High resolution two-dimensional resistance tomography
The disclosed 2-D and 3-D tomographic resistance imaging method improves tomographic resistance image resolution by adopting an orthogonal basis with the maximum number of elements N to describe the maximum resolution resistivity map ρ(r), where this number of elements N is set according to the number of electrodes Q; by defining the orthogonal basis according to any known constraints in the problem, thereby enhancing the resolution where it is needed; by positioning electrodes to be sensitive to these basis functions; and by choosing current I and voltage V contact electrode pairs that maximize signal-to-noise ratio.
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 the resistive sensing membrane comprises a plurality of local area resistances (r ABCD ) i to (r ABCD ) N , wherein the plurality of local area resistances (r ABCD ) i to (r ABCD ) N vary when an applied contact pressure is applied over the defined surface area of the resistive sensing membrane, wherein the applied contact pressure causes 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 comprises:
measuring a respective tetra-polar resistance of the plurality of local area resistances (r ABCD ) i to (r ABCD ) N sequentially, wherein i=1 to N, and N represents a maximum number of independent tetra-polar measurements,
wherein each respective tetra-polar resistance corresponds to a respective voltage and current ratio r( ABCD ) i =V CD /I AB , such that a respective voltage V CD is established across a first periphery contact electrode pair CD when a respective current I AB 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 reflects a local area resistance variation in a resistivity map ρ(r) of the 2-D resistance tomographic image;
wherein the resistivity map ρ(r) is related to orthogonal basis polynomial functions ϕ i (r) by an equation of ρ(r)=Σ i a i ϕ i (r), and the resistivity map ρ(r) is formed by superimposing the orthogonal basis polynomial functions ϕ i (r) having a resolution that increases with index i whose upper limit N is the same as a maximum number of independent tetra-polar resistance measurements, wherein a=(a 1 , a 2 , . . . a i , . . . ) are ordered vector of coefficients; and
displaying the 2-D resistance tomographic image through the resistivity map ρ(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 ϕ i (r) are a priori polynomial basis functions described by the Zernike polynomial equations:
Z
n
m
(
ρ
,
φ
)
=
{
R
n
m
(
ρ
)
cos
(
m
φ
)
;
for
m
even
R
n
m
(
ρ
)
sin
(
m
φ
)
;
for
m
odd
R
n
m
(
ρ
)
=
{
∑
k
=
0
n
-
m
2
(
-
1
)
k
(
n
-
k
)
!
k
!
(
n
+
m
2
-
k
)
!
(
n
-
m
2
-
k
)
!
ρ
n
-
2
k
;
for
n
-
m
even
0
;
for
n
-
m
odd
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 ϕ i (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.
4. The computer implemented method according to claim 3 , wherein the orthogonal basis functions ϕ i (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 )=Γ a
which can be diagonalized
Γ a =W T ΛW
where the matrix Λ is a diagonal matrix, and WW T =I,
Λ=diag(λ 1 ,λ 2 , . . . ,λ N )
wherein the eigenvalues W=[w 1 , w 2 , . . . , w N ] of the covariance matrix can be ordered λ 1 ≥λ 2 ≥ . . . ≥λ N , and the largest {circumflex over (N)} eigenvalues of the covariance matrix as the principle components for principle component analysis (PCA), where
Γ a PCA =W T Λ PCA W,Λ PCA =diag(λ 1 ,λ 2 , . . . ,λ {circumflex over (N)} ,0, . . . ,0)
Here W is comprised of all eigenvectors, W=[w 1 w 2 . . . w N ], Thus, the orthogonal basis then can be represented by the reduced basis w 1 , w 2 , . . . w {circumflex over (N)} ,
and the eigenvectors W of the covariance matrix with largest eigenvalues λ N 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 ϕ i (r) are determined by a combination of the a priori polynomial basis, the constrained polynomial basis, and the 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 ϕ i (r) having are chosen from a highest resolution in a constrained region is within the constrained.
7. The computer implemented method according to claim 5 , wherein in presence of the constraints, the method further comprising restricting, when constraints are present, the orthogonal basis functions ϕ i (r) to map features within only local regions of interest.
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 ϕ i (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 respective tetra-polar resistances that were measured.