computational imaging - massachusetts institute of...
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COMPUTATIONAL IMAGING
Berthold K.P. Horn
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What is Computational Imaging?
• Computation inherent in image formation
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What is Computational Imaging?
• Computation inherent in image formation
(1) Computing is getting faster and cheaper
—precision physical apparatus is not
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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
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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
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Computational Imaging System
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Examples of Computational Imaging:
(1) Synthetic Aperture Imaging
(2) Coded Aperture Imaging
(3) Diaphanography—Diffuse Tomography
(4) Exact Cone Beam Reconstruction
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(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
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Objective Lens Parameter Coupling
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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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Synthetic Aperture Imaging
• Precision illumination — Simple imaging
• Multiple images — Textured illumination
• Image detail in response to textures
• Non-uniform samples in FT space
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SAM M6
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Creating Interference Pattern
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Creating Interference Pattern
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Fourier Transform of Texture Pattern
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Interference Pattern Texture
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Synthetic Aperture Microscopy
• Interference of many Coherent Beams
• Amplitude and Phase Control of Beams
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Amplitude and Phase Control
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Amplitude and Phase Control
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Synthetic Aperture Microscopy
• Interference of many Coherent Beams
• Amplitude and Phase Control of Beams
• On the fly calibration
• Non-uniform inverse FT Least Squares
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Wavenumber Calibration using FT
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Hough Transform Calibration
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Least Squares Match in FT
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Fourier Transform of Texture Pattern
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Uneven Fourier Sampling
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Polystyrene Micro Beads (1µm)
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Resolution Enhancement
• Reflective Optics Illumination
Vaccum UV — Short Wavelength
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Reflective Optics M6
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Resolution Enhancement
• Reflective Optics Illumination
Vaccum UV — Short Wavelength
• Fluorescence Mode
Resolution Determined by Illumination
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Synthetic Aperture Lithography
• Create pattern — controlled interference
Example: Two Dots
Example: Straight Line
• Destructive interference “safe zone”
Example: Bessel Ring
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(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.
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Coded Aperture Imaging
• Can’t refract or reflect gamma rays
• Pinhole — tradeoff resolution and SNR
• Multiple pinholes
• Complex masks can “cast shadows”
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Masks — Fresnel Camera
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Coded Aperture Principle
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Decoding Method Rationale
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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
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Mask Design — Inverse Systems
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Maximizing SNR
minn∑i=1
w2i subject to
n∑i=1
wi = 1
yields wi = 1n
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Masks — Legri URA
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Masks — XRT Coarse
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Mask Design — 1D
Definition: q is a quadratic residue (mod p)if ∃n s.t. n2 ≡ q(mod p)
Legendre symbol(ap
)={
1 if a is quadratic residue−1 otherwise
Correlation with zero shift (p − 1)/2Correlation with non-zero shift (p − 1)/4
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Mask Design
• Auto Correlation
a(i) = (p − 1)4
(1+ δ(i))
• Power Spectrum
A(j) = (p − 1)4
(δ(j)+ 1)
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Masks — Hexagonal
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Coded Aperture Extensions
• Artifacts due to Finite Distance
• Mask / Countermask Combination
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Coded Aperture Backprojection
Reconstruction Animation
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Coded Aperture Extensions
• Artifacts due to Finite Distance
• Mask / Countermask Combination
• Multiple Detector Array Positions
• “Synthetic Aperture” radiography
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Coded Aperture Applications
• Detection of Fissile Material
• Large Area Detector Myth
• Signal and Background Amplified
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Spatially Varying Background
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Large Area Alone Doesn’t Help
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Imaging and Large Area Do!
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Coded Aperture Example
• Imaging — 1/R instead of 1/R2
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Coded Aperture Detector Array
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Computational Imaging System
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Coded Aperture Example
Three weak, distant radioactive sources
Reconstruction Animation
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Coded Aperture Applications
• Detection of Fissile Material
• Imaging — 1/R instead of 1/R2
• Increasing Gamma Camera Resolution
• Replacing Rats with Mice
.
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(3) DIAPHANOGRAPHY
(Diffuse Optical Tomography)
• Highly Scattering — Low Absorption
• Many Sources — Many Detectors
with: Xiaochun Yang, Richard Lanza,Charles Sodini, and John Wyatt.
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Diaphanography
• Randomization of Direction
• Scalar Flux Density
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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)
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Diaphanography
• Approximation: Diffusion Equation
• Leaky Resistive Sheet Analog (2D)
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Diaphanography
• “Invert” Diffusion Equation
• Regions of Influence
.
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(4) EXACT CONE BEAM ALGORITHM
• Faster Scanning—Fewer Motion Artifacts
• Lower Exposure—Uniform Resolution
with: Xiaochun Yang
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Exact Cone Beam Reconstruction
• Faster Scanning—Fewer Motion Artifacts
• Lower Exposure—Uniform Resolution
• Parallel Beam → Fan Beam
• Planar Fan → Cone Beam
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Parallel Beam to Fan Beam
Coordinate Transform in 2D Radon Space
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Cone Beam Geometry — 3D
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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
∫ ∫1rf(x,y)dx dy
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Radon’s Formula in 3D
f(x) = − 18π2
∫S2
∂2R f(l,β)∂l2
∣∣∣∣∣l=x·β
dβ
where
R f(l,β) =∫f(x) δ(x · β− l)dV
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Grangeat’s Trick
∂∂z
∫ ∫f(x,y, z)dx dy =
∂∂θ
∫ ∫f(r ,φ,θ)dr dφ
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Exact Cone Beam Reconstruction
• Data Sufficiency Condition
• Good “Orbit” for Radiation Source
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Radon Space — 2D
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Circular Orbit is Inadequate (3D)
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Data Insufficiency
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Good Source Orbit
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Exact Cone Beam Reconstruction
• Data Sufficiency Condition
• Good “Orbit” for Radiation Source
• Practical Issue: Spiral CT Scanners
• Practical Issue: “Long Body” Problem
.
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COMPUTATIONAL IMAGING
(1) Synthetic Aperture Imaging
(2) Coded Aperture Imaging
(3) Diaphanography—Diffuse Tomography
(4) Exact Cone Beam Reconstruction
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COMPUTATIONAL IMAGING