FS 501A

Fourier Methods and Imaging


3 credit hours
fall semester

instructor
	R. P. Millane

prerequisites
	Knowledge of a scientific programming language and consent of instructor.

description
Sophisticated imaging techniques are becoming increasingly important tools
 in many areas of science and technology, including biology, medicine,
 materials science, astronomy, earth sciences, chemistry and engineering.
  Many classical imaging techniques such as microscopy, crystallography,
 interferometry and tomography, as well as emerging techniques such as
 confocal microscopy, magnetic resonance imaging (MRI) and speckle 
interferometry, are grounded in Fourier theory.  Fourier techniques are 
used to describe how imaging instruments and experiments produce measurable
 data (in the form of images or diffraction patterns or projections or 
spectra), and are used computationally to produce (or restore or reconstruct) 
desired images form measured data.

The objectives of the course are to introduce theoretical computational
 aspects of Fourier transforms, to develop a unified description of a
 wide variety of imaging techniques, and to examine computational algorithms
 that are used for image restoration and reconstruction.  Practical 
computational exercises in Fourier processing, using MATLAB, are a significant
 part of the course.  Topics covered include:  Fourier transform properties,
 Fourier transform computations, Fourier description of diffraction and image
 formation, Algorithms for image restoration and reconstruction, and 
Applications to imaging problems in microscopy, crystallography, magnetic 
resonance imaging, interferometry, and tomography.

 
text


outline
1.	Fourier transform theory
definitions
spatial frequency
convolution and correlation
support and aliasing
multidimensional transforms
projections
Fourier series
sampling
interpolation
2.	Computing the Fourier transform
discrete Fourier transform (DFT)
fast Fourier transform (FFT)
algorithms
multidimensional transforms
practical aspects
3.	Physics of diffraction and image formation
Fourier optics
blurring 
crystalline, amorphous, and semi-ordered specimens
microscopy
diffraction
interferometry
projections
4.	Algorithms for image restoration and reconstruction
filtering
use of a priori information
iterative transform algorithms
image deconvolution (deblurring)
resolution/noise tradeoffs and regularization
blind deconvolution
filtering and averaging for noise suppression
periodic and nonperiodic (crystalline and noncrystalline) images
image reconstruction from diffraction data
image reconstruction from diffraction amplitudes (phase retrieval)
image reconstruction from projections
5.	A selection of applications from:
light, confocal, and electron microscopy
x-ray and electron crystallography
FT magnetic resonance spectroscopy
magnetic resonance imaging (in medical and biological imaging)
optical and radio interferometry (in astronomical imaging)
x-ray, ultrasonic, seismic, and positron emission tomography (in medical and geophysical imaging)

For further information regarding the Biomedical Engineering Program at Purdue University contact the Biomedical Engineering Graduate Office at (317) 494-5730 bmeprogram@ecn.purdue.edu