module #include "diplib/linear.h"
Linear filters Linear smoothing, sharpening and derivative filters.
Classes

struct dip::
OneDimensionalFilter  Describes a 1D filter
Aliases

using dip::
OneDimensionalFilterArray = std::vector<OneDimensionalFilter>  An array of 1D filters
Functions

auto dip::
SeparateFilter (dip::Image const& filter) > dip::OneDimensionalFilterArray  Separates a linear filter (convolution kernel) into a set of 1D filters that can be applied using
dip::SeparableConvolution
. 
void dip::
SeparableConvolution (dip::Image const& in, dip::Image& out, dip::OneDimensionalFilterArray const& filterArray, dip::StringArray const& boundaryCondition = {}, dip::BooleanArray process = {})  Applies a convolution with a filter kernel (PSF) that is separable.

void dip::
ConvolveFT (dip::Image const& in, dip::Image const& filter, dip::Image& out, dip::String const& inRepresentation = S::SPATIAL, dip::String const& filterRepresentation = S::SPATIAL, dip::String const& outRepresentation = S::SPATIAL)  Applies a convolution with a filter kernel (PSF) by multiplication in the Fourier domain.

void dip::
GeneralConvolution (dip::Image const& in, dip::Image const& filter, dip::Image& out, dip::StringArray const& boundaryCondition = {})  Applies a convolution with a filter kernel (PSF) by direct implementation of the convolution sum.

void dip::
Uniform (dip::Image const& in, dip::Image& out, dip::Kernel const& kernel = {}, dip::StringArray const& boundaryCondition = {})  Applies a convolution with a kernel with uniform weights, leading to an average (mean) filter.

void dip::
GaussFIR (dip::Image const& in, dip::Image& out, dip::FloatArray sigmas = {1.0}, dip::UnsignedArray derivativeOrder = {0}, dip::StringArray const& boundaryCondition = {}, dip::dfloat truncation = 3)  Finite impulse response implementation of the Gaussian filter and its derivatives

void dip::
GaussFT (dip::Image const& in, dip::Image& out, dip::FloatArray sigmas = {1.0}, dip::UnsignedArray derivativeOrder = {0}, dip::dfloat truncation = 3)  Fourier implementation of the Gaussian filter and its derivatives

void dip::
GaussIIR (dip::Image const& in, dip::Image& out, dip::FloatArray sigmas = {1.0}, dip::UnsignedArray derivativeOrder = {0}, dip::StringArray const& boundaryCondition = {}, dip::UnsignedArray filterOrder = {}, dip::String const& designMethod = S::DISCRETE_TIME_FIT, dip::dfloat truncation = 3)  Infinite impulse response implementation of the Gaussian filter and its derivatives

void dip::
Gauss (dip::Image const& in, dip::Image& out, dip::FloatArray sigmas = {1.0}, dip::UnsignedArray derivativeOrder = {0}, dip::String const& method = S::BEST, dip::StringArray const& boundaryCondition = {}, dip::dfloat truncation = 3)  Convolution with a Gaussian kernel and its derivatives

void dip::
FiniteDifference (dip::Image const& in, dip::Image& out, dip::UnsignedArray derivativeOrder = {0}, dip::String const& smoothFlag = S::SMOOTH, dip::StringArray const& boundaryCondition = {}, dip::BooleanArray process = {})  Finite difference derivatives

void dip::
SobelGradient (dip::Image const& in, dip::Image& out, dip::uint dimension = 0, dip::StringArray const& boundaryCondition = {})  The Sobel derivative filter

void dip::
Derivative (dip::Image const& in, dip::Image& out, dip::UnsignedArray derivativeOrder, dip::FloatArray sigmas = {1.0}, dip::String const& method = S::BEST, dip::StringArray const& boundaryCondition = {}, dip::dfloat truncation = 3)  Computes derivatives

void dip::
Dx (dip::Image const& in, dip::Image& out, dip::FloatArray sigmas = {1.0})  Computes the first derivative along x, see
dip::Derivative
. 
void dip::
Dy (dip::Image const& in, dip::Image& out, dip::FloatArray sigmas = {1.0})  Computes the first derivative along y, see
dip::Derivative
. 
void dip::
Dz (dip::Image const& in, dip::Image& out, dip::FloatArray sigmas = {1.0})  Computes the first derivative along z, see
dip::Derivative
. 
void dip::
Dxx (dip::Image const& in, dip::Image& out, dip::FloatArray sigmas = {1.0})  Computes the second derivative along x, see
dip::Derivative
. 
void dip::
Dyy (dip::Image const& in, dip::Image& out, dip::FloatArray sigmas = {1.0})  Computes the second derivative along y, see
dip::Derivative
. 
void dip::
Dzz (dip::Image const& in, dip::Image& out, dip::FloatArray sigmas = {1.0})  Computes the second derivative along z, see
dip::Derivative
. 
void dip::
Dxy (dip::Image const& in, dip::Image& out, dip::FloatArray sigmas = {1.0})  Computes the first derivative along x and y, see
dip::Derivative
. 
void dip::
Dxz (dip::Image const& in, dip::Image& out, dip::FloatArray sigmas = {1.0})  Computes the first derivative along x and z, see
dip::Derivative
. 
void dip::
Dyz (dip::Image const& in, dip::Image& out, dip::FloatArray sigmas = {1.0})  Computes the first derivative along y and y, see
dip::Derivative
. 
void dip::
Gradient (dip::Image const& in, dip::Image& out, dip::FloatArray sigmas = {1.0}, dip::String const& method = S::BEST, dip::StringArray const& boundaryCondition = {}, dip::BooleanArray process = {}, dip::dfloat truncation = 3)  Computes the gradient of the image, resulting in an Nvector image, if the input was Ndimensional.

void dip::
GradientMagnitude (dip::Image const& in, dip::Image& out, dip::FloatArray sigmas = {1.0}, dip::String const& method = S::BEST, dip::StringArray const& boundaryCondition = {}, dip::BooleanArray process = {}, dip::dfloat truncation = 3)  Computes the gradient magnitude of the image, equivalent to
dip::Norm( dip::Gradient( in ))
. 
void dip::
GradientDirection (dip::Image const& in, dip::Image& out, dip::FloatArray sigmas = {1.0}, dip::String const& method = S::BEST, dip::StringArray const& boundaryCondition = {}, dip::BooleanArray process = {}, dip::dfloat truncation = 3)  Computes the direction of the gradient of the image, equivalent to
dip::Angle( dip::Gradient( in ))
. 
void dip::
Curl (dip::Image const& in, dip::Image& out, dip::FloatArray sigmas = {1.0}, dip::String const& method = S::BEST, dip::StringArray const& boundaryCondition = {}, dip::BooleanArray process = {}, dip::dfloat truncation = 3)  Computes the curl (rotation) of the 2D or 3D vector field
in
. 
void dip::
Divergence (dip::Image const& in, dip::Image& out, dip::FloatArray sigmas = {1.0}, dip::String const& method = S::BEST, dip::StringArray const& boundaryCondition = {}, dip::BooleanArray process = {}, dip::dfloat truncation = 3)  Computes the divergence of the vector field
in
. 
void dip::
Hessian (dip::Image const& in, dip::Image& out, dip::FloatArray sigmas = {1.0}, dip::String const& method = S::BEST, dip::StringArray const& boundaryCondition = {}, dip::BooleanArray process = {}, dip::dfloat truncation = 3)  Computes the Hessian of the image, resulting in a symmetric NxN tensor image, if the input was Ndimensional.

void dip::
Laplace (dip::Image const& in, dip::Image& out, dip::FloatArray sigmas = {1.0}, dip::String const& method = S::BEST, dip::StringArray const& boundaryCondition = {}, dip::BooleanArray process = {}, dip::dfloat truncation = 3)  Computes the Laplacian of the image, equivalent to
dip::Trace( dip::Hessian( in ))
, but more efficient. 
void dip::
Dgg (dip::Image const& in, dip::Image& out, dip::FloatArray sigmas = {1.0}, dip::String const& method = S::BEST, dip::StringArray const& boundaryCondition = {}, dip::BooleanArray process = {}, dip::dfloat truncation = 3)  Computes the second derivative in the gradient direction.

void dip::
LaplacePlusDgg (dip::Image const& in, dip::Image& out, dip::FloatArray sigmas = {1.0}, dip::String const& method = S::BEST, dip::StringArray const& boundaryCondition = {}, dip::BooleanArray process = {}, dip::dfloat truncation = 3)  Adds the second derivative in the gradient direction to the Laplacian.

void dip::
LaplaceMinusDgg (dip::Image const& in, dip::Image& out, dip::FloatArray sigmas = {1.0}, dip::String const& method = S::BEST, dip::StringArray const& boundaryCondition = {}, dip::BooleanArray process = {}, dip::dfloat truncation = 3)  Subtracts the second derivative in the gradient direction from the Laplacian.

void dip::
Sharpen (dip::Image const& in, dip::Image& out, dip::dfloat weight = 1.0, dip::FloatArray sigmas = {1.0}, dip::String const& method = S::BEST, dip::StringArray const& boundaryCondition = {}, dip::dfloat truncation = 3)  Sharpens
in
by subtracting the Laplacian of the image. 
void dip::
UnsharpMask (dip::Image const& in, dip::Image& out, dip::dfloat weight = 1.0, dip::FloatArray sigmas = {1.0}, dip::String const& method = S::BEST, dip::StringArray const& boundaryCondition = {}, dip::dfloat truncation = 3)  Sharpens
in
by subtracting the smoothed image. 
void dip::
GaborFIR (dip::Image const& in, dip::Image& out, dip::FloatArray sigmas, dip::FloatArray const& frequencies, dip::StringArray const& boundaryCondition = {}, dip::BooleanArray process = {}, dip::dfloat truncation = 3)  Finite impulse response implementation of the Gabor filter

void dip::
GaborIIR (dip::Image const& in, dip::Image& out, dip::FloatArray sigmas, dip::FloatArray const& frequencies, dip::StringArray const& boundaryCondition = {}, dip::BooleanArray process = {}, dip::IntegerArray filterOrder = {}, dip::dfloat truncation = 3)  Recursive infinite impulse response implementation of the Gabor filter

void dip::
Gabor2D (dip::Image const& in, dip::Image& out, dip::FloatArray sigmas = {5.0,5.0}, dip::dfloat frequency = 0.1, dip::dfloat direction = dip::pi, dip::StringArray const& boundaryCondition = {}, dip::dfloat truncation = 3)  2D Gabor filter with direction parameter

void dip::
LogGaborFilterBank (dip::Image const& in, dip::Image& out, dip::FloatArray const& wavelengths = {3.0,6.0,12.0,24.0}, dip::dfloat bandwidth = 0.75, dip::uint nOrientations = 6, dip::String const& inRepresentation = S::SPATIAL, dip::String const& outRepresentation = S::SPATIAL)  Applies a logGabor filter bank

void dip::
NormalizedConvolution (dip::Image const& in, dip::Image const& mask, dip::Image& out, dip::FloatArray const& sigmas = {1.0}, dip::String const& method = S::BEST, dip::StringArray const& boundaryCondition = {S::ADD_ZEROS}, dip::dfloat truncation = 3)  Computes the normalized convolution with a Gaussian kernel: a Gaussian convolution for missing or uncertain data.

void dip::
NormalizedDifferentialConvolution (dip::Image const& in, dip::Image const& mask, dip::Image& out, dip::uint dimension = 0, dip::FloatArray const& sigmas = {1.0}, dip::String const& method = S::BEST, dip::StringArray const& boundaryCondition = {S::ADD_ZEROS}, dip::dfloat truncation = 3)  Computes the normalized differential convolution with a Gaussian kernel: a derivative operator for missing or uncertain data.

void dip::
MeanShiftVector (dip::Image const& in, dip::Image& out, dip::FloatArray sigmas = {1.0}, dip::String const& method = S::BEST, dip::StringArray const& boundaryCondition = {}, dip::dfloat truncation = 3)  Computes the mean shift vector for each pixel in the image
Class documentation
struct dip::OneDimensionalFilter
Describes a 1D filter
The weights are in filter
. If isComplex
, then the values in filter
are interpreted as real/imaginary
pairs. In this case, filter
must have an even length, with each two consecutive elements representing a
single filter weight. The filter.data()
pointer can thus be cast to dip::dcomplex
.
The origin is placed either at the index given by origin
if it’s nonnegative, or at index
filter.size() / 2
if origin
is negative. Note that filter.size() / 2
is either the middle pixel
if the filter is odd in length, or the pixel to the right of the center if it is even in length:
size of filter 
origin 
origin location 

any  1 
x 0 x x x x 
any  5 
x x x x x 0 
any odd value  1 
x x 0 x x 
any even value  1 
x x x 0 x x 
Note that, if positive, origin
must be an index to one of the samples in the filter
array:
origin < filter.size()
.
symmetry
indicates the filter shape: "general"
(or an empty string) indicates no symmetry.
"even"
indicates even symmetry, "odd"
indicates odd symmetry, and "conj"
indicates complex conjugate
symmetry. In these three cases, the filter represents the left half of the full filter,
with the rightmost element at the origin (and not repeated). The full filter is thus always odd in size.
"deven"
, "dodd"
and "dconj"
are similar, but duplicate the rightmost element, yielding
an evensized filter. The origin for the symmetric filters is handled identically to the general filter case.
The following table summarizes the result of using various symmetry
values. The filter
array in all
cases has n elements represented in this example as [a,b,c].
symmetry 
resulting array  resulting array length 

"general" 
[a,b,c]  n 
"even" 
[a,b,c,b,a]  2n  1 
"odd" 
[a,b,c,b,a]  2n  1 
"conj" 
[a,b,c,b^{*},a^{*}]  2n  1 
"deven" 
[a,b,c,c,b,a]  2n 
"dodd" 
[a,b,c,c,b,a]  2n 
"dconj" 
[a,b,c,c^{*},b^{*},a^{*}]  2n 
The convolution is applied to each tensor component separately, which is always the correct behavior for linear filters.
Variables  

std::vector<dfloat> filter  Filter weights. 
dip::sint origin  Origin of the filter if nonnegative. 
dip::String symmetry 
Filter shape: "" == "general" , "even" , "odd" , "conj" , "deven" , "dodd" or "dconj" .

bool isComplex 
If true, filter contains complex data.

Function documentation
dip::OneDimensionalFilterArray
dip::SeparateFilter (dip::Image const& filter)
Separates a linear filter (convolution kernel) into a set of 1D filters that can be applied using
dip::SeparableConvolution
.
If filter
does not represent a separable kernel, the output dip::OneDimensionalFilterArray
object is
empty (it’s empty
method returns true, and it’s size
method return 0).
void
dip::SeparableConvolution (dip::Image const& in,
dip::Image& out,
dip::OneDimensionalFilterArray const& filterArray,
dip::StringArray const& boundaryCondition = {},
dip::BooleanArray process = {})
Applies a convolution with a filter kernel (PSF) that is separable.
filter
is an array with exactly one dip::OneDimensionalFilter
element for each dimension of in
.
Alternatively, it can have a single element, which will be used unchanged for each dimension.
For the dimensions that are not processed (process
is false
for those dimensions),
the filter
array can have nonsensical data or a zerolength filter weights array.
Any filter
array that is zero size or the equivalent of {1}
will not be applied either.
boundaryCondition
indicates how the boundary should be expanded in each dimension. See dip::BoundaryCondition
.
process
indices which dimensions to process, and can be {}
to indicate all dimensions are to be processed.
void
dip::ConvolveFT (dip::Image const& in,
dip::Image const& filter,
dip::Image& out,
dip::String const& inRepresentation = S::SPATIAL,
dip::String const& filterRepresentation = S::SPATIAL,
dip::String const& outRepresentation = S::SPATIAL)
Applies a convolution with a filter kernel (PSF) by multiplication in the Fourier domain.
filter
is an image, and must be equal in size or smaller than in
. If both in
and filter
are real, out
will be real too, otherwise it will have a complex type.
As elsewhere, the origin of filter
is in the middle of the image, on the pixel to the right of
the center in case of an evensized image.
If in
or filter
is already Fourier transformed, set inRepresentation
or filterRepresentation
to "frequency"
. Similarly, if outRepresentation
is "frequency"
, the output will not be
inversetransformed, so will be in the frequency domain.
void
dip::GeneralConvolution (dip::Image const& in,
dip::Image const& filter,
dip::Image& out,
dip::StringArray const& boundaryCondition = {})
Applies a convolution with a filter kernel (PSF) by direct implementation of the convolution sum.
filter
is an image, and must be equal in size or smaller than in
. filter
must be realvalued.
As elsewhere, the origin of filter
is in the middle of the image, on the pixel to the right of
the center in case of an evensized image.
Note that this is a really expensive way to compute the convolution for any filter
that has more than a
small amount of nonzero values. It is always advantageous to try to separate your filter into a set of 1D
filters (see dip::SeparateFilter
and dip::SeparableConvolution
). If this is not possible, use
dip::ConvolveFT
with larger filters to compute the convolution in the Fourier domain.
Also, if all nonzero filter weights have the same value, dip::Uniform
implements a more efficient
algorithm. If filter
is a binary image, dip::Uniform
is called.
boundaryCondition
indicates how the boundary should be expanded in each dimension. See dip::BoundaryCondition
.
void
dip::Uniform (dip::Image const& in,
dip::Image& out,
dip::Kernel const& kernel = {},
dip::StringArray const& boundaryCondition = {})
Applies a convolution with a kernel with uniform weights, leading to an average (mean) filter.
The size and shape of the kernel is given by kernel
, which you can define through a default
shape with corresponding sizes, or through a binary image. See dip::Kernel
.
boundaryCondition
indicates how the boundary should be expanded in each dimension. See dip::BoundaryCondition
.
void
dip::GaussFIR (dip::Image const& in,
dip::Image& out,
dip::FloatArray sigmas = {1.0},
dip::UnsignedArray derivativeOrder = {0},
dip::StringArray const& boundaryCondition = {},
dip::dfloat truncation = 3)
Finite impulse response implementation of the Gaussian filter and its derivatives
Convolves the image with a 1D Gaussian kernel along each dimension. For each dimension,
provide a value in sigmas
and derivativeOrder
. The zerothorder derivative is a plain
smoothing, no derivative is computed. Derivatives with order up to 3 can be computed with
this function. For higherorder derivatives, use dip::GaussFT
.
The value of sigma determines the smoothing effect. For values smaller than about 0.8, the
result is an increasingly poor approximation to the Gaussian filter. Use dip::GaussFT
for
very small sigmas. Conversely, for very large sigmas it is more efficient to use dip::GaussIIR
,
which runs in a constant time with respect to the sigma. Dimensions where sigma is 0 or
negative are not processed, even if the derivative order is nonzero.
For the smoothing filter (derivativeOrder
is 0), the size of the kernel is given by
2 * std::ceil( truncation * sigma ) + 1
. The default value for truncation
is 3, which assures a good
approximation of the Gaussian kernel without unnecessary expense. It is possible to reduce
computation slightly by decreasing this parameter, but it is not recommended. For derivatives,
the value of truncation
is increased by 0.5 * derivativeOrder
.
boundaryCondition
indicates how the boundary should be expanded in each dimension. See dip::BoundaryCondition
.
void
dip::GaussFT (dip::Image const& in,
dip::Image& out,
dip::FloatArray sigmas = {1.0},
dip::UnsignedArray derivativeOrder = {0},
dip::dfloat truncation = 3)
Fourier implementation of the Gaussian filter and its derivatives
Convolves the image with a Gaussian kernel by multiplication in the Fourier domain.
For each dimension, provide a value in sigmas
and derivativeOrder
. The value of sigma determines
the smoothing effect. The zerothorder derivative is a plain smoothing, no derivative is computed.
The values of sigmas
are translated to the Fourier domain, and a Fourierdomain Gaussian is computed.
Frequencies above std::ceil(( truncation + 0.5 * derivativeOrder ) * FDsigma )
are set to 0. It is a relatively
minute computational difference if truncation
were to be infinity, so it is not worth while to try to
speed up the operation by decreasing truncation
.
Dimensions where sigma is 0 or negative are not smoothed. Note that it is possible to compute a derivative without smoothing in the Fourier domain.
void
dip::GaussIIR (dip::Image const& in,
dip::Image& out,
dip::FloatArray sigmas = {1.0},
dip::UnsignedArray derivativeOrder = {0},
dip::StringArray const& boundaryCondition = {},
dip::UnsignedArray filterOrder = {},
dip::String const& designMethod = S::DISCRETE_TIME_FIT,
dip::dfloat truncation = 3)
Infinite impulse response implementation of the Gaussian filter and its derivatives
Convolves the image with an IIR 1D Gaussian kernel along each dimension. For each dimension,
provide a value in sigmas
and derivativeOrder
. The zerothorder derivative is a plain
smoothing, no derivative is computed. Derivatives with order up to 4 can be computed with this
function. For higherorder derivatives, use dip::GaussFT
.
The value of sigma determines the smoothing effect. For smaller values, the result is an increasingly poor approximation to the Gaussian filter. This function is efficient only for very large sigmas. Dimensions where sigma is 0 or negative are not processed, even if the derivative order is nonzero.
boundaryCondition
indicates how the boundary should be expanded in each dimension. See dip::BoundaryCondition
.
The filterOrder
and designMethod
determine how the filter is implemented. By default,
designMethod
is “discrete time fit”. This is the method described in van Vliet et al. (1998).
filterOrder
can be between 1 and 5, with 3 producing good results, and increasing order producing
better results. When computing derivatives, a higher filterOrder
is necessary. By default,
filterOrder
is 3 + derivativeOrder
, capped at 5. The alternative designMethod
is “forward backward”.
This is the method described in Young and van Vliet (1995). Here filterOrder
can be between 3 and 5.
void
dip::Gauss (dip::Image const& in,
dip::Image& out,
dip::FloatArray sigmas = {1.0},
dip::UnsignedArray derivativeOrder = {0},
dip::String const& method = S::BEST,
dip::StringArray const& boundaryCondition = {},
dip::dfloat truncation = 3)
Convolution with a Gaussian kernel and its derivatives
Convolves the image with a Gaussian kernel. For each dimension, provide a value in sigmas
and
derivativeOrder
. The value of sigma determines the smoothing effect. The zerothorder derivative
is a plain smoothing, no derivative is computed. Dimensions where sigma is 0 or negative are not
smoothed. Only the “FT” method can compute the derivative along a dimension where sigma is zero or negative.
How the convolution is computed depends on the value of method
:
"FIR"
: Finite impulse response implementation, seedip::GaussFIR
."IIR"
: Infinite impulse response implementation, seedip::GaussIIR
."FT"
: Fourier domain implementation, seedip::GaussFT
."best"
: Picks the best method, according to the values ofsigmas
andderivativeOrder
: if any
derivativeOrder
is larger than 3, use the FT method,  else if any
sigmas
is smaller than 0.8, use the FT method,  else if any
sigmas
is larger than 10, use the IIR method,  else use the FIR method.
 if any
boundaryCondition
indicates how the boundary should be expanded in each dimension. See dip::BoundaryCondition
.
void
dip::FiniteDifference (dip::Image const& in,
dip::Image& out,
dip::UnsignedArray derivativeOrder = {0},
dip::String const& smoothFlag = S::SMOOTH,
dip::StringArray const& boundaryCondition = {},
dip::BooleanArray process = {})
Finite difference derivatives
Computes derivatives using the finite difference method. Set a derivativeOrder
for each dimension.
Derivatives of order up to 2 can be computed with this function. The zerothorder derivative implies either
a smoothing is applied (smoothFlag == "smooth"
) or the dimension is not processed at all.
The smoothing filter is [1,2,1]/4
(as in the Sobel filter), the first order derivative is [1,0,1]/2
(central difference), and the second order derivative is [1,2,1]
(which is the composition of twice the
noncentral difference [1,1]
). Thus, computing the first derivative twice does not yield the same result
as computing the second derivative directly.
boundaryCondition
indicates how the boundary should be expanded in each dimension. See dip::BoundaryCondition
.
Set process
to false for those dimensions that should not be filtered.
void
dip::SobelGradient (dip::Image const& in,
dip::Image& out,
dip::uint dimension = 0,
dip::StringArray const& boundaryCondition = {})
The Sobel derivative filter
This function applies the generalization of the Sobel derivative filter to arbitrary dimensions. Along the
dimension dimension
, the central difference is computed, and along all other dimensions, the triangular
smoothing filter [1,2,1]/4
is applied.
boundaryCondition
indicates how the boundary should be expanded in each dimension. See dip::BoundaryCondition
.
This function calls dip::FiniteDifference
.
void
dip::Derivative (dip::Image const& in,
dip::Image& out,
dip::UnsignedArray derivativeOrder,
dip::FloatArray sigmas = {1.0},
dip::String const& method = S::BEST,
dip::StringArray const& boundaryCondition = {},
dip::dfloat truncation = 3)
Computes derivatives
This function provides an interface to the various derivative filters in DIPlib.
For each dimension, provide a value in sigmas
and derivativeOrder
. The value of sigma determines
the smoothing effect. The zerothorder derivative is a plain smoothing, no derivative is computed.
If method
is "best"
, "gaussfir"
or "gaussiir"
, dimensions where sigma is 0 or negative are not processed,
even if the derivative order is nonzero. That is, sigma must be positive for the dimension(s) where
the derivative is to be computed.
method
indicates which derivative filter is used:
"best"
: A Gaussian derivative, seedip::Gauss
."gaussfir"
: The FIR implementation of the Gaussian derivative, seedip::GaussFIR
."gaussiir"
: The IIR implementation of the Gaussian derivative, seedip::GaussIIR
."gaussft"
: The FT implementation of the Gaussian derivative, seedip::GaussFT
."finitediff"
: A finite difference derivative, seedip::FiniteDifference
.
A finite difference derivative is an approximation to the derivative operator on the discrete grid. In contrast, convolving an image with the derivative of a Gaussian provides the exact derivative of the image convolved with a Gaussian:
Thus (considering the regularization provided by the Gaussian smoothing is beneficial) it is always better to use Gaussian derivatives than finite difference derivatives.
boundaryCondition
indicates how the boundary should be expanded in each dimension. See dip::BoundaryCondition
.
void
dip::Gradient (dip::Image const& in,
dip::Image& out,
dip::FloatArray sigmas = {1.0},
dip::String const& method = S::BEST,
dip::StringArray const& boundaryCondition = {},
dip::BooleanArray process = {},
dip::dfloat truncation = 3)
Computes the gradient of the image, resulting in an Nvector image, if the input was Ndimensional.
Each tensor component corresponds to the first derivative along the given dimension: out[ 0 ]
is the
derivative along x (dimension with index 0), out[ 1 ]
is the derivative along y (dimension with index 1),
etc.
The input image must be scalar.
Set process
to false for those dimensions along which no derivative should be taken. For example, if in
is
a 3D image, and process
is {true,false,false}
, then out
will be a scalar image, containing only the
derivative along the x axis.
By default uses Gaussian derivatives in the computation. Set method = "finitediff"
for finite difference
approximations to the gradient. See dip::Derivative
for more information on the other parameters.
void
dip::GradientMagnitude (dip::Image const& in,
dip::Image& out,
dip::FloatArray sigmas = {1.0},
dip::String const& method = S::BEST,
dip::StringArray const& boundaryCondition = {},
dip::BooleanArray process = {},
dip::dfloat truncation = 3)
Computes the gradient magnitude of the image, equivalent to dip::Norm( dip::Gradient( in ))
.
For nonscalar images, applies the operation to each image channel. See dip::Gradient
for information on the parameters.
void
dip::GradientDirection (dip::Image const& in,
dip::Image& out,
dip::FloatArray sigmas = {1.0},
dip::String const& method = S::BEST,
dip::StringArray const& boundaryCondition = {},
dip::BooleanArray process = {},
dip::dfloat truncation = 3)
Computes the direction of the gradient of the image, equivalent to dip::Angle( dip::Gradient( in ))
.
The input image must be scalar. For a 2D gradient, the output is scalar also, containing the angle of the
gradient to the xaxis. For a 3D gradient, the output has two tensor components, containing the azimuth and
inclination. See dip::Angle
for an explanation.
See dip::Gradient
for information on the parameters.
void
dip::Curl (dip::Image const& in,
dip::Image& out,
dip::FloatArray sigmas = {1.0},
dip::String const& method = S::BEST,
dip::StringArray const& boundaryCondition = {},
dip::BooleanArray process = {},
dip::dfloat truncation = 3)
Computes the curl (rotation) of the 2D or 3D vector field in
.
Curl is defined as by , for a 3vector
(the vector image in
), resulting in a 3vector with components:
For the 2D case, is assumed to be zero, and only the zcomponent of the curl is computed, yielding a scalar output.
in
is expected to be a 2D or 3D image with a 2vector or a 3vector tensor representation, respectively.
However, the image can have more dimensions if they are excluded from processing through process
.
See dip::Gradient
for information on the parameters.
void
dip::Divergence (dip::Image const& in,
dip::Image& out,
dip::FloatArray sigmas = {1.0},
dip::String const& method = S::BEST,
dip::StringArray const& boundaryCondition = {},
dip::BooleanArray process = {},
dip::dfloat truncation = 3)
Computes the divergence of the vector field in
.
Divergence is defined as
with the vector image in
. This concept naturally extends to any number
of dimensions.
in
is expected to have as many dimensions as tensor components. However, the image
can have more dimensions if they are excluded from processing through process
.
See dip::Gradient
for information on the parameters.
void
dip::Hessian (dip::Image const& in,
dip::Image& out,
dip::FloatArray sigmas = {1.0},
dip::String const& method = S::BEST,
dip::StringArray const& boundaryCondition = {},
dip::BooleanArray process = {},
dip::dfloat truncation = 3)
Computes the Hessian of the image, resulting in a symmetric NxN tensor image, if the input was Ndimensional.
The Hessian of input image is given by , with tensor components
Each tensor component corresponds to one of the secondorder derivatives. Note that is a symmetric matrix (order of differentiation does not matter). Duplicate entries are not stored in the symmetric tensor image.
Image dimensions for which process
is false do not participate in the set of dimensions that form the
Hessian matrix. Thus, a 5D image with only two dimensions selected by the process
array will yield a 2by2
Hessian matrix.
By default this function uses Gaussian derivatives in the computation. Set method = "finitediff"
for
finite difference approximations to the gradient. See dip::Derivative
for more information on the other
parameters.
The input image must be scalar.
void
dip::Laplace (dip::Image const& in,
dip::Image& out,
dip::FloatArray sigmas = {1.0},
dip::String const& method = S::BEST,
dip::StringArray const& boundaryCondition = {},
dip::BooleanArray process = {},
dip::dfloat truncation = 3)
Computes the Laplacian of the image, equivalent to dip::Trace( dip::Hessian( in ))
, but more efficient.
The Laplacian of input image is written as , and given by
See dip::Gradient
for information on the parameters.
If method
is “finitediff”, it does not add second order derivatives, but instead computes a convolution
with a 3x3(x3x…) kernel where all elements are 1 and the middle element is (with the number
of image dimensions). That is, the kernel sums to 0. For a 2D image, this translates to the wellknown kernel:
void
dip::Dgg (dip::Image const& in,
dip::Image& out,
dip::FloatArray sigmas = {1.0},
dip::String const& method = S::BEST,
dip::StringArray const& boundaryCondition = {},
dip::BooleanArray process = {},
dip::dfloat truncation = 3)
Computes the second derivative in the gradient direction.
The second derivative in the gradient direction is computed by Raleigh quotient of the Hessian matrix and the gradient vector:
This function is equivalent to:
Image g = dip::Gradient( in, ... ); Image H = dip::Hessian( in, ... ); Image Dgg = dip::Transpose( g ) * H * g; Dgg /= dip::Transpose( g ) * g;
See dip::Derivative
for how derivatives are computed, and the meaning of the parameters. See dip::Gradient
or dip::Hessian
for the meaning of the process
parameter
void
dip::LaplacePlusDgg (dip::Image const& in,
dip::Image& out,
dip::FloatArray sigmas = {1.0},
dip::String const& method = S::BEST,
dip::StringArray const& boundaryCondition = {},
dip::BooleanArray process = {},
dip::dfloat truncation = 3)
Adds the second derivative in the gradient direction to the Laplacian.
This function computes dip::Laplace( in ) + dip::Dgg( in )
, but avoiding computing the second derivatives twice.
The zerocrossings of the result correspond to the edges in the image, just as they do for the individual Laplace and Dgg operators. However, the localization is improved by an order of magnitude with respect to the individual operators.
See dip::Laplace
and dip::Dgg
for more information.
void
dip::LaplaceMinusDgg (dip::Image const& in,
dip::Image& out,
dip::FloatArray sigmas = {1.0},
dip::String const& method = S::BEST,
dip::StringArray const& boundaryCondition = {},
dip::BooleanArray process = {},
dip::dfloat truncation = 3)
Subtracts the second derivative in the gradient direction from the Laplacian.
This function computes dip::Laplace( in )  dip::Dgg( in )
, but avoiding computing the second derivatives twice.
For twodimensional images, this is equivalent to the second order derivative in the direction perpendicular to the gradient direction.
See dip::Laplace
and dip::Dgg
for more information.
void
dip::Sharpen (dip::Image const& in,
dip::Image& out,
dip::dfloat weight = 1.0,
dip::FloatArray sigmas = {1.0},
dip::String const& method = S::BEST,
dip::StringArray const& boundaryCondition = {},
dip::dfloat truncation = 3)
Sharpens in
by subtracting the Laplacian of the image.
The actual operation applied is:
out = in  dip::Laplace( in ) * weight;
See dip::Laplace
and dip::Gradient
for information on the parameters.
void
dip::UnsharpMask (dip::Image const& in,
dip::Image& out,
dip::dfloat weight = 1.0,
dip::FloatArray sigmas = {1.0},
dip::String const& method = S::BEST,
dip::StringArray const& boundaryCondition = {},
dip::dfloat truncation = 3)
Sharpens in
by subtracting the smoothed image.
The actual operation applied is:
out = in * ( 1+weight )  dip::Gauss( in ) * weight;
See dip::Gauss
and dip::Gradient
for information on the parameters.
void
dip::GaborFIR (dip::Image const& in,
dip::Image& out,
dip::FloatArray sigmas,
dip::FloatArray const& frequencies,
dip::StringArray const& boundaryCondition = {},
dip::BooleanArray process = {},
dip::dfloat truncation = 3)
Finite impulse response implementation of the Gabor filter
Convolves the image with an FIR 1D Gabor kernel along each dimension. For each dimension,
provide a value in sigmas
and frequencies
.
The value of sigma determines the amount of local averaging. For smaller values, the result is more
precise spatially, but less selective of frequencies. Dimensions where sigma is 0 or negative are not processed.
Frequencies are in the range [0, 0.5), with 0.5 being the frequency corresponding to a period of 2 pixels.
The output is complexvalued. Typically, the magnitude is the interesting part of the result.
boundaryCondition
indicates how the boundary should be expanded in each dimension. See dip::BoundaryCondition
.
Set process
to false for those dimensions that should not be filtered. This is equivalent to setting
sigmas
to 0 for those dimensions.
This function is relatively slow compared to dip::GaborIIR
, even for small sigmas. Prefer to use the IIR
implementation.
void
dip::GaborIIR (dip::Image const& in,
dip::Image& out,
dip::FloatArray sigmas,
dip::FloatArray const& frequencies,
dip::StringArray const& boundaryCondition = {},
dip::BooleanArray process = {},
dip::IntegerArray filterOrder = {},
dip::dfloat truncation = 3)
Recursive infinite impulse response implementation of the Gabor filter
Convolves the image with an IIR 1D Gabor kernel along each dimension. For each dimension,
provide a value in sigmas
and frequencies
.
The value of sigma determines the amount of local averaging. For smaller values, the result is more
precise spatially, but less selective of frequencies. Dimensions where sigma is 0 or negative are not processed.
Frequencies are in the range [0, 0.5), with 0.5 being the frequency corresponding to a period of 2 pixels.
The output is complexvalued. Typically, the magnitude is the interesting part of the result.
boundaryCondition
indicates how the boundary should be expanded in each dimension. See dip::BoundaryCondition
.
Set process
to false for those dimensions that should not be filtered. This is equivalent to setting
sigmas
to 0 for those dimensions.
The order
parameter is not yet implemented. It is ignored and assumed 0 for each dimension.
void
dip::Gabor2D (dip::Image const& in,
dip::Image& out,
dip::FloatArray sigmas = {5.0,5.0},
dip::dfloat frequency = 0.1,
dip::dfloat direction = dip::pi,
dip::StringArray const& boundaryCondition = {},
dip::dfloat truncation = 3)
2D Gabor filter with direction parameter
Convolves the 2D image with a Gabor kernel. This is a convenience wrapper around dip::GaborIIR
.
The value of sigma determines the amount of local averaging, and can be different for each dimension.
For smaller values, the result is more precise spatially, but less selective of frequencies.
frequency
is in the range [0, 0.5), with 0.5 being the frequency corresponding to a period of 2 pixels.
direction
is the filter direction, in the range [0, 2π].
The output is complexvalued. Typically, the magnitude is the interesting part of the result.
boundaryCondition
indicates how the boundary should be expanded in each dimension. See dip::BoundaryCondition
.
To use cartesian frequency coordinates, see dip::GaborIIR
.
void
dip::LogGaborFilterBank (dip::Image const& in,
dip::Image& out,
dip::FloatArray const& wavelengths = {3.0,6.0,12.0,24.0},
dip::dfloat bandwidth = 0.75,
dip::uint nOrientations = 6,
dip::String const& inRepresentation = S::SPATIAL,
dip::String const& outRepresentation = S::SPATIAL)
Applies a logGabor filter bank
A logGabor filter is a Gabor filter computed on the logarithm of the frequency, leading to a shorter tail of the Gaussian, in the frequency domain, towards the lower frequencies. The origin (DC component) is thus never included in the filter. This gives it better scale selection properties than the traditional Gabor filter.
This function generates a filter bank with wavelengths.size()
times nOrientations
filters. The width of
the filters in the angular axis is determined by the number of orientations used, and their locations are
always equally distributed over π radian, starting at 0. The radial location (scales) of the filters
is determined by wavelengths
(in pixels), which determines the center for each scale filter. The widths of
the filters in this direction are determined by the bandwidth
parameter; the default value of 0.75 corresponds
approximately to one octave, 0.55 to two octaves, and 0.41 to three octaves.
wavelengths.size()
and nOrientations
must be at least 1.
If nOrientations
is 1, no orientation filtering is applied, the filters become purely real. These filters
can be defined for images of any dimensionality. For more than one orientation, the filters are complexvalued
in the spatial domain, and can only be created for 2D images. See dip::MonogenicSignal
for a generalization
to arbitrary dimensionality.
If in
is not forged, its sizes will be used to generate the filters, which will be returned. Thus, this is
identical to (but slightly cheaper than) using a delta pulse image as input.
The filters are always generated in the frequency domain. If outRepresentation
is "spatial"
, the inverse
Fourier transform will be applied to bring the result back to the spatial domain. Otherwise, it should be
"frequency"
, and no inverse transform will be applied. Likewise, inRepresentation
specifies whether in
has already been converted to the frequency domain or not.
Out will be a tensor image with wavelengths.size()
tensor rows and nFrequencyScales
tensor columns.
The data type will be either singleprecision float or singleprecision complex, depending on the selected
parameters.
void
dip::NormalizedConvolution (dip::Image const& in,
dip::Image const& mask,
dip::Image& out,
dip::FloatArray const& sigmas = {1.0},
dip::String const& method = S::BEST,
dip::StringArray const& boundaryCondition = {S::ADD_ZEROS},
dip::dfloat truncation = 3)
Computes the normalized convolution with a Gaussian kernel: a Gaussian convolution for missing or uncertain data.
The normalized convolution is a convolution that handles missing or uncertain data. mask
is an image, expected
to be in the range [0,1], that indicates the confidence in each of the values of in
. Missing values are indicated
by setting the corresponding value in mask
to 0.
The normalized convolution is then Convolution( in * mask ) / Convolution( mask )
.
This function applies convolutions with a Gaussian kernel, using dip::Gauss
. See that function for the meaning
of the parameters. boundaryCondition
defaults to "add zeros"
, the normalized convolution then takes pixels
outside of the image domain as missing values.
void
dip::NormalizedDifferentialConvolution (dip::Image const& in,
dip::Image const& mask,
dip::Image& out,
dip::uint dimension = 0,
dip::FloatArray const& sigmas = {1.0},
dip::String const& method = S::BEST,
dip::StringArray const& boundaryCondition = {S::ADD_ZEROS},
dip::dfloat truncation = 3)
Computes the normalized differential convolution with a Gaussian kernel: a derivative operator for missing or uncertain data.
The normalized convolution is a convolution that handles missing or uncertain data. mask
is an image, expected
to be in the range [0,1], that indicates the confidence in each of the values of in
. Missing values are indicated
by setting the corresponding value in mask
to 0.
The normalized differential convolution is defined here as the derivative of the normalized convolution with a Gaussian kernel:
is the convolution operator, is in
, is mask
, and is the Gaussian kernel
The derivative is computed along dimension
.
This function uses dip::Gauss
. See that function for the meaning of the parameters. boundaryCondition
defaults
to "add zeros"
, the normalized convolution then takes pixels outside of the image domain as missing values.
void
dip::MeanShiftVector (dip::Image const& in,
dip::Image& out,
dip::FloatArray sigmas = {1.0},
dip::String const& method = S::BEST,
dip::StringArray const& boundaryCondition = {},
dip::dfloat truncation = 3)
Computes the mean shift vector for each pixel in the image
The output image is a vector image, indicating the step to take to move the window center to its center of
mass. Repeatedly following the vector will lead to a local maximum of the image in
. in
must be scalar
and realvalued.
The mean shift at a given location is then given by
where is the image, is a windowing function, and indicates convolution.
We use a Gaussian window with sizes given by sigmas
. A Gaussian window causes slower convergence than a
uniform window, but yields a smooth trajectory and more precise results (according to Comaniciu
and Meer, 2002). It also allows us to rewrite the above (with the Gaussian window with
parameter ) as
Thus, we can write this filter as dip::Gradient(in, sigmas) / dip::Gauss(in, sigmas) * sigmas * sigmas
.
See dip::Derivative
for more information on the parameters. Do not use method = "finitediff"
,
as it will lead to nonsensical results.