Transforms module #include "diplib/transform.h"
The Fourier and other transforms.
Modules
 module Lowlevel transform support
 Lowlevel functionality for computing the Discrete Fourier Transform.
Functions

void dip::
FourierTransform (dip::Image const& in, dip::Image& out, dip::StringSet const& options = {}, dip::BooleanArray process = {})  Computes the forward and inverse Fourier Transform

void dip::
InverseFourierTransform (dip::Image const& in, dip::Image& out, dip::StringSet options = {}, dip::BooleanArray process = {})  Inverse Fourier Transform. Convenience function that calls
dip::FourierTransform
after adding “inverse” tooptions
. 
auto dip::
OptimalFourierTransformSize (dip::uint size, dip::String const& which = S::LARGER, dip::String const& purpose = S::REAL) > dip::uint  Returns the next larger (or smaller) multiple of small integers. An image of this size is more efficient for FFT computations.

void dip::
RieszTransform (dip::Image const& in, dip::Image& out, dip::String const& inRepresentation = S::SPATIAL, dip::String const& outRepresentation = S::SPATIAL, dip::BooleanArray process = {})  Computes the Riesz transform of a scalar image.

void dip::
StationaryWaveletTransform (dip::Image const& in, dip::Image& out, dip::uint nLevels = 4, dip::StringArray const& boundaryCondition = {}, dip::BooleanArray const& process = {})  Computes a stationary wavelet transform (also called àtrous wavelet decomposition).

void dip::
HaarWaveletTransform (dip::Image const& in, dip::Image& out, dip::uint nLevels = 4, dip::String const& direction = S::FORWARD, dip::BooleanArray process = {})  Computes the Haar wavelet transform or its inverse.
Function documentation
void
dip::FourierTransform (dip::Image const& in,
dip::Image& out,
dip::StringSet const& options = {},
dip::BooleanArray process = {})
Computes the forward and inverse Fourier Transform
The Fourier transform as implemented here places the origin (frequency 0) in the middle of
the image. If the image has N
pixels along a dimension, then the origin will be at pixel N/2
along that dimension, where N/2
is the integer division, and hence truncates the result for
odd values of N
. For example, an image of 256 pixels wide will have the origin at pixel 128
(right of the center), whereas an image of 255 pixels will have the origin at pixel 127
(dead in the middle). The same is true for the spatial domain, which is only obvious when
computing the Fourier transform of a convolution kernel.
As it is commonly defined, the Fourier transform is not normalized, and the inverse transform
is normalized by 1/size
for each dimension. This normalization is necessary for the sequence of
forward and inverse transform to be idempotent. However, it is possible to change where the
normalization is applied. For example, DIPlib 2 used identical
normalization for each of the two transforms. The advantage of using the common
definition without normalization in the forward transform is that it is straightforward to
transform an image and a convolution kernel, multiply them, and apply the inverse transform, as
an efficient way to compute the convolution. With any other normalization, this process would
require an extra multiplication by a constant to undo the normalization in the forward transform
of the convolution kernel.
This function will compute the Fourier Transform along the dimensions indicated by process
. If
process
is an empty array, all dimensions will be processed (normal multidimensional transform).
options
is a set of strings that indicate how the transform is applied:
 “inverse”: compute the inverse transform; not providing this string causes the the forward transform to be computed.
 “real”: assumes that the (complex) input is conjugate symmetric, and returns a realvalued result. Can only be used together with “inverse”.
 “fast”: pads the input to a “nice” size (multiple of small integers), which can be processed faster.
 “corner”: sets the origin to the topleft corner of the image (both in the spatial and the frequency domain). This yields a standard DFT (Discrete Fourier Transform).
 “symmetric”: the normalization is made symmetric, where both forward and inverse transforms
are normalized by the same amount. Each transform is multiplied by
1/sqrt(size)
for each dimension. This makes the transform identical to how it was in DIPlib 2.
For tensor images, each plane is transformed independently.
With the “fast” mode, the input might be padded. This padding causes the output to be interpolated. This is not always a problem when computing convolutions or correlations, but could introduce e.g. edge effects in the result of the convolution. The normalization applied relates to the output sizes, not the input sizes. Padding is applied as follows:
 Forward transform: If
"corner"
is given, the padding is to the right of each image line. Otherwise it is split evenly on both sides, in such a way that the origin remains in the middle pixel. Added pixels replicate the border pixel (the “zero order” boundary condition, seedip::BoundaryCondition
). Its effect is similar to padding with zeros, but with reduced edge effects.  Inverse transform: Padding is with zeros (“add zeros” boundary condition), and naturally add higherfrequency
components. If
"corner"
is given, the padding is thus in the middle of each image line. Otherwise, the padding is symmetrically on both ends of the line.
dip::uint
dip::OptimalFourierTransformSize (dip::uint size,
dip::String const& which = S::LARGER,
dip::String const& purpose = S::REAL)
Returns the next larger (or smaller) multiple of small integers. An image of this size is more efficient for FFT computations.
Pad an image with zeros to the next larger size or crop the image to the next smaller size to improve FFT performance.
The largest value that can be returned depends on the FFT implementation used. When using FFTW,
the largest FFT that can be computed is 2^{31}1 (the maximum value for an int
). Otherwise any
dip::uint
can be returned. If the result is larger than the maximum FFT size, an exception
will be thrown.
By default, which
is "larger"
, in which case it returns the next larger value. Set it
to "smaller"
to obtain the next smaller value instead.
purpose
is either "real"
(the default) or "complex"
, and should be set according to the
type of transform that will be computed. This determines what are considered “small primes”.
For PocketFFT this changes depending on whether the transform is complextocomplex
(complexvalued transform), or is realtocomplex or complextoreal (realvalued transform).
See dip::GetOptimalDFTSize
.
void
dip::RieszTransform (dip::Image const& in,
dip::Image& out,
dip::String const& inRepresentation = S::SPATIAL,
dip::String const& outRepresentation = S::SPATIAL,
dip::BooleanArray process = {})
Computes the Riesz transform of a scalar image.
The Riesz transform is the multidimensional generalization of the Hilbert transform, and identical to it for onedimensional images. It is computed through the Fourier domain by
where is the input image and is the coordinate vector.
out
is a vector image with one element per image dimension. If process
is given, it specifies which
dimensions to include in the output vector image. in
must be scalar.
inRepresentation
and outRepresentation
can be "spatial"
or "frequency"
, and indicate in which domain
the input image is, and in which domain the output image should be.
If inRepresentation
is "frequency"
, the input image must already be in the frequency domain, and will not
be transformed again. Likewise, if outRepresentation
is "frequency"
, the output image will not be transformed
to the spatial domain. Use these flags to prevent redundant backandforth transformations if other processing
in the frequency domain is necessary.
void
dip::StationaryWaveletTransform (dip::Image const& in,
dip::Image& out,
dip::uint nLevels = 4,
dip::StringArray const& boundaryCondition = {},
dip::BooleanArray const& process = {})
Computes a stationary wavelet transform (also called àtrous wavelet decomposition).
For an ndimensional input image, creates an (n+1)dimensional output image where each
slice corresponds to one level of the wavelet transform. The first slice is the lowest level
(finest detail), and subsequent slices correspond to increasingly coarser levels. The last
slice corresponds to the residue. There are nLevels + 1
slices in total.
The filter used to smooth the image for the first level is [1/16, 1/4, 3/8, 1/4, 1/16]
,
applied to each dimension in sequence through dip::SeparableConvolution
.
For subsequent levels, zeros are inserted into this filter.
boundaryCondition
is passed to dip::SeparableConvolution
to determine how to extend the
input image past its boundary. process
can be used to exclude some dimensions from the
filtering.
in
can have any number of dimensions, any number of tensor elements, and any data type.
out
will have the smallest signed data type that can hold all values of in
(see
dip::DataType::SuggestSigned
). Note that the first nLevels
slices will contain negative
values, even if in
is purely positive, as these levels are the difference between two
differently smoothed images.
Summing the output image along its last dimension will yield the input image:
dip::Image img = ...; dip::Image swt = StationaryWaveletTransform( img ); dip::BooleanArray process( swt.Dimensionality(), false ); process.back() = true; img == Sum( swt, {}, process ).Squeeze();
void
dip::HaarWaveletTransform (dip::Image const& in,
dip::Image& out,
dip::uint nLevels = 4,
dip::String const& direction = S::FORWARD,
dip::BooleanArray process = {})
Computes the Haar wavelet transform or its inverse.
Recursively splits the image into lowfrequency and highfrequency components. Each step splits an ndimensional image into 2^{n} smaller blocks, the one in the topleft corner containing the lowfrequency components. The lowfrequency block is the one recursively split in the next step. The output image has the same sizes as the input image, but is of a floatingpoint type.
However, the input must have sizes multiple of 2^{nLevels}. The image will be padded with zeros for the forward transform if this is not the case. For the inverse transform, an exception will the thrown if the sizes are not as expected.
direction
can be "forward"
or "inverse"
. Applying a forward transform to any image, and an inverse transform
to the result, will yield an image identical to the input image, up to rounding errors, and potentially with some
padding to the right and bottom.
process
can be used to exclude some dimensions from the filtering.