Notes

For a single dimension array x , dct(x, norm='ortho') is equal to MATLAB dct(x) .

For norm="ortho" both the dct and idct are scaled by the same overall factor in both directions. By default, the transform is also orthogonalized which for types 1, 2 and 3 means the transform definition is modified to give orthogonality of the DCT matrix (see below).

For norm="backward" , there is no scaling on dct and the idct is scaled by 1/N where N is the "logical" size of the DCT. For norm="forward" the 1/N normalization is applied to the forward dct instead and the idct is unnormalized.

There are, theoretically, 8 types of the DCT, only the first 4 types are implemented in SciPy.'The' DCT generally refers to DCT type 2, and 'the' Inverse DCT generally refers to DCT type 3.

Type I

There are several definitions of the DCT-I; we use the following (for norm="backward" )

$$y_k = x_0 + (-1)^k x_{N-1} + 2 \sum_{n=1}^{N-2} x_n \cos\left( \frac{\pi k n}{N-1} \right)$$

If orthogonalize=True , x[0] and x[N-1] are multiplied by a scaling factor of $\sqrt{2}$ , and y[0] and y[N-1] are divided by $\sqrt{2}$ . When combined with norm="ortho" , this makes the corresponding matrix of coefficients orthonormal ( O @ O.T = np.eye(N) ).

Type II

There are several definitions of the DCT-II; we use the following (for norm="backward" )

$$y_k = 2 \sum_{n=0}^{N-1} x_n \cos\left(\frac{\pi k(2n+1)}{2N} \right)$$

If orthogonalize=True , y[0] is divided by $\sqrt{2}$ which, when combined with norm="ortho" , makes the corresponding matrix of coefficients orthonormal ( O @ O.T = np.eye(N) ).

Type III

There are several definitions, we use the following (for norm="backward" )

$$y_k = x_0 + 2 \sum_{n=1}^{N-1} x_n \cos\left(\frac{\pi(2k+1)n}{2N}\right)$$

If orthogonalize=True , x[0] terms are multiplied by $\sqrt{2}$ which, when combined with norm="ortho" , makes the corresponding matrix of coefficients orthonormal ( O @ O.T = np.eye(N) ).

The (unnormalized) DCT-III is the inverse of the (unnormalized) DCT-II, up to a factor :None:None:`2N`. The orthonormalized DCT-III is exactly the inverse of the orthonormalized DCT-II.

Type IV

There are several definitions of the DCT-IV; we use the following (for norm="backward" )

$$y_k = 2 \sum_{n=0}^{N-1} x_n \cos\left(\frac{\pi(2k+1)(2n+1)}{4N} \right)$$

orthogonalize has no effect here, as the DCT-IV matrix is already orthogonal up to a scale factor of 2N .

Parameters

x : array_like

The input array.

type : {1, 2, 3, 4}, optional

Type of the DCT (see Notes). Default type is 2.

n : int, optional

Length of the transform. If n < x.shape[axis] , x is truncated. If n > x.shape[axis] , x is zero-padded. The default results in n = x.shape[axis] .

axis : int, optional

Axis along which the dct is computed; the default is over the last axis (i.e., axis=-1 ).

norm : {"backward", "ortho", "forward"}, optional

Normalization mode (see Notes). Default is "backward".

overwrite_x : bool, optional

If True, the contents of x can be destroyed; the default is False.

workers : int, optional

Maximum number of workers to use for parallel computation. If negative, the value wraps around from os.cpu_count() . See ~scipy.fft.fft for more details.

orthogonalize : bool, optional

Whether to use the orthogonalized DCT variant (see Notes). Defaults to True when norm=="ortho" and False otherwise.

versionadded

Returns

y : ndarray of real

The transformed input array.

Return the Discrete Cosine Transform of arbitrary type sequence x.

See Also

idct

Inverse DCT

Examples

The Type 1 DCT is equivalent to the FFT (though faster) for real, even-symmetrical inputs. The output is also real and even-symmetrical. Half of the FFT input is used to generate half of the FFT output:

>>> from scipy.fft import fft, dct
... fft(np.array([4., 3., 5., 10., 5., 3.])).real
array([ 30.,  -8.,   6.,  -2.,   6.,  -8.])

>>> dct(np.array([4., 3., 5., 10.]), 1)
array([ 30.,  -8.,   6.,  -2.])

See :

Back References

The following pages refer to to this document either explicitly or contain code examples using this.

scipy.fft._realtransforms.dct scipy.fft._realtransforms.dctn scipy.fft._realtransforms.idct

Local connectivity graph

Hover to see nodes names; edges to Self not shown, Caped at 50 nodes.

Using a canvas is more power efficient and can get hundred of nodes ; but does not allow hyperlinks; , arrows or text (beyond on hover)

SVG is more flexible but power hungry; and does not scale well to 50 + nodes.

All aboves nodes referred to, (or are referred from) current nodes; Edges from Self to other have been omitted (or all nodes would be connected to the central node "self" which is not useful). Nodes are colored by the library they belong to, and scaled with the number of references pointing them

---

GitHub : /scipy/fft/_realtransforms.py#264
type: <class 'uarray._Function'>
Commit: