BMME 560 & BME 590I Medical Imaging: X-ray, CT, and Nuclear Methods Tomography Part 3

Today Tomography –Filtered backprojection

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

Simple Backprojection An LSI system model for projection followed by simple backprojection: Fourier Transform filter Inverse Fourier Transform Projection Simple backprojection

Simple Backprojection Example True image Simple backprojection

Filtered Backprojection An LSI reconstruction method: Fourier Transform filter Inverse Fourier Transform Projection Simple backprojection filter

Filtered Backprojection An LSI reconstruction method: Fourier Transform filter Inverse Fourier Transform Projection Simple backprojection filter Filtered Backprojection

Example True image Simple backprojection Filtered backprojection

Filtered Backprojection Example True image Simple backprojection Filtered backprojection

In theory, the filtered backprojection estimate should be equal to the true image. Our discrete implementation introduces some errors. The biggest problem is noise.

Filtered backprojection Look at the frequency-domain property of that ramp filter Frequency Magnitude There is a mathematical problem here

Filtered backprojection Consider the practical problem –The projection of an object is limited in resolution by physical constraints. –The noise is manifested in the detector and its spectrum is not limited. What does the ramp filter do to the signal and to the noise?

Filtered Backprojection So, in practice, we have to truncate (roll-off, apodize) the ramp filter to control noise. Frequency Magnitude Practical ramp filter Ideal ramp filter

Key Point There is an essential and inescapable tradeoff between noise and resolution in every imaging system. Noise variance Resolution (FWHM)

Filtered Backprojection Example True image Filtered backprojection Noise-free Filtered backprojection Noisy

Filtered backprojection Three low-pass filter cutoffs.2 cyc/pix.15 cyc/pix.1 cyc/pix

Filtered backprojection We have a couple of equivalent ways to implement filtered backprojection –Backproject, then apply a 2D ramp filter –Apply 1D ramp filter to each projection, then backproject –Convolve each projection with an approximation of the PSF of the ramp filter, then backproject (called convolution backprojection).

Filter of the backprojection 2D Fourier Transform Inverse 2D Fourier Transform Simple backprojection Ramp filter

Backprojection of filtered projections 1D Fourier Transform Inverse 1D Fourier Transform Simple backprojection Ramp filter At this point, we have created a set of modified projections.

Convolution Backprojection Simple backprojection Convolution Of projections The convolution kernel

Convolution Backprojection The convolution kernel is an approximation of the inverse Fourier transform of the ramp filter. t This was Cormack’s original approach.

Convolution Backprojection In a discrete implementation, we can take the IFFT of the ramp filter. Note the similarity in shape to the continuous version.

Sampling Since the projection space is two-dimensional, we have to sample in each dimension: –Spatial sampling (in t) –Angular sampling (in  ) What are the effects of sampling in each?

Sampling Spatial sampling: The pixel spacing in the detector –Finer sampling = finer resolution in the reconstruction Angular sampling: The number of angles over the desired arc (or the angular increment between projections)

Spatial Sampling 1x 1.33x2x4x Projection spacing Insufficient spatial sampling results in?

Angular Sampling Rule of thumb for the number of projections needed: Number of angles over 180 degrees = ____________

Angular Sampling 128x128 image (object: 75 pixel diameter), 360-degree arc 256 views128 views96 views 64 views32 views

Limitations To get filtered backprojection, we made a key assumption: –The projections are all perfect Radon projections They are complete, i.e., the entire object is viewed in every projection. They are perfect line integrals. The sampling is sufficient. There is no noise.

Limitations In the real world, we do not always meet these assumptions (tradeoffs): –Completeness: usually not a problem, but some special geometries suffer truncation –Perfect line integrals: Beam hardening, scatter, metals, nonuniformity, body attenuation (ECT), detector blurring all degrade this. –Sampling sufficiency: Acquisition time can be reduced with fewer angles, larger detector pixels –Noise: Always with us

Limitations Beam hardening example This object experiences different energy spectra depending on the direction of projection.

Limitations Deviations from perfect Radon projections result in inconsistent data –The mathematics do not care if the data is inconsistent; it is still possible to perform filtered backprojection. –The interpretation of the results may be a problem if there are inconsistencies.

Limitations Example: Detector is too small Two-sided truncationOne-sided truncation

Limitations We have also assumed that we know the geometry of the system perfectly. Calibration is essential in real systems. –Hospital staff perform frequent quality control tests and calibrations to ensure system drifts are accounted for.

Limitations Center-of-rotation error Perfect0.5 pixel shift1 pixel shift

Limitations Bottom line –Inconsistencies are created when real projections deviate from Radon projections. Usually due to the physical limitations or cost/performance tradeoffs of the system –Images are reconstructed, inconsistencies are manifested as recognizable image artifacts.

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 Geometry Most commonly used in CT systems –X-ray production is focused on a point anyway. Also found in SPECT –Provides magnification, at cost of field of view.

Fan-beam Geometry What arc do we need? Is 180 enough? –Consider: Each fan-beam projection line can be mapped to a projection line in the Radon space. –Each projection line maps to a (t,  ) pair in the sinogram space.

Fan-beam Geometry Sinogram mapping of a 180-degree arc with a fan-beam Each “line” is one rotational position. Redundant angles Missing angles

Fan-beam Geometry Rule for the arc required to obtain complete sinogram data for fan-beam: The minimum arc still requires compensation; it may be easier to do 360-degrees. Rotational arc = ________________________

Fan-beam Geometry arc 360 arc

Fan-beam Geometry Options for analytical reconstruction –Rebinning (resorting): Estimate parallel-beam data from fan-beam and use filtered backprojection –Fan-beam filtered backprojection: Modified version of FBP specifically for fan-beam

Fan-beam Rebinning Convert fan-beam data to equivalent parallel-beam data f(x,y) y x Note that each projection ray in fan-beam is an element of the parallel- beam projection set at some (t,  ) pair. Resort projection rays and interpolate.

Fan-beam Filtered Backprojection Fan-beam projection data Weighting based on projection angle Modified projection data Apply modified ramp filter Modified projection data Backproject along fan, weighted by distance from focus Reconstructed image * Also the basis for the Feldkamp algorithm