Transform Methods in Image Processing

S. Allen Broughton

MHC Faculty Seminar in Mathematics
Spring Term 2001 @ Mount Holyoke College

Abstract

This web page contains the notes and supporting scripts for a series of lectures on transform methods in image processing. The talks focus on various vector and matrix models of processing with a view towards:

discretely and finitely sampled signals such as audio clips and images
computational methods, employing chiefly Matlab tools

Lectures

Lecture 1: Introduction to signal and image processing and various transform examples

Lecture Notes in pdf format - printout of power point file: (32 pages 121K) MHCSemLect1.pdf
Matlab Scripts used - Image processing toolbox used, all scripts are small text files.

Edge detection script edgedet.m
Blurring 8-bit color vs monochrome eightbit.m
Denoising a signal via truncation of the DFT dftdemo.m
1D analysis and synthesis wave forms for the DFT waveformsDFT.m
1D analysis and synthesis of signal with the Discrete Cosine Transform analsynDCT.m
1D analysis and synthesis of signal with the Discrete Wavelet Transform analsynDWT.m
DFT performance fftperform.m
Split a matlab image with color index map into its three components RGBOpic.m
Split a jpeg image into three three RGB colour matrices jpg2rgb.m

Maple Scripts

Various model waveforms, periodic exponential, discrete exponential waveforms on Z_N, windowed exponential waveforms, and continuous wavelets modelwaveforms.mws (99K)
The Windowed Fourier Transform wft.mws (550K)
The Continuous Wavelet Transform cwt.mws (401K)

Images from Demos

image restoration restore.html
modelwaveforms modelwaveforms.html

Lecture 2: The discrete Fourier transform (DFT) and filtering/convolution (signal and images)

Lecture Notes pdf format - printout of LaTeX (24 pages 345K) MHCSemLect2.pdf
Matlab Scripts

Visualization of "frequency contributions" of model wave forms to a signal and the associated DFT dft1demo3.m
Polar decomposition of 2D waveforms wavefft2cband.m
High and low pass frequency filtering demo (smoothing and differencing) hilopassdemo.m
Demonstration of smoothing an image with a moving mask. smoothimage.m
1D and 2D implementations of circular convolution circconv.m, circconv2.m
Frequency response demonstration - filtering a pure wave form filterwavedemo.m
Plot magnitude and phase frequency response for a filter freqrespplot.m
Restoration script ansmid3.m

Maple Scripts

1D filtering demo filter1d.mws

Lecture 3: Finish up Lecture 2

Image discussed after seminar on fft performace as a functionof N =datasize (fftperform.pdf )

Course Notes and Links

The Mathematics of Image Processing - Course Notes for MA439 taught at Rose-Hulman

Please be aware that the notes are a work in progress.

Chapter 1. Overview MIPN1int.pdf (351K)

Part 1 Fourier Methods and JPEG compression
Chapter 2. Vector space models for signals and images. MIPN2siv.pdf (798K)
Chapter 3. Discrete Fourier Transforms (DFT) of signals and images MIPN3dft.pdf (445K)
Chapter 4. Discrete Cosine Transform (DCT) and JPEG compression MIPN4dct.pdf (799K)
Chapter 5. Convolution and filtering of signals and images MIPN5cnv.pdf (816K)

Part 2. Wavelet and other transforms, wavelet compression.
Chapter 6. Localization and windowing MIPN6loc.pdf (121K)
Chapter 7. Filter banks and the discrete wavelet transform (DWT) MIPN7dwt.pdf (491K)
References MIPNBib.pdf (40K)

MIPApp.pdf