Lecture 13Lecture 13

Last Week Summary Last Week Summary Sampled SystemsSampled Systems

Image Degradation and The Need To Improve Image Degradation and The Need To Improve Image QualityImage Quality

Signal vs. NoiseSignal vs. NoiseImage FiltersImage Filters

Image Reconstruction in Computed TomographyImage Reconstruction in Computed Tomography(Filtered Backprojection, Algabraic)(Filtered Backprojection, Algabraic)

So Far, Have DiscussedSo Far, Have Discussed

• Positron Emission and AnnihalationPositron Emission and Annihalation

• CyclotronCyclotron

• Reaction EquationsReaction Equations

• PET Detector and Detection Process PET Detector and Detection Process ( (PMT, Configuration, Detector MaterialsPMT, Configuration, Detector Materials))

• PET Image Formation PET Image Formation

PROBLEMSPROBLEMS

• These systems “sample” data.These systems “sample” data.

• They sample from specific angles, then sample They sample from specific angles, then sample into squares (pixels of voxels).into squares (pixels of voxels).

• Many problems arise from sampling Many problems arise from sampling

PROBLEMS ALSO ARISE FROM:PROBLEMS ALSO ARISE FROM:

• System NoiseSystem Noise

• Positron DetectionPositron Detection

• AttenuationAttenuation

• Low StatisticsLow Statistics

Image Degredation In PETImage Degredation In PETSingles EventsSingles Events

Image Degradation In PET RandomsImage Degradation In PET Randoms

Statistical AspectsStatistical Aspects

• Not only does the sampling pose a problem, Not only does the sampling pose a problem, but the statistical density, or lack thereof but the statistical density, or lack thereof (especially in nuclear medicine) poses a (especially in nuclear medicine) poses a “noise problem”.“noise problem”.

• Noise can be thought of as the small, Noise can be thought of as the small, random fluctuations that appear across the random fluctuations that appear across the image. image.

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

Sampling Theory Sampling Theory

• The idea is to obtain enough samples so that the The idea is to obtain enough samples so that the essential information contained in the image is not essential information contained in the image is not lost.lost.

• This can be proven mathematically, and is This can be proven mathematically, and is fundamental in communications theory, signal fundamental in communications theory, signal processing, electrical engineering, and other fields.processing, electrical engineering, and other fields.

• The closer the data points (the closer the samples), The closer the data points (the closer the samples), the better the approximation.the better the approximation.

0

20

40

60

80

100

120

0.06 0.13 0.19 0.25 0.31 0.38 0.44 0.5 0.56 0.63 0.69 0.75 0.81 0.88 0.94

0.0625

0.125

0.1875

0.25

0.3125

0.375

0.4375

0.5

0.5625

0.625

0.6875

0.75

0.8125

0.875

0.9375

Example: Continuous FunctionExample: Continuous Function

0

20

40

60

80

100

120

0.06 0.13 0.19 0.25 0.31 0.38 0.44 0.5 0.56 0.63 0.69 0.75 0.81 0.88 0.94

0.0625

0.125

0.1875

0.25

0.3125

0.375

0.4375

0.5

0.5625

0.625

0.6875

0.75

0.8125

0.875

0.9375

Sampled Points: Discrete FunctionSampled Points: Discrete Function

0

20

40

60

80

100

120

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Series1

Discrete function “Approximates” Discrete function “Approximates” Continuous FunctionContinuous Function

0

20

40

60

80

100

120

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Series1

Sampling The Information From Sampling The Information From The Heart BeatThe Heart Beat

Fourier AnalysisFourier AnalysisDecompose a Function into sine Decompose a Function into sine

and cosine waves.and cosine waves.

Bar Phantoms “Approximate” Fourier Bar Phantoms “Approximate” Fourier TransformTransform

Plot of Amplitude vs. FrequencyPlot of Amplitude vs. Frequency

Fourier Transform

0

1

2

3

4

5

6

1 2 3 4 5

Frequency

Am

pit

ud

e