Lecture 13  Last Week Summary  Sampled Systems  Image Degradation and The Need To Improve Image Quality  Signal vs. Noise  Image Filters  Image Reconstruction in Computed Tomography (Filtered Backprojection, Algabraic)

So Far, Have Discussed Positron Emission and Annihalation Positron Emission and Annihalation Cyclotron Cyclotron Reaction Equations Reaction Equations PET Detector and Detection Process ( PMT, Configuration, Detector Materials ) PET Detector and Detection Process ( PMT, Configuration, Detector Materials ) PET Image Formation PET Image Formation

PROBLEMS These systems “sample” data. These systems “sample” data. They sample from specific angles, then sample into squares (pixels of voxels). They sample from specific angles, then sample into squares (pixels of voxels). Many problems arise from sampling Many problems arise from sampling

PROBLEMS ALSO ARISE FROM: System Noise System Noise Positron Detection Positron Detection Attenuation Attenuation Low Statistics Low Statistics

Image Degredation In PET Singles Events

Image Degradation In PET Randoms

Statistical Aspects Not only does the sampling pose a problem, but the statistical density, or lack thereof (especially in nuclear medicine) poses a “noise problem”. Not only does the sampling pose a problem, but the statistical density, or lack thereof (especially in nuclear medicine) poses a “noise problem”. Noise can be thought of as the small, random fluctuations that appear across the image. Noise can be thought of as the small, random fluctuations that appear across the image.

One of the great challenges in medical imaging is to find the signal, in a sea of noise.

Sampling Theory The idea is to obtain enough samples so that the essential information contained in the image is not lost. The idea is to obtain enough samples so that the essential information contained in the image is not lost. This can be proven mathematically, and is fundamental in communications theory, signal processing, electrical engineering, and other fields. This can be proven mathematically, and is fundamental in communications theory, signal processing, electrical engineering, and other fields. The closer the data points (the closer the samples), the better the approximation. The closer the data points (the closer the samples), the better the approximation.

Example: Continuous Function

Sampled Points: Discrete Function

Discrete function “Approximates” Continuous Function

Sampling The Information From The Heart Beat

Fourier Analysis Decompose a Function into sine and cosine waves.

Bar Phantoms “Approximate” Fourier Transform

Plot of Amplitude vs. Frequency