BMME 560 & BME 590I Medical Imaging: X-ray, CT, and Nuclear Methods Tomography Part 4

Today Tomography –Other geometries Fan-beam variants Cone-beam Pinhole Ring –Other reconstruction methods Matrix-based Iterative reconstruction

Tomographic Reconstruction The problem f(x,y) p(t,  ) t t y  x s Given p(t,  ) for 0<  <  Find f(x,y)

Other Geometries We have discussed only parallel (Radon) projections so far. There are other projection geometries that are more practical in certain situations. –Fan-beam –Cone-beam –Pinhole –Ring

Fan-beam Reconstruction f(x,y) y x t y x

Fan-beam Variants There are a few variations on fan-beam –Asymmetric fan –Varifocal fan –Multi-slice fan-beam

Asymmetric Fan f(x,y) y x y x Asymmetric Symmetric

Varifocal fan-beam f(x,y) y x y x Varifocal

3D Imaging So far, all of our treatments have been with taking 1D projections of a 2D object. Traditional CT would take one slice at a time and step the patient through the CT to build up the 3D image from individual 2D images. Modern CTs take more than one slice at a time.

Multi-slice Fan-beam: 3D t f(x,y,z) y x y z z Stack up a few detectors

Multi-slice Fan-beam: 3D As long as the oblique angle is not too severe, can use fan-beam reconstruction without much distortion. As the oblique angle becomes more significant, distortions become more significant. This becomes a cone-beam.

Cone-beam Imaging Issues: Data sufficiency Inexact reconstruction methods Field of view

Cone-beam Data Sufficiency Consider a point in the focal plane To this point, projections look like a fan beam.

Cone-beam Data Sufficiency Consider a point above or below the focal plane To this point, projections look like a tilted fan beam. Tilted fan beam is insufficient to completely reconstruct a 3D object.

Cone-beam Data Sufficiency Consider a point off axis and above or below the focal plane To this point, projections look like tilted fan beams at multiple oblique angles.

Data Insufficiency Artifact Single Dual Triple Circular-orbit cone-beam

Sufficient Data Orbits Require complex motion of camera and/or subject Difficult to reconstruct: Can be done iteratively Some analytical methods exist

Cone-Beam Tomography Options for reconstruction: Feldkamp algorithm: Like FBP Fourier rebinning: Estimate parallel-beam from cone-beam Iterative algorithms

Feldkamp Algorithm Same as fan-beam filtered backprojection, but backproject along tilted fans.

Pinhole Tomography In SPECT, we can create a pinhole projection. Subject Only the photons that pass through the pinhole are detected. Detector

Ring System Exclusive to PET Detection occurs between pairs of detectors Every pair creates a “line of response” (LOR)

Ring System 2D PET y  x t Each line of response represents a ( t,  ) pair. How can you reconstruct this? Is it sufficient?

Ring System 3D PET Now, lines of response are both within a slice (“directs”) and across slices (“obliques”). How could you reconstruct this?

Other Approaches to Reconstruction FBP develops problems when: –Data is inconsistent –Geometries do not translate well to parallel-beam (especially 3D geometries: ring, cone-beam, pinhole) –Data is incomplete More general methods perform better in these situations

Matrix Methods The forward problem is reformulated as a matrix equation: A vector of all projection values over t and . A vector of all image- space pixels. A matrix defining the contribution of each image-space pixel to each projection bin. Now, the problem is: Given g and H, solve for f.

Matrix Methods Solving the problem may be as simple as inverting the matrix: If H represents the Radon projection process, then H -1 would represent FBP. However, we can compute an H for virtually any projection geometry and/or situation.

Matrix Methods Two problems: –The matrix is BIG (For CT, 10 6 x 10 6 ?) –For many geometries, the inverse does not exist or is poorly conditioned Inversion is subject to big numerical errors, especially with noise. Iterative methods can be used to approximate the solution to the matrix equation.

Iterative Reconstruction Essentially, an iterative approach to the matrix inversion. Current Estimate Project Estimated Projections Compare Projection-Space Error Backproject Image-Space Error Update Measured Projections

ART Reconstruction A classic iterative reconstruction method: Algebraic Reconstruction Technique Illustrate with this old example:

ART Reconstruction Start with an initial estimate and project it along one direction

ART Reconstruction Compute errors and backproject errors

ART Reconstruction Project the result at another angle

ART Reconstruction Backproject those errors and update

ART Reconstruction Then, start over again if necessary This case has converged after one pass through all of the data, but that does not always happen, especially with noisy data.

ART Reconstruction Sensitive to noise and the result may depend on the order of angles Can be done mulitplicatively (MART) Important for historical perspective; variations of it are used in research.

Iterative Reconstruction Essentially, an iterative approach to the matrix inversion. Current Estimate Project Estimated Projections Compare Projection-Space Error Backproject Image-Space Error Update Measured Projections

Statistical-based Iterative Algorithms More mathematically rigorous than ART Can model any geometry and any physical degradations Better at limiting noise than FBP and ART Widely used in SPECT and PET –OS-EM: Ordered-subset expectation maximization Algorithms exist for transmission CT, but are not widely used.

Iterative Reconstruction Essentially, an iterative approach to the matrix inversion. Current Estimate Project Estimated Projections Compare Projection-Space Error Backproject Image-Space Error Update Measured Projections

Statistical-based Iterative Algorithms Chief disadvantages –Long processing times, 10x or more compared to FBP; too long for X-ray CT –More complex implementation