# # Fast discrete cosine transform algorithms (Python) # # Copyright (c) 2020 Project Nayuki. (MIT License) # https://www.nayuki.io/page/fast-discrete-cosine-transform-algorithms # # Permission is hereby granted, free of charge, to any person obtaining a copy of # this software and associated documentation files (the "Software"), to deal in # the Software without restriction, including without limitation the rights to # use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of # the Software, and to permit persons to whom the Software is furnished to do so, # subject to the following conditions: # - The above copyright notice and this permission notice shall be included in # all copies or substantial portions of the Software. # - The Software is provided "as is", without warranty of any kind, express or # implied, including but not limited to the warranties of merchantability, # fitness for a particular purpose and noninfringement. In no event shall the # authors or copyright holders be liable for any claim, damages or other # liability, whether in an action of contract, tort or otherwise, arising from, # out of or in connection with the Software or the use or other dealings in the # Software. # import numpy # DCT type II, unscaled. Algorithm by Byeong Gi Lee, 1984. # See: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.118.3056&rep=rep1&type=pdf#page=34 def transform(vector): if vector.ndim != 1: raise ValueError() n = vector.size if n == 1: return vector.copy() elif n == 0 or n % 2 != 0: raise ValueError() else: half = n // 2 gamma = vector[ : half] delta = vector[n - 1 : half - 1 : -1] alpha = transform(gamma + delta) beta = transform((gamma - delta) / (numpy.cos(numpy.arange(0.5, half + 0.5) * (numpy.pi / n)) * 2.0)) result = numpy.zeros_like(vector) result[0 : : 2] = alpha result[1 : : 2] = beta result[1 : n - 1 : 2] += beta[1 : ] return result # DCT type III, unscaled. Algorithm by Byeong Gi Lee, 1984. # See: https://www.nayuki.io/res/fast-discrete-cosine-transform-algorithms/lee-new-algo-discrete-cosine-transform.pdf def inverse_transform(vector, root=True): if vector.ndim != 1: raise ValueError() if root: vector = vector.copy() vector[0] /= 2 n = vector.size if n == 1: return vector elif n == 0 or n % 2 != 0: raise ValueError() else: half = n // 2 alpha = vector[0 : : 2].copy() beta = vector[1 : : 2].copy() beta[1 : ] += vector[1 : n - 1 : 2] inverse_transform(alpha, False) inverse_transform(beta , False) beta /= numpy.cos(numpy.arange(0.5, half + 0.5) * (numpy.pi / n)) * 2.0 vector[ : half] = alpha + beta vector[n - 1 : half - 1 : -1] = alpha - beta return vector