src/openpower/decoder/isa/fastdctlee.py

   1 #
   2 # Fast discrete cosine transform algorithms (Python)
   3 #
   4 # Copyright (c) 2020 Project Nayuki. (MIT License)
   5 # https://www.nayuki.io/page/fast-discrete-cosine-transform-algorithms
   6 #
   7 # License for original fastdctlee.py by Nayuki:
   8 #
   9 # Permission is hereby granted, free of charge, to any person obtaining
  10 # a copy of this software and associated documentation files (the
  11 # "Software"), to deal in the Software without restriction, including
  12 # without limitation the rights to use, copy, modify, merge, publish,
  13 # distribute, sublicense, and/or sell copies of the Software, and to
  14 # permit persons to whom the Software is furnished to do so, subject to
  15 # the following conditions:
  16 # - The above copyright notice and this permission notice shall be included in
  17 #   all copies or substantial portions of the Software.
  18 # - The Software is provided "as is", without warranty of any kind, express or
  19 #   implied, including but not limited to the warranties of merchantability,
  20 #   fitness for a particular purpose and noninfringement. In no event shall the
  21 #   authors or copyright holders be liable for any claim, damages or other
  22 #   liability, whether in an action of contract, tort or otherwise,
  23 #   arising from, out of or in connection with the Software or the use
  24 #   or other dealings in the Software.
  25 #
  26 #
  27 # Modifications made to create an in-place iterative DCT - SPDX: LGPLv3+
  28 # Copyright (c) 2021 Luke Kenneth Casson Leighton <lkcl@lkcl.net>
  29 #
  30 # The modifications made are firstly to create an iterative schedule,
  31 # rather than the more normal recursive algorithm.  Secondly, the
  32 # two butterflys are also separated out: inner butterfly does COS +/-
  33 # whilst outer butterfly does the iterative summing.
  34 #
  35 # However, to avoid data copying some additional tricks are played:
  36 # - firstly, the data is LOADed in bit-reversed order (which is normally
  37 #   covered by the recursive algorithm due to the odd-even reconstruction)
  38 #   but then to reference the data in the correct order an array of
  39 #   bit-reversed indices is created, as a level of indirection.
  40 #   the data is bit-reversed but so are the indices, making it all A-Ok.
  41 # - secondly, normally in DCT a 2nd target (copy) array is used where
  42 #   the top half is read in reverse order (7 6 5 4) and written out
  43 #   to the target 4 5 6 7.  the plan was to do this in two stages:
  44 #   write in-place in order 4 5 6 7 then swap afterwards (7-4), (6-5).
  45 #
  46 # End result is that once the first butterfly is done - bear in mind
  47 # it's in-place - the data is in the right order so that a second
  48 # dead-straightforward iterative sum can be done: again, in-place.
  49 # Really surprising.
  50
  51 import math
  52 from copy import deepcopy
  53
  54 # bits of the integer 'val'.
  55 def reverse_bits(val, width):
  56     result = 0
  57     for _ in range(width):
  58         result = (result << 1) | (val & 1)
  59         val >>= 1
  60     return result
  61
  62
  63 # DCT type II, unscaled. Algorithm by Byeong Gi Lee, 1984.
  64 # See: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.118.3056&rep=rep1&type=pdf#page=34
  65 # original (recursive) algorithm by Nayuki
  66 def transform(vector, indent=0):
  67     idt = "   " * indent
  68     n = len(vector)
  69     if n == 1:
  70         return list(vector)
  71     elif n == 0 or n % 2 != 0:
  72         raise ValueError()
  73     else:
  74         half = n // 2
  75         alpha = [(vector[i] + vector[-(i + 1)]) for i in range(half)]
  76         beta  = [(vector[i] - vector[-(i + 1)]) /
  77                  (math.cos((i + 0.5) * math.pi / n) * 2.0)
  78                     for i in range(half)]
  79         alpha = transform(alpha)
  80         beta  = transform(beta )
  81         result = []
  82         for i in range(half - 1):
  83             result.append(alpha[i])
  84             result.append(beta[i] + beta[i + 1])
  85         result.append(alpha[-1])
  86         result.append(beta [-1])
  87         return result
  88
  89
  90 # modified (iterative) algorithm by lkcl, based on Nayuki original
  91 def transform(vector, indent=0):
  92     idt = "   " * indent
  93     n = len(vector)
  94     if n == 1:
  95         return list(vector)
  96     elif n == 0 or n % 2 != 0:
  97         raise ValueError()
  98     else:
  99         half = n // 2
 100         alpha = [0] * half
 101         beta = [0] * half
 102         print (idt, "xf", vector)
 103         print (idt, "coeff", n, "->", end=" ")
 104         for i in range(half):
 105             t1, t2 = vector[i], vector[n-i-1]
 106             k = (math.cos((i + 0.5) * math.pi / n) * 2.0)
 107             print (i, n-i-1, "i/n", (i+0.5)/n, ":", k, end= " ")
 108             alpha[i] = t1 + t2
 109             beta[i] = (t1 - t2) * (1/k)
 110         print ()
 111         print (idt, "n", n, "alpha", end=" ")
 112         for i in range(0, n, 2):
 113             print (i, i//2, alpha[i//2], end=" ")
 114         print()
 115         print (idt, "n", n, "beta", end=" ")
 116         for i in range(0, n, 2):
 117             print (i, beta[i//2], end=" ")
 118         print()
 119         alpha = transform(alpha, indent+1)
 120         beta  = transform(beta , indent+1)
 121         result = [0] * n
 122         for i in range(half):
 123             result[i*2] = alpha[i]
 124             result[i*2+1] = beta[i]
 125         print(idt, "merge", result)
 126         for i in range(half - 1):
 127             result[i*2+1] += result[i*2+3]
 128         print(idt, "result", result)
 129         return result
 130
 131
 132 # totally cool *in-place* DCT algorithm
 133 def transform2(vec):
 134
 135     vec = deepcopy(vec)
 136     # Initialization
 137     n = len(vec)
 138     print ()
 139     print ("transform2", n)
 140     levels = n.bit_length() - 1
 141
 142     # reference (read/write) the in-place data in *reverse-bit-order*
 143     ri = list(range(n))
 144     ri = [ri[reverse_bits(i, levels)] for i in range(n)]
 145
 146     # and pretend we LDed the data in bit-reversed order as well
 147     vec = [vec[ri[i]] for i in range(n)]
 148
 149     size = n
 150     while size >= 2:
 151         halfsize = size // 2
 152         tablestep = n // size
 153         ir = list(range(0, n, size))
 154         print ("  xform", size, ir)
 155         for i in ir:
 156             k = 0
 157             j = list(range(i, i + halfsize))
 158             jr = list(range(i+halfsize, i + size))
 159             jr.reverse()
 160             print ("  xform jr", j, jr)
 161             for ci, (jl, jh) in enumerate(zip(j, jr)):
 162                 t1, t2 = vec[ri[jl]], vec[ri[jh]]
 163                 coeff = (math.cos((ci + 0.5) * math.pi / size) * 2.0)
 164                 # normally DCT would use jl+halfsize not jh, here.
 165                 # to be able to work in-place, the idea is to perform a
 166                 # swap afterwards.
 167                 vec[ri[jl]] = t1 + t2
 168                 vec[ri[jh]] = (t1 - t2) * (1/coeff)
 169                 print ("coeff", size, i, k, "ci", ci,
 170                         "jl", ri[jl], "jh", ri[jh],
 171                        "i/n", (k+0.5)/size, coeff, vec[ri[jl]], vec[ri[jh]])
 172                 k += tablestep
 173             # instead of using jl+halfsize, perform a swap here.
 174             # use half of j/jr because actually jl+halfsize = reverse(j)
 175             hz2 = halfsize // 2 # can be zero which stops reversing 1-item lists
 176             for ci, (jl, jh) in enumerate(zip(j[:hz2], jr[:hz2])):
 177                 jlh = jl+halfsize
 178                 # swap
 179                 tmp = vec[ri[jlh]]
 180                 vec[ri[jlh]] = vec[ri[jh]]
 181                 vec[ri[jh]] = tmp
 182                 print ("     swap", size, i, ri[jlh], ri[jh])
 183         size //= 2
 184
 185     print("post-swapped", ri)
 186     print("transform2 pre-itersum", vec)
 187
 188     n = len(vec)
 189     size = n // 2
 190     while size >= 2:
 191         halfsize = size // 2
 192         ir = list(range(0, halfsize))
 193         print ("itersum", halfsize, size, ir)
 194         for i in ir:
 195             jr = list(range(i+halfsize, i+n-halfsize, size))
 196             print ("itersum    jr", i+halfsize, i+size, jr)
 197             for jh in jr:
 198                 vec[jh] += vec[jh+size]
 199                 print ("    itersum", size, i, jh, jh+size)
 200         size //= 2
 201
 202     print("transform2 result", vec)
 203
 204     return vec
 205
 206
 207 # DCT type III, unscaled. Algorithm by Byeong Gi Lee, 1984.
 208 # See: https://www.nayuki.io/res/fast-discrete-cosine-transform-algorithms/lee-new-algo-discrete-cosine-transform.pdf
 209 def inverse_transform(vector, root=True, indent=0):
 210     idt = "   " * indent
 211     if root:
 212         vector = list(vector)
 213         vector[0] /= 2
 214     n = len(vector)
 215     if n == 1:
 216         return vector, [0]
 217     elif n == 0 or n % 2 != 0:
 218         raise ValueError()
 219     else:
 220         half = n // 2
 221         alpha = [vector[0]]
 222         beta  = [vector[1]]
 223         for i in range(2, n, 2):
 224             alpha.append(vector[i])
 225             beta.append(vector[i - 1] + vector[i + 1])
 226         print (idt, "n", n, "alpha 0", end=" ")
 227         for i in range(2, n, 2):
 228             print (i, end=" ")
 229         print ("beta 1", end=" ")
 230         for i in range(2, n, 2):
 231             print ("%d+%d" % (i-1, i+1), end=" ")
 232         print()
 233         inverse_transform(alpha, False, indent+1)
 234         inverse_transform(beta , False, indent+1)
 235         for i in range(half):
 236             x = alpha[i]
 237             y = beta[i] / (math.cos((i + 0.5) * math.pi / n) * 2)
 238             vector[i] = x + y
 239             vector[-(i + 1)] = x - y
 240             print (idt, " v[%d] = alpha[%d]+beta[%d]" % (i, i, i))
 241             print (idt, " v[%d] = alpha[%d]-beta[%d]" % (n-i-1, i, i))
 242         return vector
 243
 244
 245 def inverse_transform2(vector, root=True):
 246     n = len(vector)
 247     if root:
 248         vector = list(vector)
 249     if n == 1:
 250         return vector
 251     elif n == 0 or n % 2 != 0:
 252         raise ValueError()
 253     else:
 254         half = n // 2
 255         alpha = [0]
 256         beta  = [1]
 257         for i in range(2, n, 2):
 258             alpha.append(i)
 259             beta.append(("add", i - 1, i + 1))
 260         inverse_transform2(alpha, False)
 261         inverse_transform2(beta , False)
 262         for i in range(half):
 263             x = alpha[i]
 264             y = ("cos", beta[i], i)
 265             vector[i] = ("add", x, y)
 266             vector[-(i + 1)] = ("sub", x, y)
 267         return vector
 268
 269
 270 # does the outer butterfly in a recursive fashion, used in an
 271 # intermediary development of the in-place DCT.
 272 def transform_itersum(vector, indent=0):
 273     idt = "   " * indent
 274     n = len(vector)
 275     if n == 1:
 276         return list(vector)
 277     elif n == 0 or n % 2 != 0:
 278         raise ValueError()
 279     else:
 280         half = n // 2
 281         alpha = [0] * half
 282         beta = [0] * half
 283         for i in range(half):
 284             t1, t2 = vector[i], vector[i+half]
 285             alpha[i] = t1
 286             beta[i] = t2
 287         alpha = transform_itersum(alpha, indent+1)
 288         beta  = transform_itersum(beta , indent+1)
 289         result = [0] * n
 290         for i in range(half):
 291             result[i*2] = alpha[i]
 292             result[i*2+1] = beta[i]
 293         print(idt, "iter-merge", result)
 294         for i in range(half - 1):
 295             result[i*2+1] += result[i*2+3]
 296         print(idt, "iter-result", result)
 297         return result
 298
 299
 300 # prints out an "add" schedule for the outer butterfly, recursively,
 301 # matching what transform_itersum does.
 302 def itersum_explore(vector, indent=0):
 303     idt = "   " * indent
 304     n = len(vector)
 305     if n == 1:
 306         return list(vector)
 307     elif n == 0 or n % 2 != 0:
 308         raise ValueError()
 309     else:
 310         half = n // 2
 311         alpha = [0] * half
 312         beta = [0] * half
 313         for i in range(half):
 314             t1, t2 = vector[i], vector[i+half]
 315             alpha[i] = t1
 316             beta[i] = t2
 317         alpha = itersum_explore(alpha, indent+1)
 318         beta  = itersum_explore(beta , indent+1)
 319         result = [0] * n
 320         for i in range(half):
 321             result[i*2] = alpha[i]
 322             result[i*2+1] = beta[i]
 323         print(idt, "iter-merge", result)
 324         for i in range(half - 1):
 325             result[i*2+1] = ("add", result[i*2+1], result[i*2+3])
 326         print(idt, "iter-result", result)
 327         return result
 328
 329
 330 # prints out the exact same outer butterfly but does so iteratively.
 331 # by comparing the output from itersum_explore and itersum_explore2
 332 # and by drawing out the resultant ADDs as a graph it was possible
 333 # to deduce what the heck was going on.
 334 def itersum_explore2(vec, indent=0):
 335     n = len(vec)
 336     size = n // 2
 337     while size >= 2:
 338         halfsize = size // 2
 339         ir = list(range(0, halfsize))
 340         print ("itersum", halfsize, size, ir)
 341         for i in ir:
 342             jr = list(range(i+halfsize, i+n-halfsize, size))
 343             print ("itersum    jr", i+halfsize, i+size, jr)
 344             for jh in jr:
 345                 vec[jh] = ("add", vec[jh], vec[jh+size])
 346                 print ("    itersum", size, i, jh, jh+size)
 347         size //= 2
 348
 349     return vec
 350
 351 if __name__ == '__main__':
 352     n = 16
 353     vec = list(range(n))
 354     levels = n.bit_length() - 1
 355     vec = [vec[reverse_bits(i, levels)] for i in range(n)]
 356     ops = itersum_explore(vec)
 357     for i, x in enumerate(ops):
 358         print (i, x)
 359
 360     n = 16
 361     vec = list(range(n))
 362     levels = n.bit_length() - 1
 363     ops = itersum_explore2(vec)
 364     for i, x in enumerate(ops):
 365         print (i, x)