Diffraction tomography systems and methods with arbitrary source and detector array surfaces

Claim 3

A method according to claim 1 further comprising: preprocessing said obtained signals before said converting step wherein said signals converted by said converting step are preprocessed signals, and wherein said preprocessing includes transforming said obtained signals with a general transform.

Claim 4

A method according to claim 2 wherein: said sources of energy are plane wave sources; and said general transform transforms a determined scattered field into a plane wave scattering amplitude for input into said filtered backpropagation technique.

Claim 5

A method according to claim 3 wherein: said preprocessed signals are filtered according to the filtered backpropagation technique ##EQU30## wherein s o is a unit propagation vector of the incident plane waves, s is a unit vector which may assume all directions, r is the vector from a coordinate system origin to a particular point of reconstruction in said coordinate system, γ(s;s o ) is the plane wave scattering amplitude, χ and χ o are respectively the angles formed by s and s o with a fixed reference direction, and O(r;χ o ) is the partial reconstruction of the object.

Claim 6

A method according to claim 4 wherein said object is two-dimensional further comprising: summing said partial reconstructions over a plurality of view angles to obtain a full two-dimensional reconstruction of said object.

Claim 7

A method according to claim 5 wherein: said detectors are located along a circular boundary.

Claim 8

A method according to claim 6 wherein: said energy sources are located along a circular boundary.

Claim 9

A method according to claim 5 wherein: said detectors are located along a boundary of weak curvature.

Claim 10

A method according to claim 5 wherein: said detectors are located along a boundary which approximates two infinite parallel lines.

Claim 11

A method according to claim 4 wherein said preprocessing further comprises: transforming said obtained signals from analog to digital form; Fourier transforming said digital signals in time; and determining the scattered field from said Fourier tansformed signals.

Claim 12

A method according to claim 2 wherein: said sources of energy are cylindrical or spherical wave sources; said general transform transforms a scattered field into a cylindrical wave response; and said cylindrcial wave response is slant stacked to provide the plane wave scattering amplitude for input into said filtered backpropagation technique.

Claim 13

A method according to claim 11 wherein: said preprocessed signals are filtered according to the filtered backpropagation technique ##EQU31## wherein s o is a unit propagation vector of the incident waves, s is a unit vector which may assume all directions, r is the vector from the origin of a coordinate system to a particular point of reconstruction in said coordinate system, γ(s;s o ) is the plane wave scattering amplitude, χ and χ o are respectively the angles formed by s and s o with a fixed reference direction, and O(r;χ o ) is the partial reconstruction of the object.

Claim 14

A method according to claim 12 wherein said object is two-dimensional further comprising: summing said partial reconstructions over a plurality of view angles to obtain a full two-dimensional reconstruction of said object.

Claim 15

A method according to claim 13 wherein said preprocessing further comprises: transforming said obtained signals from analog to digital form; Fourier transforming said digital signals in time; and determining the scattered field from said Fourier tansformed signals.

Claim 16

A method according to claim 14 wherein: said detectors are located along a circular boundary.

Claim 17

A method according to claim 15 wherein: said energy sources are located along a circular boundary.

Claim 18

A method according to claim 14 wherein: said detectors are located along a boundary of weak curvature.

Claim 19

A method according to claim 14 wherein: said detectors are located along a boundary which approximates two infinite parallel lines.

Claim 20

A method according to claim 1 further comprising: preprocessing said obtained signals before said converting step wherein said signals converted by said converting step are preprocessed signals.

Claim 21

A method according to claim 19 wherein said sources of energy are cylindrical or spherical wave sources and said proprocessing comprises: transforming said obtained signals from analog to digital form; Fourier transforming said digital signals in time; and determining the scattered field from said Fourier transformed signals.

Claim 22

A method according to claim 20 further comprising: summing said partial reconstructions to obtain a full reconstruction by integrating over the point locations of said sources.

Claim 23

A method according to claim 20 wherein: said object is two-dimensional; said sources and detectors are located along circular boundaries; and said scattered field is filtered according to a filtered backpropagation technique ##EQU32## where ##EQU33## and wherein O(r,β) is a partial reconstruction of said object, r is the vector from the origin of a coordinate system to a particular point of reconstruction in the coordinate system, β is the angle formed by R o with an arbitrary reference where R o is the cylindrical or spherical wave source location, σ is the angle made by r' with an arbitrary reference direction where r' is a detector point location, Ψ(σ,β) is the scattered field, χ and χ o are respectively the angles formed by s and s o with fixed reference directions where s is a unit vector which may assume all directions and s o is a unit propagation vector of the incident waves, and F R is a function given by ##EQU34## where H n is the n'th order Hankel function of the first kind, and k is the wave numbeer, and R is the radius of the detector boundary and R o is the radius of the source boundary.

Claim 24

A method according to claim 20 wherein: said object is two-dimensional; said detectors are located along a boundary which approximates two infinite parallel lines; and said scattered field is filtered according to a filtered backpropagation technique ##EQU35## where ##EQU36## and wherein O(r,l o ) is a partial reconstruction of said object, r is the vector from the origin of a coordinate system to a particular point of reconstruction in the coordinate system, l o is the location of R o along a straight boundary line where R o is the cylindrical or spherical wave source location, l' is the location of r' along the detector surface boundary where r' is the detector point location, Ψ(l';l o ) is the scattered field, χ and χ o are respectively the angles formed by s and s o with fixed reference directions where s is a unit vector which may assume all directions and s o is a unit propagation vector of the incident waves, n' is the unit outward normal vector to the detecting surface, and n o is the unit outward normal vector to the source surface.

Claim 25

A method according to claim 20 wherein: said object is two-dimensional; said detectors are located along a boundary of weak curvature; and said scattered field is filtered according to a filtered backpropagation technique ##EQU37## where ##EQU38## and wherein O(r,l o ) is a partial reconstruction of said object, r is the vector from the origin of a coordinate system to a particular point in the coordinate system, l o is the location of R o along a straight boundary line where R o is the cylindrical or spherical wave source location, l' is the location of r' along the detector surface boundary where r' is the detector point location, Ψ(r';R o ) is the scattered field, χ and χ o are respectively the angles formed by s and s o with fixed reference directions where s is a unit vector which may assume all directions and s o is a unit propagation vector of the incident waves, n' is the unit outward normal vector to the detecting surface, and n o is the unit outward normal vector to the source surface.

Claim 28

A method according to claim 25 wherein: said preprocessed signals are filtered according to the filtered backpropagation technique ##EQU39## wherein s o is a unit propagation vector of the incident waves, s is a unit vector which may assume all directions, r is the vector from the origin of a coordinate system to a particular point of reconstruction in said coordinate system, γ(s;s o ) is the plane wave scattering amplitude, χ and χ o are respectively the angles formed by s and s o with a fixed reference direction, and O(r;χ o ) is the partial reconstruction of the object.

Claim 29

A method according to claim 26 wherein said object is a two-dimensional object, further comprising: summing said partial reconstructions over a plurality of view angles to obtain a full two-dimensional reconstruction of said object.

Claim 30

A method according to claim 26 wherein said sources of energy are plane wave sources, and said preprocessing step comprises: transforming said produced signals from analog to digital form; Fourier transforming said digital signals in time; determining the scattered field from said Fourier transformed signals; and determining the plane wave scattering amplitude by transforming said scattered field with a general transform.

Claim 31

A method according to claim 28 wherein: said detectors and sources are located along circular boundaries.

Claim 32

A method according to claim 28 wherein: said detectors and sources are located along boundaries of weak curvature.

Claim 33

A method according to claim 28 wherein: said detectors and sources are located along boundaries which approximate two infinite parallel lines.

Claim 34

A method according to claim 26 wherein said sources of energy are cylindrical or spherical wave sources and said preprocessing step comprises: transforming said produced signals from analog to digital form; Fourier transforming said digital signals in time; determining the scattered field from said Fourier transformed signals; transforming the scattered field into a cylindrical wave response according to a general transform; and determining the plane wave scattering amplitude by slant stacking said cylindrical wave response.

Claim 35

A method according to claim 32 wherein: said detectors and sources are located along circular boundaries.

Claim 36

A method according to claim 32 wherein: said detectors and sources are located along boundaries of weak curvature.

Claim 37

A method according to claim 32 wherein said detectors and sources are located along boundaries which approximate two infinite parallel lines.

Claim 38

A method according to claim 25 wherein said sources of energy are cylindrical or spherical wave sources and said preprocessing comprises: transforming said produced signals from analog to digital form; Fourier transforming said digital signals in time; and determining the scattered field from said Fourier transformed signals.

Claim 39

A method according to claim 36 further comprising: summing said partial reconstructions by integrating over the point locations of said sources of energy to obtain a full reconstruction of said object.

Claim 40

A method according to claim 36 wherein: said sources and detectors are located along circular boundaries; and said scattered field is filtered according to the filtered backpropagation technique ##EQU40## where ##EQU41## and wherein O(r,β) is a partial reconstruction of said object, r is the vector from the origin of a coordinate system to a particular point located in said coordinate system, β is the angle formed by R o with an arbitrary reference where R o is the cylindrical or spherical wave source location, σ is the angle made by r' with arbitrary reference direction where r' is a detector point location, Ψ(σ,β) is the scattered field, χ and χ o are respectively the angles formed by s and s o with fixed reference directions where s is a unit vector which may assume all directions and s o is a unit propagation vector of the incident waves, and F R is a function given by ##EQU42## where H n is the n'th order Hankel function of the first kind, and k is the wavenumber. and R is the radius of the detector boundary and R is the radius of the source boundary.

Claim 41

A method according to claim 36 wherein: said detectors are located along a boundary which approximates two infinite parallel lines; and said scattered field is filtered according to the filtered backpropagation technique ##EQU43## where ##EQU44## and wherein O(r,l o ) is a partial reconstruction of said object, r is the vector from the origin of a coordinate system to a particular point in the coordinate system, l o is the location of R o along a straight boundary line where R o is the cylindrical or spherical wave source location, l' is the location of r' along the detector surface boundary where r' is the detector point location, Ψ(l o ;l') is the scattered field, χ and χ o are respectively the angles formed by s and s o with fixed reference directions where s is a unit vector which may assume all directions and s o is a unit propagation vector of the incident waves, n' is the unit outward normal vector to the detecting surface, and n o is the unit outward normal vector to the source surface.

Claim 42

A method according to claim 36 wherein: said detectors are located along a boundary of weak curvature; and said scattered field is filtered according to a filtered backpropagation technique ##EQU45## where ##EQU46## and wherein O(r,l o ) is a partial reconstruction of said object, r is the vector from the origin of a coordinate system to a particular point in said coordinate system, l o is the location of R o along a straight boundary line where R o is the cylindrical or spherical wave source location, l' is the location of r' along the detector surface boundary where r' is the detector point location, Ψ(r';R o ) is the scattered field, χ and χ o are respectively the angles formed by s and s o with fixed reference directions where s is a unit vector which may assume all directions and s o is a unit propagation vector of the incident waves, n' is the unit outward normal vector to the detecting surface, and n o is the unit outward normal vector to the source surface.

Claim 45

A system according to claim 41 wherein: said processing means includes means for filtering and backpropagating said preprocessed signals according to the filtered backpropagation technique ##EQU47## wherein s o is a unit propagation vector of the incident waves, s is a unit vector which may assume all directions, r is the vector from the origin of a coordinate system to a particular point of reconstruction in said coordinate system, γ(s;s o ) is the plane wave scattering amplitude, χ and χ o are respectively the angles formed by s and s o with a fixed reference direction, and O(r;χ o ) is the partial reconstruction of the object.