G Uppercase bold: vectorial (multichannel) discrete image, matrix, tensor

k¯, n¯, … A bar indicates a unit vector k˜, k˜, x˜, … A tilde indicates a dimensionless or normalized quantity Gˆu,v , Gˆ, … A hat indicates a quantity in the Fourier domain B, R, … Calligraphic letters indicate representation-independent


x = [x,y]T = [x1, x2]T Image coordinates in the spatial domain k =

[ kx, ky

]T = [k1, k2]T Image coordinates in the Fourier domain X = [X, Y , Z]T = [X1, X2, X3]T World coordinates K, L, M , N Extension of images in t, z, y , and x directions k, l, m, n Indices of images in t, z, y , and x directions k′, l′, m′, n′ Indices in sums of images in t, z, y , and x directions r , s, u, v Indices of images in Fourier domain in t, z, y , and x di-


r Size of masks for neighborhood operators p,q Index for component in multichannel image or a quanti-

zation level

P Number of components in a multichannel image Q Number of quantization levels w Index for a coordinate in a W -dimensional image W Dimension of an image

Symbols Description

· Pointwise multiplication of two images ∗ Convolution Correlation ,⊕ Morphological erosion and dilation operators ◦,• Morphological opening and closing operators ⊗ Morphological hit-miss operator ∨,∧ Boolean or and and operators ∪,∩ Union and intersection of sets ⊂,⊆ Set is subset, subset or equal Shift operator

⇓s Sample or reduction operator: take only every sth pixel, row, etc.