Overview: two parts to this presentation Physics-Derived Basis Pursuit for Buried Object Identification Fast 3D Blind Deconvolution of Even Point Spread Functions Andrew E. Yagle, January 2005

Voltage Response Buried Object Metal Detector Coil Physics Derived Basis Pursuit in Buried Object Identification Springfield, VA, January 2005 Jay A. Marble and Andrew E. Yagle Part I of this Presentation

Air Ground Primary Magnetic Field Secondary Magnetic Field Air Ground Source Induced Sources Vertical Dipole Horizontal Dipole Transmitted FieldsFields at Buried Object Response Induced Sources (approximation) Metal Detector Phenomenology Coil An Electromagnetic Induction (EMI) Metal Detector utilizes a coil of wire to generate a magnetic field. This magnetic field interacts with a buried object inducing “swirling” electrical (eddy) currents. These induced currents form secondary sources, which can be modeled as vertical and horizontal dipoles. The metal detector coil is then switched from a transmitter to a receiver. + -

EMI Phenomenology Air Ground Primary Magnetic Field Buried Sphere Current Source Electronics & Sampler Data Storage Simplified EMI System Concept Air Ground Source Secondary Magnetic Field Source H-field Incident Field at Object Metal Object Reaction

Air Ground Source Source H-field (x,y,-d) (x,y,h) EMI Phenomenology

Metal Object Reaction Secondary Magnetic Field prpr pzpz EMI Phenomenology * Model assumes a solid spherical target.

Induced Magnetic Sources pxpx pzpz * Model no longer assumes a solid spherical target. H 0x – Horizontal magnetic field at the center of the target produced by the source magnetic dipole. H xz – Vertical magnetic field at the receive coil produced by the horizontal induced magnetic dipole. H 0z – Vertical magnetic field at the center of the target produced by the source magnetic dipole. H zz – Vertical magnetic field at the receive coil produced by the vertical induced magnetic dipole. Target Magnetic Polarizability Vector EMI Phenomenology

Vertical DipoleHorizontal Dipole Physics Derived Basis Functions The  Basis Function The  W Basis Function The vertical dipole produces the  basis. The horizontal dipole produces the W basis.

Physics Derived Basis Functions d – depth of buried object a – Polarizability of object in Z-direction. b – Polarizability of object in X-direction. The spatial signal is composed of the  (x) and W(x) basis functions. The basis functions are parameterized by depth. Any object at the same depth will have the same basis. The object’s shape affects the weighting coeffs “a” and “b”.

Sphere at 0.0m All objects simulated at 0.25m depth. These 3 signals come from identical spheres at different depths. Canonical Depths Sphere at 0.25mSphere at 1.0m

Canonical Depths a  Component bW Component a  Component bW Component a  Component bW Component Sphere at 0.0m These 3 signals come from identical spheres at different depths. Sphere at 0.25mSphere at 1.0m

dd WdWd 2D Signal Subspace Higher Dimensional Space 2D Plane Spanned by  d and W d The natural  d and W d bases are non-orthogonal. An interesting fact is that the  d and W d bases form an angle of 62° regardless of object depth d. All metal objects at this depth will exist in this signal subspace. dd WdWd 62°

Subspaces for Objects at Different Depths The bases of objects at a second depth,  d2 and W d2, span a second plane that is non-orthogonal to the plane spanned by the first depth bases,  d1 and W d1.  d1 W d1  d2 W d2 First Depth Subspace Second Depth Subspace

SphereCylinderFlat Plate All objects simulated at 0.25m depth. All 3 of these signals are represented by a vector in the same 2D subspace. Canonical Shapes

SphereCylinderFlat Plate All objects simulated at 0.25m depth. All 3 of these signals are represented by a vector in the same 2D subspace. Canonical Shapes a  Component bW Component

Effect of Object Shape, Size, and Content dd WdWd 62° 45°-Sphere 30°-Cylinder 10°-Flat Plate Larger or More Conductive Sphere Larger or More Conductive Cylinder Larger or More Conductive Flat Plate abangle sphere1145° cylinder10.530° flat plate ° 2D Subspace for Objects at Depth “d” The object’s polarizability (the a and b coeffs) determines the angle of the signal in the 2D subspace. Increasing the object’s size increases the weightings, a and b. More conductive metal also increases the weightings, a and b.

Subspace Identification Using Projections

Deep Sphere Deep Sphere Deep Sphere Subspace Identification Performance Table 3a: Norm After Projection into Subspace (No Noise) SpheresFlat PlatesCylinders ShallowMidDeepShallowMidDeepShallowMidDeep Shallow Mid Deep Table 3b: Norm After Projection into Subspace (Noise Var: 0.01) SpheresFlat PlatesCylinders ShallowMidDeepShallowMidDeepShallowMidDeep Shallow Mid Deep Table 3c: Norm After Projection into Subspace (Noise Var: 0.05) SpheresFlat PlatesCylinders ShallowMidDeepShallowMidDeepShallowMidDeep Shallow Mid Deep Mid Depth Sphere Mid Depth Sphere Mid Depth Sphere

600Hz to 60kHz New EMI Modality Georgia Tech EMI

Fast 3D Blind Deconvolution of Even Point Spread Functions Andrew Yagle and Siddharth Shah The University of Michigan, Ann Arbor Part II of this Presentation

Motivation Many blind deconvolution algorithms need an initial PSF estimate Can be tedious and problematic to measure PSF accurately Blind Deconvolution (Don’t need PSF) BUT But Blind Deconvolution algorithms tend to be slow ! Also many still need initial PSF estimate What we need A fast algorithm that performs blind deconvolution

We will show: An algorithm that performs blind deconvolution that is Fast Parallelizable A linear algebraic formulation Non iterative (at least for the Least Squares solution)

Assumptions 1. Point Spread Function or PSF h(x,y,z) is even in 3-D. Reasonable in optics. PSFs are symmetric in x, y, and z. Hence h(x,y,z) = h(-x,-y,-z) 2. We know the PSF or image support size. 3. Image has compact support. Potential Problem: Asymmetric PSFs due to optical aberrations Can these be solved too ? YES (later)

Formulation y(i 1,i 2,i 3 ) = h(i 1,i 2,i 3 ) *** u(i 1,i 2,i 3 ) + n(i 1,i 2,i 3 ) DATA PSF OBJECT NOISE where u(i 1,i 2,i 3 ) ≠ 0 for 0 ≤ i 1,i 2,i 3 ≤ M-1 h(i 1,i 2,i 3 ) ≠ 0 for 0 ≤ i 1,i 2,i 3 ≤ L-1 y(i 1,i 2,i 3 ) ≠ 0 for 0 ≤ i 1,i 2,i 3 ≤ N-1 N=L+M-1 h(i 1,i 2,i 3 ) = h(L-i 1,L-i 2,L-i 3 ) (even PSF) n(i 1,i 2,i 3 ) is zero mean white Gaussian noise. PROBLEM Given only data y(i 1,i 2,i 3 ) reconstruct the object u(i 1,i 2,i 3 ) and the PSF h(i 1,i 2,i 3 )

1-D Solution Equating coefficients, we get the following matrix Toeplitz Structure

2-D Problem or Example Solve:

2-D Example Solution Toeplitz Block Toeplitz structure Size of matrix (2M + L- 2) 2 X (2M 2 )

3-D Solution Equating coefficients we would get a doubly nested Toeplitz matrix Matrix size: (2M + L-2) 3 X (2M) 3 Q: So we have solved the 3D problem ? A: Not quite !! If M=5 and L=3 then the matrix size is 4913 X1024 It will be intractable to use this method “as is” in 3D !

Fourier Decomposition Using conjugate symmetry The point ? The last equation is decoupled into a set of M 2D problems ! 0 ≤ k ≤ M-1

2-D to 1-D So we broke down a huge 3D problem to M simpler 2D problems What next ? Substitute for x k in each 2D problem and you would get M 1D problems in z To summarize We broke up a large 3D problem into M 2 1D problems

2-D scale factors Note that each 1D problem will be correct upto a scale factor. Decoupling 2D to 1D Each row is solved to a scaled factor. How do we get the whole 2D solution correctly ? Solve along columns and compare coefficients 1D FT along columns 1D FT along rows solve c 1 U(1,:) c 2 U(2,:) c 3 U(3,:) c 4 U(4,:) solve d 1 U(:,1) d 2 U(:,2) d 3 U(:,3) d 4 U(:,4) Compare and scale U(x,y)

3-D scale factors We just learned how a 2D problem could be correctly scaled Decouple 3D to 2D Solve 3D Decouple 2D to 1D Solve 1D Scale 1D Sol ns Scale 2D Sol ns 2D solutions c1 c2 d1 d2 U(x,y,z) Decouple along z Solve 2D problems Decouple along x Compare and scale 2D problems

Stochastic Case In presence of noise the nullspace of the toeplitz structure no longer exists. We can find “nearest” nullspace using Least Squares (LS) (fast) Can use structure of matrix to solve by structured least squares (STLS) (slow but more accurate) We can show that such norm minimization will give us the Maximum Likelihood Estimate of the object u(x,y,z)

Simulations Synthetic bead image (30X30X30), (3X3X3) PSF, no noise

Stochastic case: STLN vs. LS Least Squares v/s STLN comparison Least squares does well at high SNRs but at low and medium SNRS STLN is better. 7X7X7 image. 3X3X3 PSF. 50 iterations per SNR

Comparison with Lucy Richardson SNR MSE Time Our algorithm gave a lower MSE. In LR Final accuracy even in absence of noise depends on initial PSF estimate. Our algorithm need only a fixed amount of time to solve independent of the SNR. LR needs more time and time to solve depends on the SNR.