Linear Algebra and Image Processing

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

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

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

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

Vector Space Properties Also true: Linear combination Linearly independent vectors

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

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

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

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

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

Numerical Linear Algebra Further reading QR LU SVD …

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

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

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

Histogram Importance Brightness and contrast Information theory Image matching Local features

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

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

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

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?

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