# zoom_x2_3D.m

E. Koenig, 08/04/2010

Augmentation of information of a 3D field by a factor 2.

## Contents

- References
- Description
- Use
- Inputs
- Outputs
- Example

## References

`[1] Koenig, E., Chainais, P.: Virtual resolution enhancement of scale invariant textured images using stochastic processes, Proceedings of ICIP 2009 (2009)`

`[2] Chainais, P.: Infinitely divisible cascades to model the statistics of natural images, IEEE Trans. on P.A.M.I. 29(12), 2105-2119 (2007)`

## Description

The size of the initial field is multiplied by two while new small scale information is added below the initial voxel resolution. The initial 3D field is supposed to be scale invariant with known scale invariance parameters. Furthermore, it can be modelized by a 3D compound Poisson cascade (CPC) [2].

This field is seen as an integrated 3D CPC modified in variance by alpha and in mean by I_0:

The high resolution field must have the same scale invariance properties as the initial field but on a larger range of scales. The new information added must be scale invariant. It is created by the stochastic model of CPC [2].

The augmentation method is divided into 5 parts:

- interpolation: interpolates the initial field to the new resolution. This step adds no information to the initial field. It is realized using splines.
- fractional derivation: apply a fractional derivation to access the underlying CPC (Q) of the initial field.
- augmentation of information: add new details (created by CPC) with the same scale invariant properties as the initial field.
- fractional integration: apply a fractional integration to get the new augmentaed field with the right scale invariant properties.
- conservation: ensures that a zoom out to the initial resolution gives the initial field back.

For more information on this method see [1].

Needs the functions:

- interp_x2_3D
- derivfrac3D
- givexpIDC2D
- cpc_3D to be compiled, in the folder CPC_3D
- integfrac3D
- zoom_out_x2_3D

## Use

[I_aug, seed_out] = zoom_x2_3D(I, law, param, H, alpha_ref, I_0_ref, seed_in);

## Inputs

- I: input 3D field
- law: distribution law used to create the details with good properties:

            - 1: log-normal
            - 2: exponential
            - 3: log-Poisson

- param: parameters of the law. The law and its parameters are determined by a previous analysis of the scale invariance properties of the initial field.
- H: parameter of the fractional derivation and integration.
- alpha_ref: if known, this is the value of the modified variance of the initial field. 0 if not known.
- I_0_ref: if known, this is the value of the modified mean of the initial field. 0 if not known.
- seed_in: value of the initial seed used to create the stochastic details.

## Outputs

- I_aug: augmented 3D field. This field has the same scale invariance properties as the initial field but on a larger range of scales, and a zoom out to the initial resolution yields the initial field back.
- seed_out: last seed used during the process. It can be used to iterate the augmentation process whithout having redundances when creating small scale details.

## Example

For a complete example of use, see the test program test_zoom_3D.