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ter Haar Romeny, EMBS Berder 2004 Gaussian derivative kernels The algebraic expressions for the first 4 orders:

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ter Haar Romeny, EMBS Berder 2004 The factors in front of the Gaussian are the Hermite polynomials.

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ter Haar Romeny, EMBS Berder 2004 They are named after Charles Hermite, a brilliant French mathematician (1822-1901). The Hermite polynomials explode, but the Gaussian hull suppresses any order.

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ter Haar Romeny, EMBS Berder 2004 The Gaussian kernel is not the envelope of the signal! The Hermite polynomials are orthogonal functions, the Gaussian derivative functions are not!

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ter Haar Romeny, EMBS Berder 2004 Gaussian derivatives in the Fourier domain: They act like bandfilters.

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ter Haar Romeny, EMBS Berder 2004 Zero crossings of Gaussian derivative functions Range estimation by Zernicke (1931)

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ter Haar Romeny, EMBS Berder 2004 Correlation between Gaussian derivative kernels correlated uncorrelated

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ter Haar Romeny, EMBS Berder 2004 One can analytically calculate the formula

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ter Haar Romeny, EMBS Berder 2004 Natural limits on observations Taking smaller and smaller kernels gives deviating results. First order derivative of a test image with slope = 1 image as function of scale There is a fundamental relation between the order of differentiation, scale of the operator and the accuracy required.

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ter Haar Romeny, EMBS Berder 2004 The Fourier transform of the derivative of a function is – i times the Fourier transform of the function: A smaller kernel in the spatial domain gives rise to a wider kernel in the Fourier domain

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ter Haar Romeny, EMBS Berder 2004 The error is defined as the amount of the energy (the square) of the kernel that is 'leaking' relative to the total area under the curve (note the integration ranges) Solution:

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ter Haar Romeny, EMBS Berder 2004 Fundamental result:

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1 “Figures and images used in these lecture notes by permission, copyright 1997 by Alan V. Oppenheim and Alan S. Willsky” Signals and Systems Spring 2003.

1 “Figures and images used in these lecture notes by permission, copyright 1997 by Alan V. Oppenheim and Alan S. Willsky” Signals and Systems Spring 2003.

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