SIMG-716 Fourier Methods for Imaging

This course presents mathematical descriptions for functions and systems and demonstrates their application to solving imaging problems.

Course Outline: (dates are estimates and subject to change, some material is tentative)

Week 1

1) Introduction and motivation

a) Three imaging “entities:” input object f[x,y], output image g[x',y'], system function h[x,y;x',y']

b) Three imaging problems (direct, inverse, system analysis/synthesis)

c) Examples of imaging systems and models

d) Necessity to constrain action of system for tractable mathematical description

2) Complex numbers and their geometric interpretation

a) complex numbers as real-valued vectors

b) representations: real+imaginary, magnitude+phase, phasor/Argand diagram

c) complex arithmetic

d) Euler relation, deMoivre's theorem, roots of complex numbers

Week 2

3) Functions

a) classifications

i) domain and range (real/complex, continuous/discrete)

ii) form (linear/nonlinear, periodic, harmonic)

iii) symmetry (even/odd)

b) Representations obtained by projecting onto different sets of basis functions

i) inner product, relation to scalar product

ii) orthogonal/orthonormal sets of functions

iii) power series representations, Taylor series

c) 1-D real-valued functions: unit-amplitude function 1[x], null function 0[x], rectangle, triangle, signum, step, signum, cosine, chirp, Gaussian functions

d) 1-D Dirac delta function (impulse) and related functions

e) 1-D complex-valued functions

f) 1-D stochastic functions (uniform, Gaussian, and Poisson distributions)

4) 2-D functions from 1-D functions

a) separable in Cartesian coordinates

b) separable in polar coordinates Þ circular symmetry

c) 2-D Dirac delta function

i) representations in Cartesian and polar coordinates

ii) variants of 2-D Dirac delta function (line delta and CROSS)

d) representations of rotated 2-D functions

Week 3

5) Mathematical representation of systems, operators

a) Linearity property

b) Shift (space-, time-) variance and shift (space-, time-) invariant property

c) linear and shift-invariant (LSI) systems, convolution/filtering

d) Representations of systems

i) linear discrete Þ analogy with matrix -vector multiplication

ii) linear continuous Þ superposition integral

iii) LSI discrete systems Þ circulant matrix, diagonalizing transformation

iv) LSI continuous systems Þ convolution integral

v) impulse response/point-spread function of LSI continuous system

e) Crosscorrelation and autocorrelation as variants of convolution

Week 4

6) Alternative representations of functions

a) Projection of functions onto alternative sets of basis functions

i) representation in space and spatial frequency domain

b) Intuitive derivation of sine and cosine transforms

c) Hartley transform as weighted sum of cosine and sine transforms

d) Fourier transform as weighted sum of cosine and sine transforms

e) Fourier series for discrete functions

Week 5

7) Fourier transforms of 1-D functions

a) Evaluation of Fourier transforms by direct integration

b) Evaluation of Fourier transforms of special functions

c) Theorems of the Fourier transform

d) Chirp Fourier transform

MIDTERM EXAM, 10/12 (W)

Week 6

8) Fourier transforms of 2-D functions

a) Separable functions in Cartesian coordinates

b) Transforms of circularly symmetric functions (Hankel transform)

c) Radon transform

Week 7

9) Discrete functions

a) Ideal sampling at uniform intervals, sampled special functions

b) Interpretation of sampling in frequency domain, Whittaker-Shannon sampling theorem

c) Realistic sampling at uniform intervals, reduced modulation

d) Reconstruction of functions from samples, interpolation

10) Processing of discrete functions

a) Discrete Fourier transform (DFT)

i) 1-D DFT as matrix applied to 1-D discrete vector

ii) inverse DFT, normalization conventions

iii) Fourier series from the Fourier transform

b) Computation of the DFT

c) computation of spectrum at arbitrary frequency

d) Cooley-Tukey implementation of DFT Þ “fast Fourier transform (FFT)

e) 2-D FFT

Week 8

f) Practical Considerations of DFT

i) leakage, window functions

ii) Resolution in space and frequency domains, zero padding

iii) Data formats, centered and noncentered data, checkerboarding

iv) Phase in the FFT

g) convolution of discrete functions

h) normalization conventions

i) linear and circular convolution

Week 9

11) Linear Filtering

a) impulse response and transfer function, psf and OTF

b) Amplitude filters (lowpass, highpass, bandpass, bandstop)

c) Allpass (phase-only) filters

d) Deconvolution and detection filters

i) Inverse filter

ii) Wiener and Wiener-Helstrom filter

iii) Matched filter

Week 10

12) Applications of linear systems in imaging

a) Fresnel diffraction as convolution with chirp function

b) Implementation of optical chirp Fourier transform

c) Radon transform as basis for medical computed tomography (CT) and magnetic resonance imaging (MRI)

19 August 2011