Hybrid-dual-fourier tomographic algorithm for a fast three-dimensionial optical image reconstruction in turbid media

1. A method for producing a three-dimensional image of objects in a turbid medium from a hybrid dual Fourier transform, comprising:

(a) measuring intensity data at multiple detectors from multiple sources in parallel geometry;

(b) transforming data at positions of the detectors and the sources into Fourier coordinates (q d , q s ) to create a dual two-dimensional x-y spatial Fourier transform;

(c) performing a linear hybrid transformation of the Fourier coordinates (q d , q s ) of the multiple sources and multiple detectors in accordance with a predefined relationship;

(d) performing a one-dimensional inverse reconstruction for each first value of the predefined relationship; and

(e) generating a fast two-dimensional inverse Fourier transform based on each first value to produce three-dimensional image of objects in the turbid medium.

2. The method of claim 1 , wherein said predefined relationship comprises the following relationship:

{right arrow over (u)}={right arrow over (q)} d +{right arrow over (q)} s

{right arrow over (v)}={right arrow over (q)} d −{right arrow over (q)} s ;

wherein {right arrow over (u)} and {right arrow over (v)} are coordinates, and q d , q s are the Fourier coordinates of the multiple sources and multiple detectors, respectively.

3. The method of claim 1 , wherein said each first value is {right arrow over (u)} of the following relationship:

{right arrow over (u)}={right arrow over (q)} d +{right arrow over (q)} s

{right arrow over (v)}={right arrow over (q)} d −{right arrow over (q)} s ;

wherein {right arrow over (u)} and {right arrow over (v)} are coordinates, and q d and q s are the Fourier coordinates of the multiple sources and multiple detectors, respectively.

4. The method of claim 1 , wherein said predefined relationship comprises the following relationship:

u=q d +q s

v=q d −q s ;

wherein u and v are coordinates, and q d and q s are the Fourier coordinates of the multiple sources and multiple detectors, respectively.

5. The method of claim 1 , wherein said each first value is u of the following relationship:

u=q d +q s

v=q d −q s ;

wherein u and v are coordinates, and q d and q s are the Fourier coordinates of the multiple sources and multiple detectors, respectively.

6. The method of claim 1 , wherein computational burdens of image reconstruction are reduced by combining a dual two-dimensional Fast Fourier Transform inversion with a one-dimensional matrix inversion.

7. The method of claim 1 , further comprising the step of:

performing the linear hybrid transform to provide a fast three-dimensional Fourier image reconstruction for multiple sources and multiple detectors.

8. The method of claim 1 , wherein the multiple sources are selected from the group consisting of light, X-ray, micro-wave, sound, electrons, particles and mechanical vibration.

9. The method of claim 1 , wherein the detectors comprise sensors for detecting signals of one of light, sound, electricity and mechanical waves.

10. The method of claim 1 , wherein the intensity data is a function of a coordinate space selected from the group consisting of position, time, wavelength spectrum and vibration mode.

11. The method of claim 1 , wherein a hybrid dual Fourier tomographic method is implemented for at least one of continuous-wave sources, frequency domain sources and ultra-fast pulse sources.

12. The method of claim 1 , wherein steps (b) thru (e) are solved based on N-dimensional dual de-convolution.

13. The method of claim 1 , wherein the turbid medium has a cylindrical surface, and the multiple sources and multiple detectors are arranged circumferentially with respect to the cylinderical surface.

14. The method of claim 13 , wherein the hybrid dual Fourier tomographic method for the cylindrical surface is performed in accordance with the following relationships

Y ( {right arrow over (r)} d ,{right arrow over (r)} s ,z d ,z s )=∫ d{right arrow over (r)}dzW ( {right arrow over (r)} d ,{right arrow over (r)} s ,{right arrow over (r)};z d −z,z s −z ) X ( {right arrow over (r)},z );

wherein W({right arrow over (r)} d ,{right arrow over (r)} s ,{right arrow over (r)};z d −z,z s −z) is a weight function, which is function of z d −z and z s −z;

Ŷ ( q d ,q s ,{right arrow over (r)} d ,{right arrow over (r)} s )=∫ dzŴ( q d ,q s ,{right arrow over (r)},{right arrow over (r)} d ,{right arrow over (r)} s ) {circumflex over (X)} ( q d +q s ,{right arrow over (r)} )

wherein Ŷ, {circumflex over (X)}, and Ŵ are corresponding Fourier space quantities;

u=q d +q s

v=q d −q s ;

wherein u and v are coordinates, and q d , q s are the Fourier coordinates of the source and detector, respectively; and

{tilde over (Y)} ( u,v,{right arrow over (r)} d ,{right arrow over (r)} s )=∫ d{right arrow over (r)}{tilde over (W)} ( u,v,{right arrow over (r)} d ,{right arrow over (r)} s ;{right arrow over (r)} ) {tilde over (X)} ( u,{right arrow over (r)} );

wherein {tilde over (Y)}, {tilde over (X)}, and {tilde over (W)} are, respectively, Ŷ, {circumflex over (X)}, and Ŵ as functions of u and v.

15. The method of claim 1 , wherein a forward physical model of the method is based on an analytical cumulant solution to a Boltzmann photon transport equation.

16. The method of claim 15 , wherein a background Green's function of the method comprises the cumulant solution of the Boltzmann photon transport equation.

17. The method of claim 15 , wherein an accurate analytical form of a solution of the Boltzmann equation is implemented to permit computation of the forward physical model.