Transforms module

The Fourier and other transforms.

Contents

Classes

template<typename T>
class dip::DFT
An object that encapsulates the Discrete Fourier Transform (DFT).

Functions

auto dip::GetOptimalDFTSize(size_t size0) -> size_t
Returns a size equal or larger to size0 that is efficient for our DFT implementation.
void dip::FourierTransform(dip::Image const& in, dip::Image& out, dip::StringSet const& options = {}, dip::BooleanArray process = {})
Computes the forward and inverse Fourier Transform
auto dip::OptimalFourierTransformSize(dip::uint size) -> dip::uint
Returns the next higher multiple of {2, 3, 5}. The largest value that can be returned is 2125764000 (smaller than 231-1, the largest possible value of an int on most platforms).
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 à-trous wavelet decomposition).

Variables

size_t const dip::maximumDFTSize constexpr
The largest size supported by the DFT (both the internal code and FFTW use int for sizes).

Function documentation

size_t dip::GetOptimalDFTSize(size_t size0)

Returns a size equal or larger to size0 that is efficient for our DFT implementation.

Returns 0 if size0 is too large for our DFT implementation.

Prefer to use dip::OptimalFourierTransformSize in your applications, it will throw an error if the transform size is too large.

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 multi-dimensional transform).

options is an 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 real-valued result. Only to be used together with “inverse”.
  • “fast”: pads the input to a “nice” size, multiple of 2, 3 and 5, which can be processed faster. Note that “fast” causes the output to be interpolated. This is not always a problem when computing convolutions or correlations, but will introduce e.g. edge effects in the result of the convolution.
  • “corner”: sets the origin to the top-left 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 will be padded. If “corner” is given, the padding is to the right. Otherwise it is split evenly on both sides, in such a way that the origin remains in the middle pixel. For the forward transform, the padding applied is the “zero order” boundary condition (see dip::BoundaryCondition). Its effect is similar to padding with zeros, but with reduced edge effects. For the inverse transform, padding is with zeros (“add zeros” boundary condition). However, the combination of “fast”, “corner” and “inverse” is not allowed, since padding in that case is non-trivial.

dip::uint dip::OptimalFourierTransformSize(dip::uint size)

Returns the next higher multiple of {2, 3, 5}. The largest value that can be returned is 2125764000 (smaller than 231-1, the largest possible value of an int on most platforms).

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 multi-dimensional generalization of the Hilbert transform, and identical to it for one-dimensional images. It is computed through the Fourier domain by

\[ R_j f = \mathcal{F}^{-1} \left\{ -i\frac{x_j}{|x|}(\mathcal{F}f) \right\} \; , \]

where \(f\) is the input image and \(x\) 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 back-and-forth 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 à-trous wavelet decomposition).

For an n-dimensional 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 if 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 == dip.Sum( swt, {}, process ).Squeeze();

Variable documentation

size_t const dip::maximumDFTSize constexpr

The largest size supported by the DFT (both the internal code and FFTW use int for sizes).