Tomography. Forward projectionsp θ (r) are known as a Radon transform. Objective: reverse this process to form the original image
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1 C. A. Bouman: Digital Image Processing - January 9, Tomography Many medical imaging systems can only measure projections through an object with density f(x,y). Projections must be collected at every angleθ and displacement r. Forward projectionsp θ (r) are known as a Radon transform. y x r θ p (r) θ Objective: reverse this process to form the original image f(x,y). Fourier Slice Theorem is the basis of inverse Inverse can be computed using convolution back projection (CBP)
2 C. A. Bouman: Digital Image Processing - January 9, Medical Imaging Modalities Anatomical Imaging Modalities Chest X-ray Computed Tomography (CT) Magnetic Resonance Imaging (MRI) Functional Imaging Modalities Signal Photon Emission Tomography (SPECT) Positron Emission Tomography (PET) Functional Magnetic Resonance Imaging (fmri)
3 C. A. Bouman: Digital Image Processing - January 9, Multislice Helical Scan CT Multislice CT has a cone-beam structure Path of Helical Scan X-ray Source Plane of Desired Image Reconstruction Detector Array
4 C. A. Bouman: Digital Image Processing - January 9, Example: CT Scan Gantry rotates under fiberglass cover 3D helical/multislice/fan beam scan
5 C. A. Bouman: Digital Image Processing - January 9, Photon Attenuation λx Y X Ray Source Material with density u(x) Pin Hole Columnator T x x - depth into material measured in cm Y x - Number of photons at depth x λ x = E[Y x ] Number of photons is a Poisson random variable P{Y x = k} = e λ x λ k x k! As photons pass through material, they are absorbed. The rate of absorption is proportional to the number of photons and the density of the material..
6 C. A. Bouman: Digital Image Processing - January 9, Differential Equation for Photon Attenuation λx Y X Ray Source Material with density u(x) Pin Hole Columnator x T The attenuation of photons obeys the following equation dλ x dx = µ(x)λ x whereµ(x) is the density in units of cm 1. The solution to this equation is given by λ x = λ e x µ(t)dt So we see that x µ(t)dt = log log ( λx ( λ Yx λ ) )
7 C. A. Bouman: Digital Image Processing - January 9, Estimate of the Projection Integral λx Y X Ray Source Material with density u(x) Pin Hole Columnator T x A commonly used estimate of the projection integral is T ( ) µ(t)dt = YT log λ where: λ is the dosage Y T is the photon count at the detector
8 C. A. Bouman: Digital Image Processing - January 9, Positron Emission Tomography (PET) Detector i x j - emission rate A ij x j - detection rate Detector i E[y i ] = j A ij x j Subject is injected with radio-active tracer Gamma rays travel in opposite directions When two detectors detect a photon simultaneously, we know that an event has occurred along the line connecting detectors. A ring of detectors can be used to measure all angles and displacements
9 C. A. Bouman: Digital Image Processing - January 9, Example: PET/CT Scan Generally low space/time resolution Little anatomical detail couple with CT Can detect disease
10 C. A. Bouman: Digital Image Processing - January 9, Coordinate Rotation Define the counter-clockwise rotation matrix [ ] cos(θ) sin(θ) A θ = sin(θ) cos(θ) Define the new coordinate system(r,z) [ ] [ ] x r = A y θ z Geometric interpretation z y r θ x Inverse transformation [ ] r z = A θ [ x y ]
11 C. A. Bouman: Digital Image Processing - January 9, Integration Along Projections Consider the function f(x,y). y f(x,y) x We compute projections by integrating alongz for eachr. z y r θ x The projection integral for each r and θ is given by ( [ ]) r p θ (r) = f A θ dz z = f (rcos(θ) zsin(θ),rsin(θ)+zcos(θ))dz
12 C. A. Bouman: Digital Image Processing - January 9, The Radon Transform The Radon transform of the function f(x,y) is defined as p θ (r) = f (rcos(θ) zsin(θ),rsin(θ)+zcos(θ))dz The geometric interpretation is f(x,y) y x r θ p (r) θ Notice that the projection corresponding to r = goes through the point (x, y) = (, ).
13 C. A. Bouman: Digital Image Processing - January 9, Let The Fourier Slice Theorem P θ (ρ) = CTFT {p θ (r)} F(u,v) = CSFT {f(x,y)} Then P θ (ρ) = F(ρcos(θ),ρsin(θ)) P θ (ρ) isf(u,v) in polar coordinates! v ρ θ u
14 C. A. Bouman: Digital Image Processing - January 9, Proof of the Fourier Slice Theorem By definition p θ (r) = f ( A θ [ r z The CTFT ofp θ (r) is then given by P θ (ρ) = = = ]) dz p θ (r)e j2πρr dr [ ( [ ]) r f A θ z ( [ ]) r f A θ e j2πρr dzdr z We next make the change of variables [ ] [ ] r x = A z θ. y ] dz e j2πρr dr Notice that the Jacobian is A θ = 1, and thatr = xcos(θ)+ ysin(θ). This results in P θ (ρ) = = = F(ρcos(θ),ρsin(θ)) f (x,y)e j2πρ[xcos(θ)+ysin(θ)] dxdy f (x,y)e j2π[xρcos(θ)+yρsin(θ)] dxdy
15 C. A. Bouman: Digital Image Processing - January 9, Alternative Proof of the Fourier Slice Theorem First let θ =, then Then P (ρ) = = p (r) = f(r,y)dy p (r)e 2πjrρ dr [ ] f(r,y)dy e 2πjrρ dr = f(r,y)e 2πj(rρ+y) drdy = F(ρ,) By rotation property of CSFT, it must hold for any θ.
16 C. A. Bouman: Digital Image Processing - January 9, Inverse Radon Transform Physical systems measure p θ (r). From these, we compute P θ (ρ) = CTFT{p θ (r)}. v Transformed Projection at each angle θ u Next we take an inverse CSFT to formf(x,y). Problem: This requires polar to rectagular conversion. Solution: Convolution backprojection
17 C. A. Bouman: Digital Image Processing - January 9, Convolution Back Projection (CBP) Algorithm In order to compute the inverse CSFT of F(u,v) in polar coordinates, we must use the Jacobian of the polar coordinate transformation. dudv = ρ dθdρ This results in the expression f(x,y) = F(u,v)e 2πj(xu+yv) dudv = = π π [ P θ (ρ)e 2πj(xρcos(θ)+yρsin(θ)) ρ dθdρ ] ρ P θ (ρ)e 2πjρ(xcos(θ)+ysin(θ)) dρ } {{ } g θ (xcos(θ)+ysin(θ)) dθ Then g(t) is given by g θ (t) = ρ P θ (ρ)e 2πjρt dρ = CTFT 1 { ρ P θ (ρ)} = h(t) p θ (r)
18 C. A. Bouman: Digital Image Processing - January 9, whereh(t) = CTFT 1 { ρ }, and f(x,y) = π g θ (xcos(θ)+ysin(θ))dθ
19 C. A. Bouman: Digital Image Processing - January 9, Summary of CBP Algorithm 1. Measure projections p θ (r). 2. Filter the projections g θ (r) = h(r) p θ (r). 3. Back project filtered projections f(x,y) = π g θ (xcos(θ)+ysin(θ))dθ
20 C. A. Bouman: Digital Image Processing - January 9, A Closer Look at Projection Filter 1. At each angle, projections are filtered. g θ (r) = h(r) p θ (r) 2. The frequency response of the filter is given by H(ρ) = ρ 3. But real filters must be bandlimited to ρ f c for some cut-off frequencyf c. H( ρ) = ρ So ρ fc fc H(ρ) = f c [rect(f/(2f c )) Λ(f/f c )] [ h(r) = fc 2 2sinc(t2fc ) sinc 2 (tf c ) ]
21 C. A. Bouman: Digital Image Processing - January 9, A Closer Look at Back Projection Back Projection function is f(x,y) = π b θ (x,y)dθ where b θ (x,y) = g θ (xcos(θ)+ysin(θ)) Consider the set of points(x,y) such that This set looks like r = xcos(θ)+ysin(θ) y r set of points along a single backprojection x Along this lineb θ (x,y) = g θ (r).
22 C. A. Bouman: Digital Image Processing - January 9, Back Projection Continued For each angleθ back projection is constant along lines of angleθ and takes on valueg θ (r). g (r) r y r set of points along a single backprojection x Complete back projection is formed by integrating (summing) back projections for angles ranging fromto π. f(x,y) = π π M b θ (x,y)dθ M 1 m= bmπ M (x,y) Back projection smears values of g(r) back over image, and then adds smeared images for each angle.
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