1. Linear Algebra and Image Processing

2. Topics Vectors and Matrices Vector Spaces Eigenvalues and Eigenvectors
Digital Images - Basic Concepts
Histograms
Spatial Filtering

3. Vectors Scalar – single value Vector – tuple of values
Dimension – Cardinality of vector*
Standard operations
Inner product, Outer product
Usage

4. Matrices Matrix – 2D vector* Dimensions Standard operations
Matrix multiplication
Trace and determinant
Rows and columns
Matrix types
Usage

5. Vector Spaces A collection of vectors over a field
Supports addition and scalar multiplication
Satisfies:
Examples

6. Vector Space Properties
Also true:
Linear combination
Linearly independent vectors

7. Subspaces A subspace is a subset of vectors from the vector space.
It must be closed for addition and scalar multiplication
Subspaces are vector spaces themselves
Examples

8. Spanning Set and Basis
A spanning set is a set of all possible linear combinations of
A basis is a set of vectors satisfying
Spanning the space
Linearly independent
Dimension – the length of the basis
Examples

9. Eigenvalues and Eigenvectors
Eigenvector of a square matrix is a non-zero vector such that for some scalar
The scalar is the matching Eigenvalue
Number of non-zero eigenvalues = matrix rank
Examples
Importance

10. Solving for Eigenvalues
Characteristic polynomial
Roots are eigenvalues of A
Algebraic and geometric multiplicities
Diagonalization:
Importance

11. Properties of Eigenvalues
Trace – sum of eigenvalues
Determinant – product of eigenvalues
Power leads to
A is invertible for non-zero eigenvalues only
Invertible – power property holds for -1
A is hermitian – eigenvalues are real
A is unitary – eigenvalues satisfy

12. Numerical Linear Algebra
Further reading QR LU
SVD …

13. Digital Images - Basic Concepts
Digital image – A matrix of pixels
Pixel – Smallest picture element
Digital image acquisition:
Optics
Sampling
Quantization

14. Digital Image Processing
Representation - discrete signal, 1D or 2D
Discrete convolution, discrete derivatives, …
Discrete transforms (e.g. DFT, DCT)
Notable applications
Enhancement – Denoising, Inpainting, Debluring
Compression
Super-Resolution

15. Histogram Density function of the image
Statistical tool for estimation and processing
Gray levels vs. number of occurrences
Can be normalized  PDF
Global, Invariant to order of pixels

16. Histogram Importance Brightness and contrast Information theory
Image matching
Local features

17. Spatial Convolution Convolution in 1D Convolution in 2D Usage
Filtering
Edge Detection
Template matching

18. Linear Filtering Linear combination of image and filter Examples
Averaging
Gaussian
Laplacian

19. Non-Linear Filtering Not all filters can be formulated as matrices
Minimum, Maximum
Median filter
Frequency mixer
Energy transfer filter …

20. Adaptive Filtering Not all filters are space invariant
Image statistics may be local
Corruption may be location dependent
Different schemes at edges and at textures
How to create location dependent filters?

21. Examples Wallis filter – local dynamic range correction
Edge based denoising
Importance for Computer Vision