COMPUTATIONAL IMAGING. Berthold K.P. Horn
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1 COMPUTATIONAL IMAGING Berthold K.P. Horn
2 What is Computational Imaging? Computation inherent in image formation
3 What is Computational Imaging? Computation inherent in image formation (1) Computing is getting faster and cheaper precision physical apparatus is not
4 What is Computational Imaging? Computation inherent in image formation (1) Computing is getting faster and cheaper precision physical apparatus is not (2) Can t refract or reflect some radiation
5 What is Computational Imaging? Computation inherent in image formation (1) Computing is getting faster and cheaper precision physical apparatus is not (2) Can t refract or reflect some radiation (3) Detection is at times inherently coded
6 Computational Imaging System
7 Examples of Computational Imaging: (1) Synthetic Aperture Imaging (2) Coded Aperture Imaging (3) Diaphanography Diffuse Tomography (4) Exact Cone Beam Reconstruction
8 (1) SYNTHETIC APERTURE IMAGING Traditional approach: Coupling of resolution, DOF, FOV to NA Precision imaging flat illumination with: Michael Mermelstein, Jekwan Ryu, Stanley Hong, and Dennis Freeman
9 Objective Lens Parameter Coupling
10 Synthetic Aperture Imaging Traditional approach: Coupling of resolution, DOF, FOV to NA Precision imaging flat illumination New approach: Precision illumination Simple imaging Multiple images Textured illumination
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15 Synthetic Aperture Imaging Precision illumination Simple imaging Multiple images Textured illumination Image detail in response to textures Non-uniform samples in FT space
16 SAM M6
17 Creating Interference Pattern
18 Creating Interference Pattern
19 Fourier Transform of Texture Pattern
20 Interference Pattern Texture
21 Synthetic Aperture Microscopy Interference of many Coherent Beams Amplitude and Phase Control of Beams
22 Amplitude and Phase Control
23 Amplitude and Phase Control
24 Synthetic Aperture Microscopy Interference of many Coherent Beams Amplitude and Phase Control of Beams On the fly calibration Non-uniform inverse FT Least Squares
25 Wavenumber Calibration using FT
26 Hough Transform Calibration
27 Least Squares Match in FT
28 Fourier Transform of Texture Pattern
29 Uneven Fourier Sampling
30 Polystyrene Micro Beads (1µm)
31 Resolution Enhancement Reflective Optics Illumination Vaccum UV Short Wavelength
32 Reflective Optics M6
33 Resolution Enhancement Reflective Optics Illumination Vaccum UV Short Wavelength Fluorescence Mode Resolution Determined by Illumination
34 Synthetic Aperture Lithography Create pattern controlled interference Example: Two Dots Example: Straight Line Destructive interference safe zone Example: Bessel Ring.
35 (2) CODED APERTURE IMAGING Can t refract or reflect gamma rays Pinhole tradeoff resolution and SNR with: Richard Lanza, Roberto Accorsi, Klaus Ziock, and Lorenzo Fabris.
36 Coded Aperture Imaging Can t refract or reflect gamma rays Pinhole tradeoff resolution and SNR Multiple pinholes Complex masks can cast shadows
37 Masks Fresnel Camera
38 Coded Aperture Principle
39 Decoding Method Rationale
40 Coded Aperture Imaging Can t refract or reflect gamma rays Pinhole tradeoff resolution and SNR Complex masks can cast shadows Decoding by Correlation Special Masks with Flat Power Spectrum
41 Mask Design Inverse Systems
42 Maximizing SNR n n min w 2 i subject to w i = 1 i=1 i=1 yields w i = 1 n
43 Masks Legri URA
44 Masks XRT Coarse
45 Mask Design 1D Definition: q is a quadratic residue (mod p) if n s.t. n 2 q(mod p) Legendre symbol ( a ) p = 1 { 1 if a is quadratic residue otherwise Correlation with zero shift (p 1)/2 Correlation with non-zero shift (p 1)/4
46 Mask Design Auto Correlation a(i) = (p 1) 4 (1 + δ(i)) Power Spectrum A(j) = (p 1) (δ(j) + 1) 4
47 Masks Hexagonal
48 Coded Aperture Extensions Artifacts due to Finite Distance Mask / Countermask Combination
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51 Coded Aperture Backprojection Reconstruction Animation
52 Coded Aperture Extensions Artifacts due to Finite Distance Mask / Countermask Combination Multiple Detector Array Positions Synthetic Aperture radiography
53 Coded Aperture Applications Detection of Fissile Material Large Area Detector Myth Signal and Background Amplified
54 Spatially Varying Background
55 Large Area Alone Doesn t Help
56 Imaging and Large Area Do!
57 Coded Aperture Example Imaging 1/R instead of 1/R 2
58 Coded Aperture Detector Array
59 Computational Imaging System
60 Coded Aperture Example Three weak, distant radioactive sources Reconstruction Animation
61 Coded Aperture Applications Detection of Fissile Material Imaging 1/R instead of 1/R 2 Increasing Gamma Camera Resolution Replacing Rats with Mice.
62 (3) DIAPHANOGRAPHY (Diffuse Optical Tomography) Highly Scattering Low Absorption Many Sources Many Detectors with: Xiaochun Yang, Richard Lanza, Charles Sodini, and John Wyatt.
63 Diaphanography Randomization of Direction Scalar Flux Density
64 Diaphanography Approximation: Diffusion Equation v(x,y) + ρ(x,y)c(x,y) = 0 v(x,y) flux density ρ(x, y) scattering coefficient c(x,y) absorption coefficient Forward: given c(x,y) find v(x,y)
65 Diaphanography Approximation: Diffusion Equation Leaky Resistive Sheet Analog (2D)
66 Diaphanography Invert Diffusion Equation Regions of Influence.
67 (4) EXACT CONE BEAM ALGORITHM Faster Scanning Fewer Motion Artifacts Lower Exposure Uniform Resolution with: Xiaochun Yang
68 Exact Cone Beam Reconstruction Faster Scanning Fewer Motion Artifacts Lower Exposure Uniform Resolution Parallel Beam Fan Beam Planar Fan Cone Beam
69 Parallel Beam to Fan Beam Coordinate Transform in 2D Radon Space
70 Cone Beam Geometry 3D
71 Radon s Formula In 2D: ~ derivatives of line integrals In 3D: derivatives of plane integrals Can t get plane integrals from projections ( ) f (r, θ)dr dθ 1 f (x, y) dx dy r
72 Radon s Formula in 3D f(x) = 1 8π 2 2 R f (l, β) S 2 l 2 l=x β dβ where R f (l, β) = f(x) δ(x β l)dv
73 Grangeat s Trick z f (x, y, z) dx dy = f (r, φ, θ) dr dφ θ
74 Exact Cone Beam Reconstruction Data Sufficiency Condition Good Orbit for Radiation Source
75 Radon Space 2D
76 Circular Orbit is Inadequate (3D)
77 Data Insufficiency
78 Good Source Orbit
79 Exact Cone Beam Reconstruction Data Sufficiency Condition Good Orbit for Radiation Source Practical Issue: Spiral CT Scanners Practical Issue: Long Body Problem.
80 COMPUTATIONAL IMAGING (1) Synthetic Aperture Imaging (2) Coded Aperture Imaging (3) Diaphanography Diffuse Tomography (4) Exact Cone Beam Reconstruction
81 COMPUTATIONAL IMAGING
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