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**Linear Algebra and Image Processing**

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**Topics Vectors and Matrices Vector Spaces Eigenvalues and Eigenvectors**

Digital Images - Basic Concepts Histograms Spatial Filtering

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**Vectors Scalar – single value Vector – tuple of values**

Dimension – Cardinality of vector* Standard operations Inner product, Outer product Usage

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**Matrices Matrix – 2D vector* Dimensions Standard operations**

Matrix multiplication Trace and determinant Rows and columns Matrix types Usage

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**Vector Spaces A collection of vectors over a field**

Supports addition and scalar multiplication Satisfies: Examples

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**Vector Space Properties**

Also true: Linear combination Linearly independent vectors

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**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

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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

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**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

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**Solving for Eigenvalues**

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

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**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

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**Numerical Linear Algebra**

Further reading QR LU SVD …

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**Digital Images - Basic Concepts**

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

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**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

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**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

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**Histogram Importance Brightness and contrast Information theory**

Image matching Local features

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**Spatial Convolution Convolution in 1D Convolution in 2D Usage**

Filtering Edge Detection Template matching

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**Linear Filtering Linear combination of image and filter Examples**

Averaging Gaussian Laplacian

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**Non-Linear Filtering Not all filters can be formulated as matrices**

Minimum, Maximum Median filter Frequency mixer Energy transfer filter …

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**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?

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**Examples Wallis filter – local dynamic range correction**

Edge based denoising Importance for Computer Vision

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