Quantitative Imaging XRay CT and Transmission Tomography Joseph

Quantitative Imaging: X-Ray CT and Transmission Tomography Joseph A. O’Sullivan Samuel C. Sachs Professor Dean, UMSL/WU Joint Undergraduate Engineering Program Professor of Electrical and Systems Engineering, Biomedical Engineering, and Radiology jao@wustl. edu Supported by Washington University and NIH-NCI grant R 01 CA 75371 (J. F. Williamson, VCU, PI). P-20 Seminar, 3/12/05 R. M. Arthur J. A. O’Sullivan, Quantitative Imaging, 4/8/13

Collaborators Faculty Bruce R. Whiting David G. Politte Jeffrey F. Williamson, VCU Donald L. Snyder David Brady, Duke Richard Laforest Yuan-Chuan Tai Chemical Engineers M. Dudukovic M. Al-Dahhan R. Varma P-20 Seminar, 3/12/05 R. M. Arthur J. A. O’Sullivan, Quantitative Imaging, 4/8/13 Students Norbert Agbeko Yaqi Chen Liangjun Xie Daniel Keesing Daheng Li Josh Evans, VCU Ikenna Odinaka Debashish Pal Jasenka Benac Giovanni Lasio, VCU

Dual Energy CT Image Reconstruction • Scanners • Data Models Reconstruction Algorithms • Image Reconstruction Approaches – “Linear” Approaches – Statistical Iterative Reconstruction • Experiment P-20 Seminar, 3/12/05 R. M. Arthur J. A. O’Sullivan, Quantitative Imaging, 4/8/13 SOMATOM Definition CT Scanner ccir. wustl. edu

X-Ray CT Scanners • Manufacturers: Siemens, GE, Philips, Toshiba, Analogic SOMATOM Definition CT Scanner • “Third Generation” vs. flat panel ccir. wustl. edu • Rotate 3 times per second • Detectors sampled approximately 1000 times per rotation • 16 – 64 rows of detectors • 600+ detectors per row • Siemens has two heads • > 5. 4 E 7 samples per second • Reconstructed volumes 512 by 200 P-20 Seminar, 3/12/05 R. M. Arthur J. A. O’Sullivan, Quantitative Imaging, 4/8/13

X-Ray Transmission Tomography— Basics • Source Spectrum, Energy E • Beer’s Law and Attenuation • Mean Photons/Detector I 0 • Beam-hardening • Detector Sensitivity Spectrum • Mean (Photon) Flux J. L. Prince and J. Links, Medical Imaging Signals and Systems. Pearson Education: Upper Saddle River, NJ, 2006. P-20 Seminar, 3/12/05 R. M. Arthur J. A. O’Sullivan, Quantitative Imaging, 4/8/13

Beer’s Law • Attenuation includes absorption and scatter • Attenuation is related to survival probabilities – The probability that a photon survives going through n chunks of material is the product of the probabilities of surviving individual chunks – The limit of the product of many terms is the exponential of an integral – The integral is through the attenuation function P-20 Seminar, 3/12/05 R. M. Arthur J. A. O’Sullivan, Quantitative Imaging, 4/8/13

“Linear” Image Reconstruction • • Normalize relative to an air scan Beam-hardening correction to target energy Negative log to estimate line integrals at target energy Linear inversion, normalize (e. g. water-equivalent) Normalize & correct P-20 Seminar, 3/12/05 R. M. Arthur J. A. O’Sullivan, Quantitative Imaging, 4/8/13 Negative logarithm FBP normalize

Limitations and Implementation • Real detectors measure random data – Electronics noise, quantization, photon-counting noise • Linearized model: – Measure all line-integrals (Radon transform) – Invert this linear mapping (Inverse Radon transform or filtered back-projection (FBP)) • Today: – – Measure flux through beakers with copper sulfate Estimate attenuation relative to no copper sulfate Approximate as a linear mapping Invert this linear mapping P-20 Seminar, 3/12/05 R. M. Arthur J. A. O’Sullivan, Quantitative Imaging, 4/8/13

Inversion of Linear Mappings 1. Problem: Given measured data y, the goal is to find the image represented here by x. The linear mapping from the image to the data is A. This linear mapping is known. 2. Approximate: Find some reasonable estimate for the image that gives predicted data close to the measurements. 3. Ideally: Given linear measurements, find the estimate of the image that minimizes the sum of the square errors between the measurements and the predicted measurements. P-20 Seminar, 3/12/05 R. M. Arthur J. A. O’Sullivan, Quantitative Imaging, 4/8/13

Matrices and Vectors P-20 Seminar, 3/12/05 R. M. Arthur J. A. O’Sullivan, Quantitative Imaging, 4/8/13

Matrices and Vectors P-20 Seminar, 3/12/05 R. M. Arthur J. A. O’Sullivan, Quantitative Imaging, 4/8/13