Gradient vector flow: Difference between revisions

Revision as of 00:36, 4 December 2023

Latest revision as of 21:30, 13 February 2025

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to be a solution can be found by calculus of variations, yielding

{{NumBlk|:|<math display = "block">\mu \nabla^2 u - (u - f_x) (f_x^2 + f_y^2) = 0 \,,</math> | 2a}}

{{NumBlk|:|<math display = "block">\mu \nabla^2 v - (v - f_xf_y) (f_x^2 + f_y^2) = 0 \,,</math> | 2b}}

where <math>\textstyle\nabla^2</math> is the Laplacian operator. It is instructive to examine the form of the equations in&nbsp;(2). Each is a partial differential equation that the components <math>u</math> and <math>v</math> of <math>\mathbf{v}</math> must satisfy. If the magnitude of the edge gradient is small, then the solution of each equation is guided entirely by Laplace's equation, for example <math>\textstyle\nabla^2 u = 0</math>, which will produce a smooth scalar field entirely dependent on its boundary conditions. The boundary conditions are effectively provided by the locations in the image where the magnitude of the edge gradient is large, where the solution is driven to agree more with the edge gradients.