Linear Algebraand

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

1

v v

v v v

u v u v

u v v u

v v

Vector Space Properties

• Also true:

• Linear combination• Linearly independent vectors

1 1 1... 0 ,..., 0n n nv v

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

1,..., nv v

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

Av v A

v

Solving for Eigenvalues

• Characteristic polynomial• Roots are eigenvalues of A

• Algebraic and geometric multiplicities• Diagonalization:

• Importance

P( ) det( )A I

1P AP D

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

1,... nA 1 ,...k k knA

1

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

1 2

3 4 5

[ , ] [ , ] [ 1, ]

[ , 1] [ 1, ] [ , 1]

J m n I m n I m n

I m n I m n I m n

2 3 2

3 5 3

2 3 2

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