Biomedical Imaging I

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

2. Linear Systems

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. … … … "system" human body imaging algorithm Energy source detector(s) "system" analog / digital image quantum image energy conversion detection, storage Overview of topic • Goal: To describe a physical system with a mathematical model • Example : medical imaging

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. Signal transfer by a physical system • Signal transferred by a system • System input is a functionh(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 Input System Output S h(x) S{h(x)}

7. Linear systems • Additivity • Homogeneity • Preceding two can be combined into a single property, which is actually a definition of linearity:

8. “same” S Linear systems • Additivity • Homogeneity • LSI (linear-shift invariant) systems:

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(-m x) • Diode V-I curve I = c [exp(V/kT)-1] • Radiant energy vs. temperature P = kT4

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. 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. 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. Systematic error, or Bias Random error, or Variance, or Noise Varieties of measurement error

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) S{h(x)} Input System Output

15. a) Very noisy signal b) Not so noisy signal s s 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

16. s s SNR = 0.5 SNR = 15 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

17. Signal, S Background, b 0 Contrast • Separation of signal (image) features from background • Contrast describes relative brightness of a feature • Examples of varying contrast

22. Contrast values C1 =0.18 C1 =0.15 36% 42% 30% 36% C1 =0.12 C1 =0.13 48% 54% 42% 48%

23. output input S b 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)

24. S h(x) =d(x) S(h(x)) = irf(x) 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)

25. Deterministic process I • Contrast transfer by system • Reduction of contrast, Ci > Co

26. Deterministic process II • Image blur • Poor spatial resolution: loss of details, sharp transitions

27. Stochastic variations • Random noise added by system • Combination of all three effects severely degrades image quality

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. 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 m) • Width (standard deviation s) • Completely determined by two values m and s • Linear operations maintain the Gaussian nature

30. Extra Topic 3: FWHM (Full Width at Half Maximum)

31. Extra Topic 3: FWHM (Full Width at Half Maximum) 2

32. Extra Topic 4: Poisson distributions • Noise in imaging applications can often be described by Poisson or counting statistics • p(n,m) is the probability of n counts within a certain detector area (or pixel), when the average/expected number of counts is m • 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. Extra Topic 5: Line and edge spread functions

34. Extra Topic 6: Modulation transfer function

35. Linear systems • Additivity • Homogeneity • Preceding two can be combined into a single property, which is actually a definition of linearity:

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. Rectangular input function (Rectangular pulse) Input System Output S h(x) S{h(x)} h(x), h(t) 0 x, t

38. As the pulse narrows we also make it higher, such that the area under the pulse is constant. (Variable power, constant energy.) 0 x, t Limiting case of unit pulse 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. impulse function goes in… impulse responsefunction (irf) comes out! Impulse response function (irf) Input System Output S h(x) S{h(x)} 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. = Significance of the irf of a LS • 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·h1 + b·h2} = a·S{h1} + b·S{h2}) tells us that the overall system response is the sum of the corresponding irfs, properly scaled and shifted.

41. or t Notice that the sum of these two arguments is a constant Significance of the irf of a LS • To state the same idea mathematically, LS output given by convolution of input signal and irf: