number of iterations

2)a()a(ε rr

a

Initial guess

Estimated solution

Actual solution

14Inversion Results

ConvergenceBlock-DCT

1 step 2 steps

4 steps

6 steps

8 steps

15Inversion Results

ConvergenceWavelets

3 steps 6 steps

16Applications I: Vision Science

Basis functions of the response domain(given a masking pattern A0, a0, r0 ):

In the transform domain ( )

00

01010100

0 )) )a,(δaa i-ii-i rRrRrRi δ )((δ(

)a,(a 00iδ

0

000 )a,(a i-iRi δ )(aδ -1

00 00

0 )( )a,(a ijijRi δ)(aδ -1

In the spatial domain ( )

)a,(α )A,( 00

1100 iTi δa δA

)A,( 00iδA

If the jacobian is not diagonal, the function is not a basis function of T!

17Applications I: Vision Science

Masking Pattern (A0 )

Basis functions (space)

Basis functions (transform)

Basis functions (response)

18Applications II: Image coding

O.27 bpp

e=1.69

e=1.74

e=1

19Final remarks

•The divisive normalization model is not invertible due to the interaction between transform coefficents

•The inversion method allows to explore a number of new possibilities both in vision science and image coding

•The differential method presented here is a non conventional method to solve this nonlinear problem

•The differential method performs better than comparable gradient based seach algorithms

20Appendix I:Parameters block-DCT

Parameters Responses

jjiji

iii

iii

hRr 2

2

a

aa

100)a(

Response (contrast masking)

ij

jjiji

ij

iij

jjiji

iiij

iij h

hh

R 22

2

2

a

aa2

a

a2

100)a(

Jacobian

21Appendix II:Parameters wavelet

Image Transform

Kernel

Inhibition

Response

22Conclusions

W as complement of

Expression for the computation of W

•New tools to be used in the transform coding problem:

R-1

W

The inversion method reconstructs the image from the response in few Runge-Kutta iterations

•The response representation achieves good decorrelation results in terms of and W

•Promising compression results are obtained with a uniform quantization of the response representation