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COMP 576 – Mathematics for Image Computing

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Course Objectives

To present mathematics relevant to image processing and analysis using real image computing objectives, as illustrated by computer implementations. The image computing objectives of concern include sharpening blurred images, denoising images, rotating or scaling objects in images, changing the number of image pixels, and locating object edges and bars in images. This mathematics is important in understanding methods that improve image quality, that find, recognize, or visualize objects in images, or that register images.

Lecture Recordings (YouTube videos)

Full Playlist

  1. General course info, course overview
  2. Course overview, concl., derivatives and their tensors
  3. Derivative tensors and applications, Taylor series; Basis images
  4. Delta images, eigenimages; sinusoids, invariance, linearity
  5. Three Sinusoidal image forms. Separability; periodicity; applying shift inv., linear operator
  6. DFT by FFT, reconst’n from ampl’s and phases, coeffs by dot-product, phases and ampl’s info; other bases
  7. lin. Op. appl’n summary; convolution, kernels,transfer f’c’ns, erivatives, cascading operators
  8. Eigentheory, inp. & outp. view of convol’n. PSFs, matched filters
  9. Bar strength, 2nd derivatives w/ apertures via Hessians & DFT, DFT outp. layout, Div & conq. transf’n. Img. sampl’g & aliasing
  10. Sampling interval choice; nonlocality of DFT. Optimal image basis for compression, Parseval,
  11. Optimal compression basis is eigenvectors of A^T A. Matrix rot. & scaling decomp.
  12. RMS width, Optimal compr’n basis via eig’vectors of A A^T. Global bases. Locality and scale: apertures & LODetail
  13. Spatial scale types, basis images for multiscale. Why Gaussian aperture, Gaussian properties. Noise vs. scale
  14. Gaussian dervis. as LODs. Gabor & orthog. Wavelets. Splines. Multiscale, pyramids, residues.
  15. B-splines cont., seperable appln, spline usage, incl. for deform/n displ’ts. Pyramidal rep’ns.
  16. Pyramidal rep’ns incl. Laplacian pyr., orthogonal wavelets. Scale space. Scale choice.
  17. Units. Scaled derivatives & error propag’n. Interpolation kernels: linear, bilinear
  18. Summary for midterm
  19. Summary for midterm, concl., answers on prev. midterm
  20. prev. midterm concl., interpol’n concl., incl. via sinc
  21. Successive subdivision. Least squares.. Orthogonalization.
  22. 3D geometric transformations: transl’ns, rotat’s: matrices, Euler angles, axis & angle.
  23. Geom. Transf’ns so far, Exp and Log. Quaternions
  24. Quaternions, concl., Homogenous coordinates for affine transf’n & for perspective transf’n
  25. Persp’ve, concl. 3D slices, 3D to 2D proj’n. Deformations, thin plate splines
  26. Geometric transfs on discrete images. Height ridges. Watershed ridges.
  27. Review of whole course
  28. Review of whole course cont.

Syllabus and Course Introduction (PDF)

Syllabus: COMP 576/BMME 576, Spring 2022

Lecture Slides (PDF)