Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Matlab Image Processing
Document related concepts
no text concepts found
Transcript
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: v v v v v u v u v u v vu 1v v • Examples Vector Space Properties • Also true: • Linear combination • Linearly independent vectors 1v1 ... n vn 0 1 ,..., n 0 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 v1 ,..., vn • 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 A is a non-zero vector v such that Av v for some scalar • The scalar is the matching Eigenvalue • Number of non-zero eigenvalues = matrix rank • Examples • Importance Solving for Eigenvalues • Characteristic polynomial P( ) det( A I ) • Roots are eigenvalues of A • Algebraic and geometric multiplicities • Diagonalization: P 1 AP D • Importance Properties of Eigenvalues • Trace – sum of eigenvalues • Determinant – product of eigenvalues • Power - A 1 ,...n leads to Ak 1k ,...nk • 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 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 J [m, n] 1 I [m, n] 2 I [m 1, n] 3 I [m, n 1] 4 I [m 1, n] 5 I [m, n 1] • Examples • Averaging • Gaussian • Laplacian 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
Related documents