1. BMI I FS05 – Class 2 “Linear Systems” Slide 1 Biomedical Imaging I Class 2 – Mathematical Preliminaries: Signal Transfer and Linear Systems Theory 9/21/05

2. BMI I FS05 – Class 2 “Linear Systems” Slide 2 Linear Systems

3. BMI I FS05 – Class 2 “Linear Systems” Slide 3 Class objectives Topics you should be familiar with after lecture: Linear systems (LS): definition of linearity, examples of LS, limitations Deterministic and stochastic processes in signal transfer Contrast Noise, signal-to-noise ratio (SNR) LS theory description of imaging systems (SNR, contrast, resolution)

4. BMI I FS05 – Class 2 “Linear Systems” Slide 4 Overview of topic Goal: To describe a physical system with a mathematical model Example : medical imaging Energy sourcedetector(s)human body "system" quantum image energy conversion analog / digital image detection, storage "system" imaging algorithm ………

5. BMI I FS05 – Class 2 “Linear Systems” Slide 5 Applications Analyzing a system based on a known input and a measured output Decomposing a system into subsystems Predicting output for an arbitrary input Modeling systems Analyzing systems Correcting for signal degradation by the system to obtain a better replica of the input signal Quantifying the signal transfer fidelity of a system

6. BMI I FS05 – Class 2 “Linear Systems” Slide 6 Signal transfer by a physical system Signal transferred by a system System input is a function h(x) System operates on input function (system can be described by a mathematical operator) System output is function S{h(x)} Objective: To come up with operators that accurately model systems of interest S h(x)h(x)S{h(x)} InputSystemOutput

7. BMI I FS05 – Class 2 “Linear Systems” Slide 7 Linear systems Additivity Homogeneity Preceding two can be combined into a single property, which is actually a definition of linearity:

8. BMI I FS05 – Class 2 “Linear Systems” Slide 8 Linear systems Additivity Homogeneity LSI (linear-shift invariant) systems: “same” S

9. BMI I FS05 – Class 2 “Linear Systems” Slide 9 Examples Examples of linear systems: Spring (Hooke’s law): x = k F Resistor V-I curve (Ohm’s law): V = R I Amplifier Wave propagation Differentiation and Integration Examples of nonlinear systems: Light intensity vs. thickness of medium I = c exp(-  x) Diode V-I curve I = c [exp(V/kT)-1] Radiant energy vs. temperature P = kT 4

10. BMI I FS05 – Class 2 “Linear Systems” Slide 10 LS significance and validity Why is it desirable to deal with LS? Decomposition (analysis) and superposition (synthesis) of signals System acts individually on signal components No signal “mixing” Simplifies qualitative and quantitative measurements Real world phenomena are never truly linear Higher order effects Noise Linearization strategies Small signal behavior (e.g., transistors, pendulum) Calibration (e.g., temperature sensors)

11. BMI I FS05 – Class 2 “Linear Systems” Slide 11 Deterministic vs. stochastic systems Deterministic systems: Will always produce the exact same output if presented with identical input Examples: Idealized models Imaging algorithms Stochastic systems: Identical inputs will produce outputs that are similar but never exactly identical Examples: Any physical measurement Noisy processes (i.e., all physical processes)

12. BMI I FS05 – Class 2 “Linear Systems” Slide 12 Random data Deterministic processes: Future behavior predictable within certain margins of error from past observation and knowledge of physics of the problem. e.g., mechanics, electronics, classical physics… Random data/phenomena: … each experiment produces a unique (time history) record which is not likely to be repeated and cannot be accurately predicted in detail. Consider data records (temporal variation, spatial variation, repeated measurements, …) e.g., measurement of physical properties, statistical observations, time series analyses (e.g, neuronal recordings), medical imaging

13. BMI I FS05 – Class 2 “Linear Systems” Slide 13 Systematic error, or Bias Random error, or Variance, or Noise Varieties of measurement error

14. BMI I FS05 – Class 2 “Linear Systems” Slide 14 Noise I What is the reason for measurement-to-measurement variations in the signal? Changes can occur in the input signal in the underlying process in the measurement Noise is the presence of stochastic fluctuations in the signal Noise is not a deterministic property of the system (i.e. it cannot be predicted or corrected for) Noise does not bear any information content S h(x)h(x)S{h(x)} InputSystemOutput

15. BMI I FS05 – Class 2 “Linear Systems” Slide 15 Noise II Examples of noise: phone static, snowy TV picture, grainy film/photograph Because noise is a stochastic phenomenon, it can be described only with statistical methods   a) Very noisy signalb) Not so noisy signal

16. BMI I FS05 – Class 2 “Linear Systems” Slide 16 Signal-to-noise ratio (SNR) The data quality (information content) of is quantified by the signal- to-noise ratio (SNR) Example for possible definition: Signal = mean (average magnitude) Noise = standard deviation   SNR = 0.5SNR = 15

17. BMI I FS05 – Class 2 “Linear Systems” Slide 17 Contrast Separation of signal (image) features from background Contrast describes relative brightness of a feature Examples of varying contrast Background, b Signal, S 0

18. BMI I FS05 – Class 2 “Linear Systems” Slide 18

19. BMI I FS05 – Class 2 “Linear Systems” Slide 19

20. BMI I FS05 – Class 2 “Linear Systems” Slide 20

21. BMI I FS05 – Class 2 “Linear Systems” Slide 21

22. BMI I FS05 – Class 2 “Linear Systems” Slide 22 30% 36% 42% 48% 54% C 1 =0.18 C 1 =0.15 C 1 =0.13 C 1 =0.12 Contrast values

23. BMI I FS05 – Class 2 “Linear Systems” Slide 23 Deterministic effects Signal contrast transfer: Contrast curve System resolution: impulse response function (irf) Every point of the input produces a more blurry point in the output Limits image resolution In imaging: point spread function (psf) input output b S

24. BMI I FS05 – Class 2 “Linear Systems” Slide 24 Point spread function (PSF) Impulse response function (irf): system output for delta impulse (spatial / time domain). The irf completely describes a linear system (LS)! Imaging system: 2D point spread function (psf(x,y)) is the response of the system to a point in the object (spatial domain). The psf completely describes a (linear) imaging system! psf defines spatial resolution of imaging system (how close can two points be in the object and still be distinguishable in the image) S h(x)  d(x)S(h(x))  irf(x)

25. BMI I FS05 – Class 2 “Linear Systems” Slide 25 Deterministic process I Contrast transfer by system Reduction of contrast, C i > C o

26. BMI I FS05 – Class 2 “Linear Systems” Slide 26 Deterministic process II Image blur Poor spatial resolution: loss of details, sharp transitions

27. BMI I FS05 – Class 2 “Linear Systems” Slide 27 Stochastic variations Random noise added by system Combination of all three effects severely degrades image quality

28. BMI I FS05 – Class 2 “Linear Systems” Slide 28 Extra Topic 1: Acronyms and unfamiliar(?) terms IRF = Impulse Response Function PSF = Point Spread Function LSF = Line Spread Function ESF = Edge Spread Function MTF = Modulation Transfer Function FOV = Field of View FWHM = Full Width at Half Maximum ROI = Region of Interest Convolution Fourier Transform Gaussian (Normal) Distribution Poisson Distribution

29. BMI I FS05 – Class 2 “Linear Systems” Slide 29 Extra Topic 2: Gaussian distribution Uncorrelated noise (i.e. signal fluctuations are caused by independent, individual processes) is closely approximated by a Gaussian pdf (normal distribution) Central limit theorem Examples: Thermal noise in resistors, film graininess Normalized Location of center (mean value  ) Width (standard deviation  ) Completely determined by two values  and  Linear operations maintain the Gaussian nature

30. BMI I FS05 – Class 2 “Linear Systems” Slide 30 Extra Topic 3: FWHM (Full Width at Half Maximum)

31. BMI I FS05 – Class 2 “Linear Systems” Slide 31 Extra Topic 3: FWHM (Full Width at Half Maximum) 2

32. BMI I FS05 – Class 2 “Linear Systems” Slide 32 Extra Topic 4: Poisson distributions Noise in imaging applications can often be described by Poisson or counting statistics p(n  is the probability of n counts within a certain detector area (or pixel), when the average/expected number of counts is  SNR for Poisson-distributed processes ( N = mean number of registered quanta): Usually, the error of an estimated mean for M samples is given by

33. BMI I FS05 – Class 2 “Linear Systems” Slide 33 Extra Topic 5: Line and edge spread functions

34. BMI I FS05 – Class 2 “Linear Systems” Slide 34 Extra Topic 6: Modulation transfer function

35. BMI I FS05 – Class 2 “Linear Systems” Slide 35 Linear systems Additivity Homogeneity Preceding two can be combined into a single property, which is actually a definition of linearity:

36. BMI I FS05 – Class 2 “Linear Systems” Slide 36 LS significance and validity Why is it desirable to deal with LS? Decomposition (analysis) and superposition (synthesis) of signals System acts individually on signal components No signal “mixing” Simplifies qualitative and quantitative measurements Lets us separate S{h(x)} into two independent factors: the source, or driving, term, and the system’s impulse response function (irf)

37. BMI I FS05 – Class 2 “Linear Systems” Slide 37 Rectangular input function (Rectangular pulse) S h(x)h(x)S{h(x)} InputSystemOutput h(x), h(t) x, t 0

38. BMI I FS05 – Class 2 “Linear Systems” Slide 38 Limiting case of unit pulse x, t 0 As the pulse narrows we also make it higher, such that the area under the pulse is constant. (Variable power, constant energy.) We can imagine making the pulse steadily narrower (briefer) until it has zero width but still has unit area! A pulse of that type (zero width, unit area) is called an impulse.

39. BMI I FS05 – Class 2 “Linear Systems” Slide 39 Impulse response function (irf) S h(x)h(x)S{h(x)} InputSystemOutput impulse function goes in…impulse response function (irf) comes out! Note: irf has finite duration. Any input function whose width/duration is << that of the irf is effectively an impulse with respect to that system. But the same input might not be an impulse wrt a different system.

40. BMI I FS05 – Class 2 “Linear Systems” Slide 40 Output for arbitrary input signals is given by superposition principle (Linearity!) Think of an arbitrary input function as a sequence of impulse functions, of varying strengths (areas), tightly packed together Then the defining property of an LS, (S{a·h 1 + b·h 2 } = a·S{h 1 } + b·S{h 2 }) tells us that the overall system response is the sum of the corresponding irfs, properly scaled and shifted. Significance of the irf of a LS =

41. BMI I FS05 – Class 2 “Linear Systems” Slide 41 Significance of the irf of a LS To state the same idea mathematically, LS output given by convolution of input signal and irf: Notice that the sum of these two arguments is a constant or t