CLAIMS

We claim:

1. A computer implemented method for mapping a tomographic image over a surface, comprising:

defining a surface area of a resistive sensing membrane having Q periphery contact electrodes attached along a periphery of the defined surface area of the resistive sensing membrane, wherein Q comprises an integer higher than or equal to five, wherein a plurality of local area resistances {XABCD)I to (r i«r/ )\ that vary with an applied contact pressure over the defined surface area of the resistive sensing membrane cause a two-dimensional (2-D) resistance variation;

mapping a 2-D resistance tomographic image over the defined surface area of the resistive sensing membrane according to the plurality of local area resistances of the applied contact pressure to the defined surface area of the resistive sensing membrane, wherein the 2-D resistance tomographic image mapping comprising:

measuring the plurality of local area resistances (i UBCD)I to (r 4BCD)N sequentially, over each and every of N maximum combinations of two periphery contact electrode pairs from among the Q periphery contact electrodes, a respective tetra-polar resistance (r i« i_{>}) wherein i =1 to N, and , and N represents a maximum number of

independent tetra-polar measurements,

wherein each respective tetra-polar resistance T(ABCD)_{I} corresponds to a respective voltage and current ratio r(ABCD)i = VCD/IAB, such that a respective voltage VCD is established across a first periphery contact electrode pair CD when a respective current IAB is simultaneously passed across a second periphery contact electrode pair AB, wherein the first periphery contact electrode pair CD is different from the second periphery contact electrode pair AB, wherein the respective tetra-polar resistance r( i« i_{>})_{\} reflects a local area resistance variation in a resistivity map p(r) of the 2-D resistance tomographic image; wherein the resistivity map p( r) is related to orthogonal basis polynomial functions fί{ r) by an equation of p(r) = å; a, <fa( r), and the resistivity map p(r) is formed by superimposing the orthogonal basis polynomial functions f( r) having a resolution that increases with index i whose upper limit N is the same as the maximum number of

independent tetra-polar resistance measurements, wherein a = (a i, a , ... a;, ...) are ordered vector of coefficients; and

displaying the 2-D resistance tomographic image through the resistivity map p(r) on the defined surface.

2. The computer implemented method according to claim 1, wherein the defined surface area of the resistive sensing membrane is an arbitrary shape, and in a case when the defined surface area is circular, the orthogonal basis polynomial functions <ja( r) are a priori polynomial basis functions described by the Zernike polynomial equations:

whereby the integer n = {0,1,2, ... } ranks the resolution of the polynomial from low to high, and m satisfies— n < m < n.

3. The computer implemented method according to claim 2, wherein the orthogonal basis polynomial functions <ja( r) is a constrained polynomial basis having a subset of basis states being disallowed, wherein a remainder of allowable basis states are indexed from low to high resolutions, subtracting a resolution that increases with index i whose upper limit N is the same as the maximum number of independent tetra-polar resistance measurements.

4. The computer implemented method according to claim 3, wherein the orthogonal basis functions <fn( r) are determined by applying a principle component analysis (PCA) to a representative set of likely resistance maps a, as a way to generate basis functions which are sensitive to the most important variations in a resistivity profile, wherein the covariance matrix of the resistance map is calculated from equation:

Cov(a) = G_{a}

which can be diagonalized to

G_{a} = W AW

where the matrix L is a diagonal matrix, and WW^{T} = /.

L = diagC^, l_{2}, ... , )

wherein the eigenvalues W = [u^ w_{2} ... w_{w}] of the covariance matrix can be ordered l_{c} > l_{2} ³ ··· > A_{w}, and the largest N eigenvalues of the covariance matrix as principle components for principle component analysis (PCA), where

r^{p}_{a}^{CA} = W^{T}A^{PCA}W, A^{PCA} = diag(¾, l_{2}. ½, 0.0)

Here W is comprised of all eigenvectors, W = [u^ w_{2} ... w_{N}]. Thus, the orthogonal basis then can be represented by the reduced basis w_{l} w_{2}, ... , Wf_{j}

and the eigenvectors W of the covariance matrix with largest eigenvalues ln are used as orthogonal basis functions with index i whose upper limit N is the same as the maximum number of independent tetra-polar measurements.

5. The computer implemented method according to claim 4, wherein the orthogonal basis functions <ja( r) are determined by a combination of the priori, constrained and PCA basis functions having a resolution that increases with index i whose upper limit N is the same as the maximum number of independent tetra-polar measurements.

6. The computer implemented method according to claim 5, wherein a choice of the orthogonal basis functions f( r) having highest resolution is within the constrained.

7. The computer implemented method according to claim 5, wherein in presence of the constraints, the method comprising restricting the orthogonal basis functions f( r) to map features within only the local regions.

8. The computer implemented method according to claim 1, further comprising choosing locations of the periphery contact electrodes to have highest resolution to discern the orthogonal basis functions f{ r).

9. The computer implemented method according to claim 1, further comprising identifying what pairs of current and voltage electrodes should be measured to provide a maximally independent set of complete measurements while maximizing signals.

10. The computer implemented method according to claim 1, wherein a measured resistance vector is calculated from the measured respective tetra-polar resistances X{ABCD)[ to r( i« Y )N, if the measured resistance vector and a simulated resistance vector are different, then a predicted sample resistivity is adjusted to recalculate a resolution recursively via a quasi -Newton’s method, simulated annealing, or a similar self-consistent recursive method, until both of the measured resistance vector and the simulated resistance vector are within a specified tolerance.

11. A computer implemented method for mapping a tomographic image over a volume beneath a surface, comprising:

defining a resistive volume having Q surface contact electrodes attached on the defined surface area of the resistive volume, wherein Q comprises an integer higher than or equal to five, wherein a plurality of local volume resistances {XABCD)I to (TABCD)N that vary with depth and material compositions beneath the defined surface area of the resistive sensing membrane cause a three-dimensional (3-D) resistance variation;

mapping a 3-D resistance tomographic image over the defined resistive volume according to the plurality of local volume resistances beneath the defined surface area of the resistive volume, wherein the 3-D resistance tomographic image mapping comprising:

measuring the plurality of local volume resistances sequentially, over each and every of N maximum combinations of two periphery contact electrode pairs from among the Q surface contact electrodes, a respective tetra-polar resistance {XABCD wherein i =1 to N, and N

, and N represents a maximum number of independent tetra-polar

measurements,

wherein each respective tetra-polar resistance T(ABCD)_{I} corresponds to a respective voltage and current ratio X{ABCD)[ = W/Vl i/i, wherein a respective voltage VCD is established across a first surface contact electrode pair CD when a respective current \AB is simultaneously passed across a second surface contact electrode pair AB, wherein the first

surface contact electrode pair CD is different from the second surface contact electrode pair AB , wherein the respective tetra-polar resistance r( i«r/ )i reflects a local volume resistance variation in a resistivity map p( r) of the 3-D resistance tomographic image; wherein the resistivity map p( r) is related to orthogonal basis polynomial functions fί{ r) by an equation of p(r) = å/ a, <fa( r), and the resistivity map p(r) is formed by superimposing the orthogonal basis polynomial functions f( r) having a resolution that increases with index i whose upper limit N is the same as the maximum number of independent tetra-polar resistance measurements, wherein a = (a i, a , . . . a;,...) are ordered vector of coefficients; and

displaying the 3-D resistance tomographic image through the resistivity map p( r) beneath the defined surface.

12. The computer implemented method according to claim 1, wherein the defined volume of a resistively imaged volume is arbitrary, and in a case when the defined volume is spherical, the orthogonal basis polynomial functions <ja( r) are a priori polynomial basis functions described by spherical harmonic equations:

whereby the functions R{^{h}(c^{'}) are associated Legendre polynomials:

such that the integer l = {0,1,2, ... } ranks the resolution of the polynomial from low to high, and m satisfies—l £ m £ +l.

13. The computer implemented method according to claim 12, wherein the orthogonal basis polynomial functions <ja( r) is a constrained polynomial basis having a subset of basis states being disallowed, wherein a remainder of allowable basis states are indexed from low to high resolution, having a resolution that increases with index i whose upper limit N is the same as the maximum number of independent tetra-polar measurements.

14. The computer implemented method according to claim 13, wherein the orthogonal basis functions f( r) are determined by applying a principle component analysis (PCA) to a representative set of likely resistance maps a, as a way to generate basis functions which are sensitive to the most important variations in a resistivity profile, wherein the covariance matrix of the resistance map is calculated from equation:

Cov(a) = G_{a}

which can be diagonalized to

G_{a} = W AW

where the matrix L is a diagonal matrix, and WW^{T} = /.

L = diag(A_{1}, A_{2}, ... , )

wherein the eigenvalues W = [w_{1} w_{2} ... w_{N}] of the covariance matrix can be ordered l_{1} > l_{2} > > l_{N}, and the largest N eigenvalues of the covariance matrix as principle components for principle component analysis (PCA), where

r^{p}_{a}^{CA} = W^{T}A^{PCA}W, A^{PCA} = diag(¾, l_{2}. ½, 0.0)

Here W is comprised of all eigenvectors, W = [w_{1} w_{2} ... w_{w}]. Thus, the orthogonal basis then can be represented by the reduced basis w w_{2}, ... ,

and the eigenvectors W of the covariance matrix with largest eigenvalues ln are used as orthogonal basis functions with index i whose upper limit N is the same as the maximum number of independent tetra-polar measurements.

15. The computer implemented method according to claim 14, wherein the orthogonal basis functions <ja( r) are determined by a combination of the priori, constrained and PCA basis functions having a resolution that increases with index i whose upper limit N is the same as the maximum number of independent tetra-polar measurements.

16. The computer implemented method according to claim 15, wherein a choice of the orthogonal basis functions <ja( r) having highest resolution is within the constrained.

17. The computer implemented method according to claim 15, wherein in presence of the constraints, the method comprising restricting the orthogonal basis functions <ja( r) to map features within only the local regions.

18. The computer implemented method according to claim 11, further comprising choosing locations of the periphery contact electrodes to have highest resolution to discern the orthogonal basis functions f( r).

19. The computer implemented method according to claim 11, further comprising identifying what pairs of current and voltage electrodes should be measured to provide a maximally independent set of complete measurements while maximizing signals.

20. The computer implemented method according to claim 11, wherein a measured resistance vector is calculated from the measured respective tetra-polar resistances X{ABCD)[ to r( i« YJ)N, if the measured resistance vector and a simulated resistance vector are different, then a predicted sample resistivity is adjusted to recalculate a resolution recursively via a quasi -Newton’s method, simulated annealing, or a similar self-consistent recursive method, until both of the measured resistance vector and the simulated resistance vector are within a specified tolerance.