429 lines
15 KiB
Python
429 lines
15 KiB
Python
# Copyright 2021 The JAX Authors.
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#
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# Licensed under the Apache License, Version 2.0 (the "License");
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# you may not use this file except in compliance with the License.
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# You may obtain a copy of the License at
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#
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# https://www.apache.org/licenses/LICENSE-2.0
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#
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# Unless required by applicable law or agreed to in writing, software
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# distributed under the License is distributed on an "AS IS" BASIS,
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# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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# See the License for the specific language governing permissions and
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# limitations under the License.
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from __future__ import annotations
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from collections.abc import Sequence
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from functools import partial
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import math
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from jax import lax
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import jax.numpy as jnp
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from jax._src.util import canonicalize_axis
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from jax._src.numpy.util import promote_dtypes_complex, promote_dtypes_inexact
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from jax._src.typing import Array
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def _W4(N: int, k: Array) -> Array:
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N_arr, k = promote_dtypes_complex(N, k)
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return jnp.exp(-.5j * jnp.pi * k / N_arr)
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def _dct_interleave(x: Array, axis: int) -> Array:
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v0 = lax.slice_in_dim(x, None, None, 2, axis)
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v1 = lax.rev(lax.slice_in_dim(x, 1, None, 2, axis), (axis,))
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return lax.concatenate([v0, v1], axis)
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def _dct_ortho_norm(out: Array, axis: int) -> Array:
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factor = lax.concatenate([lax.full((1,), 4, out.dtype), lax.full((out.shape[axis] - 1,), 2, out.dtype)], 0)
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factor = lax.expand_dims(factor, [a for a in range(out.ndim) if a != axis])
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return out / lax.sqrt(factor * out.shape[axis])
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# Implementation based on
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# John Makhoul: A Fast Cosine Transform in One and Two Dimensions (1980)
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def dct(x: Array, type: int = 2, n: int | None = None,
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axis: int = -1, norm: str | None = None) -> Array:
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"""Computes the discrete cosine transform of the input
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JAX implementation of :func:`scipy.fft.dct`.
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Args:
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x: array
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type: integer, default = 2. Currently only type 2 is supported.
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n: integer, default = x.shape[axis]. The length of the transform.
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If larger than ``x.shape[axis]``, the input will be zero-padded, if
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smaller, the input will be truncated.
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axis: integer, default=-1. The axis along which the dct will be performed.
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norm: string. The normalization mode: one of ``[None, "backward", "ortho"]``.
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The default is ``None``, which is equivalent to ``"backward"``.
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Returns:
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array containing the discrete cosine transform of x
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See Also:
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- :func:`jax.scipy.fft.dctn`: multidimensional DCT
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- :func:`jax.scipy.fft.idct`: inverse DCT
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- :func:`jax.scipy.fft.idctn`: multidimensional inverse DCT
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Examples:
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>>> x = jax.random.normal(jax.random.key(0), (3, 3))
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>>> with jnp.printoptions(precision=2, suppress=True):
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... print(jax.scipy.fft.dct(x))
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[[ 6.43 3.56 -2.86]
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[-1.75 1.55 -1.4 ]
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[ 1.33 -2.01 -0.82]]
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When ``n`` smaller than ``x.shape[axis]``
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>>> with jnp.printoptions(precision=2, suppress=True):
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... print(jax.scipy.fft.dct(x, n=2))
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[[ 7.3 -0.57]
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[ 0.19 -0.36]
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[-0. -1.4 ]]
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When ``n`` smaller than ``x.shape[axis]`` and ``axis=0``
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>>> with jnp.printoptions(precision=2, suppress=True):
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... print(jax.scipy.fft.dct(x, n=2, axis=0))
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[[ 3.09 4.4 -2.81]
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[ 2.41 2.62 0.76]]
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When ``n`` larger than ``x.shape[axis]`` and ``axis=1``
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>>> with jnp.printoptions(precision=2, suppress=True):
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... print(jax.scipy.fft.dct(x, n=4, axis=1))
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[[ 6.43 4.88 0.04 -3.3 ]
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[-1.75 0.73 1.01 -2.18]
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[ 1.33 -1.05 -2.34 -0.07]]
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"""
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if type != 2:
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raise NotImplementedError('Only DCT type 2 is implemented.')
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if norm is not None and norm not in ['backward', 'ortho']:
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raise ValueError(f"jax.scipy.fft.dct: {norm=!r} is not implemented")
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axis = canonicalize_axis(axis, x.ndim)
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if n is not None:
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x = lax.pad(x, jnp.array(0, x.dtype),
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[(0, n - x.shape[axis] if a == axis else 0, 0)
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for a in range(x.ndim)])
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N = x.shape[axis]
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v = _dct_interleave(x, axis)
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V = jnp.fft.fft(v, axis=axis)
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k = lax.expand_dims(jnp.arange(N, dtype=V.real.dtype), [a for a in range(x.ndim) if a != axis])
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out = V * _W4(N, k)
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out = 2 * out.real
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if norm == 'ortho':
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out = _dct_ortho_norm(out, axis)
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return out
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def _dct2(x: Array, axes: Sequence[int], norm: str | None) -> Array:
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axis1, axis2 = map(partial(canonicalize_axis, num_dims=x.ndim), axes)
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N1, N2 = x.shape[axis1], x.shape[axis2]
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v = _dct_interleave(_dct_interleave(x, axis1), axis2)
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V = jnp.fft.fftn(v, axes=axes)
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k1 = lax.expand_dims(jnp.arange(N1, dtype=V.dtype),
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[a for a in range(x.ndim) if a != axis1])
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k2 = lax.expand_dims(jnp.arange(N2, dtype=V.dtype),
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[a for a in range(x.ndim) if a != axis2])
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out = _W4(N1, k1) * (_W4(N2, k2) * V + _W4(N2, -k2) * jnp.roll(jnp.flip(V, axis=axis2), shift=1, axis=axis2))
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out = 2 * out.real
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if norm == 'ortho':
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return _dct_ortho_norm(_dct_ortho_norm(out, axis1), axis2)
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return out
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def dctn(x: Array, type: int = 2,
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s: Sequence[int] | None=None,
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axes: Sequence[int] | None = None,
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norm: str | None = None) -> Array:
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"""Computes the multidimensional discrete cosine transform of the input
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JAX implementation of :func:`scipy.fft.dctn`.
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Args:
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x: array
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type: integer, default = 2. Currently only type 2 is supported.
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s: integer or sequence of integers. Specifies the shape of the result. If not
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specified, it will default to the shape of ``x`` along the specified ``axes``.
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axes: integer or sequence of integers. Specifies the axes along which the
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transform will be computed.
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norm: string. The normalization mode: one of ``[None, "backward", "ortho"]``.
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The default is ``None``, which is equivalent to ``"backward"``.
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Returns:
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array containing the discrete cosine transform of x
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See Also:
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- :func:`jax.scipy.fft.dct`: one-dimensional DCT
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- :func:`jax.scipy.fft.idct`: one-dimensional inverse DCT
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- :func:`jax.scipy.fft.idctn`: multidimensional inverse DCT
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Examples:
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``jax.scipy.fft.dctn`` computes the transform along both the axes by default
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when ``axes`` argument is ``None``.
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>>> x = jax.random.normal(jax.random.key(0), (3, 3))
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>>> with jnp.printoptions(precision=2, suppress=True):
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... print(jax.scipy.fft.dctn(x))
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[[ 12.01 6.2 -10.17]
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[ 8.84 9.65 -3.54]
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[ 11.25 -1.54 -0.88]]
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When ``s=[2]``, dimension of the transform along ``axis 0`` will be ``2``
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and dimension along ``axis 1`` will be same as that of input.
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>>> with jnp.printoptions(precision=2, suppress=True):
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... print(jax.scipy.fft.dctn(x, s=[2]))
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[[ 9.36 10.22 -8.53]
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[11.57 2.85 -2.06]]
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When ``s=[2]`` and ``axes=[1]``, dimension of the transform along ``axis 1`` will
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be ``2`` and dimension along ``axis 0`` will be same as that of input.
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Also when ``axes=[1]``, transform will be computed only along ``axis 1``.
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>>> with jnp.printoptions(precision=2, suppress=True):
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... print(jax.scipy.fft.dctn(x, s=[2], axes=[1]))
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[[ 7.3 -0.57]
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[ 0.19 -0.36]
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[-0. -1.4 ]]
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When ``s=[2, 4]``, shape of the transform will be ``(2, 4)``.
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>>> with jnp.printoptions(precision=2, suppress=True):
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... print(jax.scipy.fft.dctn(x, s=[2, 4]))
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[[ 9.36 11.23 2.12 -10.97]
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[ 11.57 5.86 -1.37 -1.58]]
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"""
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if type != 2:
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raise NotImplementedError('Only DCT type 2 is implemented.')
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if norm is not None and norm not in ['backward', 'ortho']:
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raise ValueError(f"jax.scipy.fft.dctn: {norm=!r} is not implemented")
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if axes is None:
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axes = range(x.ndim)
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if len(axes) == 1:
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return dct(x, n=s[0] if s is not None else None, axis=axes[0], norm=norm)
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if s is not None:
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ns = dict(zip(axes, s))
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pads = [(0, ns[a] - x.shape[a] if a in ns else 0, 0) for a in range(x.ndim)]
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x = lax.pad(x, jnp.array(0, x.dtype), pads)
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if len(axes) == 2:
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return _dct2(x, axes=axes, norm=norm)
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# compose high-D DCTs from 2D and 1D DCTs:
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for axes_block in [axes[i:i+2] for i in range(0, len(axes), 2)]:
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x = dctn(x, axes=axes_block, norm=norm)
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return x
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def idct(x: Array, type: int = 2, n: int | None = None,
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axis: int = -1, norm: str | None = None) -> Array:
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"""Computes the inverse discrete cosine transform of the input
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JAX implementation of :func:`scipy.fft.idct`.
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Args:
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x: array
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type: integer, default = 2. Currently only type 2 is supported.
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n: integer, default = x.shape[axis]. The length of the transform.
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If larger than ``x.shape[axis]``, the input will be zero-padded, if
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smaller, the input will be truncated.
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axis: integer, default=-1. The axis along which the dct will be performed.
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norm: string. The normalization mode: one of ``[None, "backward", "ortho"]``.
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The default is ``None``, which is equivalent to ``"backward"``.
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Returns:
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array containing the inverse discrete cosine transform of x
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See Also:
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- :func:`jax.scipy.fft.dct`: DCT
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- :func:`jax.scipy.fft.dctn`: multidimensional DCT
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- :func:`jax.scipy.fft.idctn`: multidimensional inverse DCT
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Examples:
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>>> x = jax.random.normal(jax.random.key(0), (3, 3))
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>>> with jnp.printoptions(precision=2, suppress=True):
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... print(jax.scipy.fft.idct(x))
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[[ 0.78 0.41 -0.39]
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[-0.12 0.31 -0.23]
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[ 0.17 -0.3 -0.11]]
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When ``n`` smaller than ``x.shape[axis]``
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>>> with jnp.printoptions(precision=2, suppress=True):
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... print(jax.scipy.fft.idct(x, n=2))
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[[ 1.12 -0.31]
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[ 0.04 -0.08]
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[ 0.05 -0.3 ]]
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When ``n`` smaller than ``x.shape[axis]`` and ``axis=0``
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>>> with jnp.printoptions(precision=2, suppress=True):
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... print(jax.scipy.fft.idct(x, n=2, axis=0))
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[[ 0.38 0.57 -0.45]
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[ 0.43 0.44 0.24]]
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When ``n`` larger than ``x.shape[axis]`` and ``axis=0``
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>>> with jnp.printoptions(precision=2, suppress=True):
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... print(jax.scipy.fft.idct(x, n=4, axis=0))
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[[ 0.1 0.38 -0.16]
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[ 0.28 0.18 -0.26]
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[ 0.3 0.15 -0.08]
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[ 0.13 0.3 0.29]]
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``jax.scipy.fft.idct`` can be used to reconstruct ``x`` from the result
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of ``jax.scipy.fft.dct``
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>>> x_dct = jax.scipy.fft.dct(x)
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>>> jnp.allclose(x, jax.scipy.fft.idct(x_dct))
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Array(True, dtype=bool)
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"""
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if type != 2:
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raise NotImplementedError('Only DCT type 2 is implemented.')
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if norm is not None and norm not in ['backward', 'ortho']:
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raise ValueError(f"jax.scipy.fft.idct: {norm=!r} is not implemented")
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axis = canonicalize_axis(axis, x.ndim)
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if n is not None:
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x = lax.pad(x, jnp.array(0, x.dtype),
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[(0, n - x.shape[axis] if a == axis else 0, 0)
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for a in range(x.ndim)])
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N = x.shape[axis]
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x, = promote_dtypes_inexact(x)
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if norm is None or norm == 'backward':
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x = _dct_ortho_norm(x, axis)
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x = _dct_ortho_norm(x, axis)
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k = lax.expand_dims(jnp.arange(N, dtype=x.dtype), [a for a in range(x.ndim) if a != axis])
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# everything is complex from here...
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w4 = _W4(N,k)
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x = x.astype(w4.dtype)
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x = x / (_W4(N, k))
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x = x * 2 * N
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x = jnp.fft.ifft(x, axis=axis)
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# convert back to reals..
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out = _dct_deinterleave(x.real, axis)
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return out
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def idctn(x: Array, type: int = 2,
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s: Sequence[int] | None=None,
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axes: Sequence[int] | None = None,
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norm: str | None = None) -> Array:
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"""Computes the multidimensional inverse discrete cosine transform of the input
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JAX implementation of :func:`scipy.fft.idctn`.
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Args:
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x: array
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type: integer, default = 2. Currently only type 2 is supported.
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s: integer or sequence of integers. Specifies the shape of the result. If not
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specified, it will default to the shape of ``x`` along the specified ``axes``.
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axes: integer or sequence of integers. Specifies the axes along which the
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transform will be computed.
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norm: string. The normalization mode: one of ``[None, "backward", "ortho"]``.
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The default is ``None``, which is equivalent to ``"backward"``.
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Returns:
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array containing the inverse discrete cosine transform of x
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See Also:
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- :func:`jax.scipy.fft.dct`: one-dimensional DCT
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- :func:`jax.scipy.fft.dctn`: multidimensional DCT
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- :func:`jax.scipy.fft.idct`: one-dimensional inverse DCT
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Examples:
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``jax.scipy.fft.idctn`` computes the transform along both the axes by default
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when ``axes`` argument is ``None``.
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>>> x = jax.random.normal(jax.random.key(0), (3, 3))
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>>> with jnp.printoptions(precision=2, suppress=True):
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... print(jax.scipy.fft.idctn(x))
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[[ 0.12 0.11 -0.15]
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[ 0.07 0.17 -0.03]
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[ 0.19 -0.07 -0.02]]
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When ``s=[2]``, dimension of the transform along ``axis 0`` will be ``2``
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and dimension along ``axis 1`` will be the same as that of input.
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>>> with jnp.printoptions(precision=2, suppress=True):
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... print(jax.scipy.fft.idctn(x, s=[2]))
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[[ 0.15 0.21 -0.18]
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[ 0.24 -0.01 -0.02]]
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When ``s=[2]`` and ``axes=[1]``, dimension of the transform along ``axis 1`` will
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be ``2`` and dimension along ``axis 0`` will be same as that of input.
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Also when ``axes=[1]``, transform will be computed only along ``axis 1``.
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>>> with jnp.printoptions(precision=2, suppress=True):
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... print(jax.scipy.fft.idctn(x, s=[2], axes=[1]))
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[[ 1.12 -0.31]
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[ 0.04 -0.08]
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[ 0.05 -0.3 ]]
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When ``s=[2, 4]``, shape of the transform will be ``(2, 4)``
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>>> with jnp.printoptions(precision=2, suppress=True):
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... print(jax.scipy.fft.idctn(x, s=[2, 4]))
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[[ 0.1 0.18 0.07 -0.16]
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[ 0.2 0.06 -0.03 -0.01]]
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``jax.scipy.fft.idctn`` can be used to reconstruct ``x`` from the result
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of ``jax.scipy.fft.dctn``
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>>> x_dctn = jax.scipy.fft.dctn(x)
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>>> jnp.allclose(x, jax.scipy.fft.idctn(x_dctn))
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Array(True, dtype=bool)
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"""
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if type != 2:
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raise NotImplementedError('Only DCT type 2 is implemented.')
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if norm is not None and norm not in ['backward', 'ortho']:
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raise ValueError(f"jax.scipy.fft.idctn: {norm=!r} is not implemented")
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if axes is None:
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axes = range(x.ndim)
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if len(axes) == 1:
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return idct(x, n=s[0] if s is not None else None, axis=axes[0], norm=norm)
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if s is not None:
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ns = dict(zip(axes, s))
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pads = [(0, ns[a] - x.shape[a] if a in ns else 0, 0) for a in range(x.ndim)]
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x = lax.pad(x, jnp.array(0, x.dtype), pads)
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# compose high-D DCTs from 1D DCTs:
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for axis in axes:
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x = idct(x, axis=axis, norm=norm)
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return x
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def _dct_deinterleave(x: Array, axis: int) -> Array:
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empty_slice = slice(None, None, None)
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ix0 = tuple(
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slice(None, math.ceil(x.shape[axis]/2), 1) if i == axis else empty_slice
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for i in range(len(x.shape)))
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ix1 = tuple(
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slice(math.ceil(x.shape[axis]/2), None, 1) if i == axis else empty_slice
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for i in range(len(x.shape)))
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v0 = x[ix0]
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v1 = lax.rev(x[ix1], (axis,))
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out = jnp.zeros(x.shape, dtype=x.dtype)
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evens = tuple(
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|
slice(None, None, 2) if i == axis else empty_slice for i in range(len(x.shape)))
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|
odds = tuple(
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|
slice(1, None, 2) if i == axis else empty_slice for i in range(len(x.shape)))
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out = out.at[evens].set(v0)
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|
out = out.at[odds].set(v1)
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|
return out
|