derivate_vector_field🔗
Short description🔗
derivate_vector_field — Convolve a vector field with derivatives of the Gaussian.
Signature🔗
derivate_vector_field( image VectorField, out image Result, real Sigma, string Component )
Description🔗
derivate_vector_field convolves the components of a vector field with
the derivatives of a Gaussian and calculates various features derived
therefrom.
derivate_vector_field only accepts vector fields of the semantic
type vector_field_relative.
The VectorField \(F(r,c)=(u(r,c),v(r,c))\) is defined as in
optical_flow_mg.
Sigma is the parameter of the Gaussian
(i.e., the amount of smoothing). If a single value is passed in
Sigma, the amount of smoothing in the column and row direction is
identical. If two values are passed in Sigma, the first value
specifies the amount of smoothing in the column direction, while the second
value specifies the amount of smoothing in the row direction. The possible
values for Component are:
-
'curl': The curl of the vector field. One application of using 'curl' is to analyze optical flow fields. Metaphorically speaking, the curl is how much a small boat would rotate if the vector field was a fluid.
\[\begin{eqnarray*} curl = \frac{\partial u(r,c)}{\partial c} - \frac{\partial v(r,c)}{\partial r} \end{eqnarray*}\] -
'divergence': The divergence of the vector field. One application of using 'divergence' is to analyze optical flow fields. Metaphorically speaking, the divergence is where the source and sink would be if the vector field was a fluid.
\[\begin{eqnarray*} div = \frac{\partial u(r,c)}{\partial r} + \frac{\partial v(r,c)}{\partial c} \end{eqnarray*}\]
When used in context of photometric stereo, the operator
derivate_vector_field offers two more parameters, which are
especially designed to process the gradient field that is returned by
photometric_stereo. In this case, we interpret the input vector
field as gradient of the underlying surface.
In the following formulas, the input vector field is therefore noted as \(G(r,c) = \nabla f = ( \frac{\partial f(r,c)}{\partial r} , \frac{\partial f(r,c)}{\partial c} )\) where the first and second component of the input is the gradient field of the surface \(f(r,c)\). In the formulas below f_rc denotes the first derivative in column direction of the first component of the gradient field.
-
'mean_curvature': Mean curvature \(H\) of the underlying surface when the input vector field
VectorFieldis interpreted as gradient field. One application of using 'mean_curvature' is to process the vector field that is returned byphotometric_stereo. After filtering the vector field, even tiny scratches or bumps can be segmented.\[\begin{eqnarray*} A & = & (1 + \frac{\partial f(r,c)^2}{\partial r}) \frac{\partial^2 f(r,c)}{\partial c^2} \\ B & = & \frac{\partial f(r,c)}{\partial r} \frac{\partial f(r,c)}{\partial c} ( \frac{\partial^2 f(r,c)}{\partial r \partial c} + \frac{\partial^2 f(r,c)}{\partial c \partial r} ) \\ C & = & (1 + \frac{\partial f(r,c)^2}{\partial c}) \frac{\partial^2 f(r,c)}{\partial r^2} \\ D & = & (1 + \frac{\partial f(r,c)^2}{\partial r} + \frac{\partial f(r,c)^2}{\partial c})^{\frac{3}{2}}\\ H & = & \frac{A - B + C}{D} \end{eqnarray*}\] -
'gauss_curvature': Gaussian curvature \(K\) of the underlying surface when the input vector field
VectorFieldis interpreted as gradient field. One application of using 'gauss_curvature' is to process the vector field that is returned byphotometric_stereo. After filtering the vector field, even tiny scratches or bumps can be segmented. If the underlying surface of the vector field is developable, the Gaussian curvature is zero.\[\begin{eqnarray*} K = \frac{\frac{\partial f(r,c)^2}{\partial r \partial r} * \frac{\partial f(r,c)^2}{\partial c \partial c} - \frac{\partial f(r,c)^2}{\partial r \partial c} * \frac{\partial f(r,c)^2}{\partial c \partial r} } {(1 + \frac{\partial f(r,c)^2}{\partial r} + \frac{\partial f(r,c)^2}{\partial c})^2} \end{eqnarray*}\]
Execution information🔗
Execution information
-
Multithreading type: reentrant (runs in parallel with non-exclusive operators).
-
Multithreading scope: global (may be called from any thread).
-
Automatically parallelized on domain level.
-
Automatically parallelized on tuple level.
Parameters🔗
VectorField (input_object) singlechannelimage(-array) → object (vector_field)
Input vector field.
Result (output_object) singlechannelimage(-array) → object (real)
Filtered result images.
Sigma (input_control) real(-array) → (real)
Sigma of the Gaussian.
Default: 1.0
Suggested values: 0.7, 1.0, 1.5, 2.0, 3.0, 4.0, 5.0
Value range: 0.01 ≤ Sigma ≤ 50.0
Component (input_control) string → (string)
Component to be calculated.
Default: 'mean_curvature'
List of values: 'curl', 'divergence', 'gauss_curvature', 'mean_curvature'
Result🔗
If the parameters are valid, the operator derivate_vector_field
returns the value 2 (H_MSG_TRUE).
The behavior in case of empty input (no input images available)
is set via the operator set_system('no_object_result',<Result>).
If necessary, an exception is raised.
Combinations with other operators🔗
Module🔗
Foundation