1. Vector Spaces, Signals, and Images.

1.1 Overview.

1.2 Some common image processing problems.

1.3 Signals and images.

1.4 Vector space models for signals and images.

1.5 Basic wave forms the analog case.

1.6 Sampling and aliasing.

1.7 Basic wave forms the discrete case.

1.8 Inner product spaces and orthogonality.

1.9 Signal and image digitization.

1.10 Infinitedimensional inner product spaces.

1.11 Matlab project.

Exercises.

**2. The Discrete Fourier Transform.**

2.1 Overview.

2.2 The time domain and frequency domain.

2.3 A motivational example.

2.4 The onedimensional DFT.

2.5 Properties of the DFT.

2.6 The fast Fourier transform.

2.7 The twodimensional DFT.

2.8 Matlab project.

Exercises.

**3. The discrete cosine transform.**

3.1 Motivation for the DCT: compression.

3.2 Initial examples thresholding.

3.3 The discrete cosine transform.

3.4 Properties of the DCT.

3.5 The twodimensional DCT.

3.6 Block transforms.

3.7 JPEG compression.

3.8 Matlab project.

Exercises.

**4. Convolution and filtering.**

4.1 Overview.

4.2 Onedimensional convolution.

4.3 Convolution theorem and filtering.

4.4 2D convolution filtering images.

4.5 Infinite and biinfinite signal models.

4.6 Matlab project.

Exercises.

**5. Windowing and Localization.**

5.1 Overview: Nonlocality of the DFT.

5.2 Localization via windowing.

5.3 Matlab project.

Exercises.

**6. Filter banks.**

6.1 Overview.

6.2 The Haar filter bank.

6.3 The general onstage twochannel filter bank.

6.4 Multistage filter banks.

6.5 Filter banks for finite length signals.

6.6 The 2D discrete wavelet transform and JPEG 2000.

6.7 Filter design.

6.8 Matlab project.

6.9 Alternate Matlab project.

Exercises.

**7. Wavelets.**

7.1 Overview.

7.2 The Haar Basis.

7.3 Haar wavelets versus the Haar filter bank.

7.4 Orthogonal wavelets.

7.5 Biorthogonal wavelets.

7.6 Matlab Project.

Exercises.