Syllabus for IMGS-616 Fourier Methods in Imaging (RIT #11857) 15 August 2013

Week 1: 8/27, 8/29

I. Signals, Operators, and Imaging Systems

A. Imaging “chain”

B. Three imaging “tasks”

C. Examples of the imaging chain and tasks

II. 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: 9/3, 9/5

III. Representations of systems, inputs, and outputs by functions

A. Symmetry properties of functions: even and odd

B. Projections of functions onto “reference” functions, scalar products, orthogonality

C. special functions

1. Support

2. Area (1-D) or volume (2-D)

3. Shifting and scaling

4. Definitions of deterministic real-valued functions

a. unit function, null function

b. rectangle

c. triangle

d. signum and step

e. Gaussian

f. linear-phase sinusoid

g. quadratic-phase sinusoid

h. Dirac delta function

i. Stochastic functions

Week 3: 9/10, 9/12

5. Complex-valued 1-D special functions

a. representations as real and imaginary parts and as magnitude and phase

b. linear- and quadratic-phase sinusoids

6. 2-D functions that are separable in Cartesian coordinates

7. 2-D functions that are separable in polar coordinates

8. 2-D Dirac delta function and its relatives (line delta and cross)

9. Rotation of 2-D functions

Week 4: 9/17, 9/19

IV. Classes of Imaging Operators

A. Linearity

B. Shift invariance

C. Crosscorrelation

D. Convolution

1. theorems

2. examples

Week 5: 9/24

Midterm Exam 1

Week 5: 9/26 and Week 6: 10/1

V. Alternative representations of functions: Fourier analysis and synthesis

A. Projection of functions onto sinusoidal “reference” functions

B. Projections onto combinations of “reference” functions

1. projection onto cosine + sine: Hartley transform

2. projection onto cosine + i×sine: Fourier transform

C. Fourier synthesis, inverse Fourier transform

D. Fourier transforms of the 1-D special functions

Week 6: 10/3 and Week 7: 10/8

VI. Theorems of the Fourier transform

A. Transform of transform

B. Scaling theorem

C. Shift theorem

D. Filter theorem

E. Modulation theorem

F. Derivative theorem

G. Fourier transform of autocorrelation, Wiener spectrum

H. Rayleigh’s and Parseval’s theorems

I. Fourier transform of periodic function

J. Fourier transform of sampled function

K. Fourier transform of discrete periodic function

L. Effect of nonlinear operations on spectra

M. Central-limit theorem

N. Uncertainty relation(s)

Week 7: 10/10

VII. Fourier transforms of multidimensional functions

A. Separable 2-D functions

B. Circularly symmetric 2-D functions, Hankel transform

Week 8: No class 10/15 (Monday class schedule), 10/17

VIII. Filtering of continuous functions

A. Magnitude filters

1. Lowpass (averagers, integrators)

2. Highpass (differencers, differentiators)

3. Bandpass (difference of averages), bandboost, bandstop filters

Week 9, 10/22 and 10/24

B. Phase filters (“allpass”)

1. constant phase

2. linear phase

3. quadratic phase

4. chirp Fourier transform by multiplication (M) and convolution (C) with quadratic-phase factors

a. multiplication + convolution + multiplication (M-C-M)

b. convolution + multiplication + convolution (C-M-C)

C. Magnitude-and-phase filters

1. causality

Week 10: 10/29

Midterm Exam 2

Week 10: 10/31, Week 11: 11/5

IX. Applications of linear filters

A. Inverse imaging task, “deconvolution”

1. Wiener filter

2. Wiener-Helstrom filter

B. Matched filtering

C. Analogies between inverse and matched filters

Week 11: 11/7 and Week 12: 11/12

X. Fourier transforms and optical imaging systems

A. propagation of light as spherical waves, Huygens’ principle, convolution with spherical waves

B. approximation to propagation of light as paraboloidal waves, Fresnel diffraction, convolution with constant-magnitude quadratic-phase factor

C. approximation to propagation of light as plane waves, Fraunhofer diffraction, shift-variant Fourier transform

D. Imaging system formed from Fraunhofer propagation + aperture + Fraunhofer propagation

E. Approximation of lens action as multiplication by apodized quadratic-phase factor

F. optical Fourier transform via C-M-C chirp transform

G. optical filtering by cascade of two C-M-C chirp transforms

H. Schlieren imaging

Week 12: 11/14 and Week 13: 11/19

XI. Sampling

A. Ideal sampling of special functions

B. Ideal sampling of d[x] and COMB[x]

C. Ideal interpolation of sampled functions

D. Nyquist limit

E. Whittaker-Shannon sampling theorem

F. Aliasing

G. Realistic Sampling (finite-sized detector elements)

H. Realistic Interpolation

I. Quantization, quantization error

Week 13: 11/21, Week 14:11/26

XII. Discrete Fourier Transform (DFT)

A. Infinite-Support DFT

B. Finite-Support DFT; “blurring” of spectrum

C. Efficient evaluation of DFT via fast Fourier transform

Week 15: 12/3 and 12/5

D. Practical considerations

1. 2D DFT

2. location of constant (DC) component

a. Centered and “uncentered” arrays

b. conversion between renderings

3. Units of measure in the two domains

a. relation of N, Dx, Dx

4. Leakage, sidebands

5. Data windows

a. Hanning

b. Hamming

c. Bartlett (triangle)

Week 16: 12/10

XIII. Review