COMP 576 – Mathematics for Image Computing
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)
- General course info, course overview
- Course overview, concl., derivatives and their tensors
- Derivative tensors and applications, Taylor series; Basis images
- Delta images, eigenimages; sinusoids, invariance, linearity
- Three Sinusoidal image forms. Separability; periodicity; applying shift inv., linear operator
- DFT by FFT, reconst’n from ampl’s and phases, coeffs by dot-product, phases and ampl’s info; other bases
- lin. Op. appl’n summary; convolution, kernels,transfer f’c’ns, erivatives, cascading operators
- Eigentheory, inp. & outp. view of convol’n. PSFs, matched filters
- Bar strength, 2nd derivatives w/ apertures via Hessians & DFT, DFT outp. layout, Div & conq. transf’n. Img. sampl’g & aliasing
- Sampling interval choice; nonlocality of DFT. Optimal image basis for compression, Parseval,
- Optimal compression basis is eigenvectors of A^T A. Matrix rot. & scaling decomp.
- RMS width, Optimal compr’n basis via eig’vectors of A A^T. Global bases. Locality and scale: apertures & LODetail
- Spatial scale types, basis images for multiscale. Why Gaussian aperture, Gaussian properties. Noise vs. scale
- Gaussian dervis. as LODs. Gabor & orthog. Wavelets. Splines. Multiscale, pyramids, residues.
- B-splines cont., seperable appln, spline usage, incl. for deform/n displ’ts. Pyramidal rep’ns.
- Pyramidal rep’ns incl. Laplacian pyr., orthogonal wavelets. Scale space. Scale choice.
- Units. Scaled derivatives & error propag’n. Interpolation kernels: linear, bilinear
- Summary for midterm
- Summary for midterm, concl., answers on prev. midterm
- prev. midterm concl., interpol’n concl., incl. via sinc
- Successive subdivision. Least squares.. Orthogonalization.
- 3D geometric transformations: transl’ns, rotat’s: matrices, Euler angles, axis & angle.
- Geom. Transf’ns so far, Exp and Log. Quaternions
- Quaternions, concl., Homogenous coordinates for affine transf’n & for perspective transf’n
- Persp’ve, concl. 3D slices, 3D to 2D proj’n. Deformations, thin plate splines
- Geometric transfs on discrete images. Height ridges. Watershed ridges.
- Review of whole course
- Review of whole course cont.
- Note: Due to wifi problems, the recording for lecture 28 has a few hiccups in it. The following audio-only recording contains most of the lecture.
- 28. (Audio only) Upcoming courses. Last assingment results. Student questions.