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     # reference list for not needing to do data-swaps, just swap what
 150     # *indices* are referenced (two levels of indirection at the moment)
 151     ji = list(range(n))
 152
 153     # start the inner butterfly
 154     size = n
 155     while size >= 2:
 156         halfsize = size // 2
 157         tablestep = n // size
 158         ir = list(range(0, n, size))
 159         print ("  xform", size, ir)
 160         for i in ir:
 161             # two lists of half-range indices, e.g. j 0123, jr 7654
 162             j = list(range(i, i + halfsize))
 163             jr = list(range(i+halfsize, i + size))
 164             jr.reverse()
 165             print ("  xform jr", j, jr)
 166             for ci, (jl, jh) in enumerate(zip(j, jr)):
 167                 t1, t2 = vec[ri[ji[jl]]], vec[ri[ji[jh]]]
 168                 coeff = (math.cos((ci + 0.5) * math.pi / size) * 2.0)
 169                 # normally DCT would use jl+halfsize not jh, here.
 170                 # to be able to work in-place, the idea is to perform a
 171                 # swap afterwards.
 172                 vec[ri[ji[jl]]] = t1 + t2
 173                 vec[ri[ji[jh]]] = (t1 - t2) * (1/coeff)
 174                 print ("coeff", size, i, "ci", ci,
 175                         "jl", ri[jl], "jh", ri[jh],
 176                        "i/n", (ci+0.5)/size, coeff, vec[ri[ji[jl]]],
 177                                                     vec[ri[ji[jh]]])
 178             # instead of using jl+halfsize, perform a swap here.
 179             # use half of j/jr because actually jl+halfsize = reverse(j)
 180             hz2 = halfsize // 2 # can be zero which stops reversing 1-item lists
 181             for ci, (jl, jh) in enumerate(zip(j[:hz2], jr[:hz2])):
 182                 jlh = jl+halfsize
 183                 # swap
 184                 if False:
 185                     tmp = vec[ri[jlh]]
 186                     vec[ri[jlh]] = vec[ri[jh]]
 187                     vec[ri[jh]] = tmp
 188                 else:
 189                     tmp1 = ji[jlh]
 190                     tmp2 = ji[jh]
 191                     ji[jlh] = tmp2
 192                     ji[jh] = tmp1
 193                 print ("     swap", size, i, ji[jlh], ji[jh])
 194         size //= 2
 195
 196     print("post-swapped", ri)
 197     print("ji-swapped", ji)
 198
 199     print("transform2 pre-itersum", vec)
 200
 201     # ok so the above in-place-ness caused the order to change.
 202     # need to work out how to invert it back into the correct order
 203     idx_search = range(n)
 204     ir = [idx_search[reverse_bits(i, levels)] for i in range(n)]
 205     print("transform2 expected bit-rev", ir)
 206     vv = [0] * n
 207     for i in range(n):
 208         vv[i] = ir[ji[i]]
 209     print("restored", vv)
 210     vv = [vv[reverse_bits(i, levels)] for i in range(n)]
 211     print("restored-inverted", vv)
 212
 213     # this is TEMPORARY!  it should be possible to establish
 214     # a *LOAD* schedule which has everything work out such that
 215     # all data ends up *in* this order at the end of the inner butterfly
 216     vec2 = deepcopy(vec)
 217     for i in range(n):
 218         vec[i] = vec2[vv[i]]
 219
 220     # now things are in the right order for the outer butterfly.
 221     n = len(vec)
 222     size = n // 2
 223     while size >= 2:
 224         halfsize = size // 2
 225         ir = list(range(0, halfsize))
 226         print ("itersum", halfsize, size, ir)
 227         for i in ir:
 228             jr = list(range(i+halfsize, i+n-halfsize, size))
 229             print ("itersum    jr", i+halfsize, i+size, jr)
 230             for jh in jr:
 231                 vec[jh] += vec[jh+size]
 232                 print ("    itersum", size, i, jh, jh+size)
 233         size //= 2
 234
 235     print("transform2 result", vec)
 236
 237     return vec
 238
 239
 240 # DCT type III, unscaled. Algorithm by Byeong Gi Lee, 1984.
 241 # See: https://www.nayuki.io/res/fast-discrete-cosine-transform-algorithms/lee-new-algo-discrete-cosine-transform.pdf
 242 def inverse_transform(vector, root=True, indent=0):
 243     idt = "   " * indent
 244     if root:
 245         vector = list(vector)
 246         vector[0] /= 2
 247     n = len(vector)
 248     if n == 1:
 249         return vector, [0]
 250     elif n == 0 or n % 2 != 0:
 251         raise ValueError()
 252     else:
 253         half = n // 2
 254         alpha = [vector[0]]
 255         beta  = [vector[1]]
 256         for i in range(2, n, 2):
 257             alpha.append(vector[i])
 258             beta.append(vector[i - 1] + vector[i + 1])
 259         print (idt, "n", n, "alpha 0", end=" ")
 260         for i in range(2, n, 2):
 261             print (i, end=" ")
 262         print ("beta 1", end=" ")
 263         for i in range(2, n, 2):
 264             print ("%d+%d" % (i-1, i+1), end=" ")
 265         print()
 266         inverse_transform(alpha, False, indent+1)
 267         inverse_transform(beta , False, indent+1)
 268         for i in range(half):
 269             x = alpha[i]
 270             y = beta[i] / (math.cos((i + 0.5) * math.pi / n) * 2)
 271             vector[i] = x + y
 272             vector[-(i + 1)] = x - y
 273             print (idt, " v[%d] = alpha[%d]+beta[%d]" % (i, i, i))
 274             print (idt, " v[%d] = alpha[%d]-beta[%d]" % (n-i-1, i, i))
 275         return vector
 276
 277
 278 def inverse_transform2(vector, root=True):
 279     n = len(vector)
 280     if root:
 281         vector = list(vector)
 282     if n == 1:
 283         return vector
 284     elif n == 0 or n % 2 != 0:
 285         raise ValueError()
 286     else:
 287         half = n // 2
 288         alpha = [0]
 289         beta  = [1]
 290         for i in range(2, n, 2):
 291             alpha.append(i)
 292             beta.append(("add", i - 1, i + 1))
 293         inverse_transform2(alpha, False)
 294         inverse_transform2(beta , False)
 295         for i in range(half):
 296             x = alpha[i]
 297             y = ("cos", beta[i], i)
 298             vector[i] = ("add", x, y)
 299             vector[-(i + 1)] = ("sub", x, y)
 300         return vector
 301
 302
 303 # does the outer butterfly in a recursive fashion, used in an
 304 # intermediary development of the in-place DCT.
 305 def transform_itersum(vector, indent=0):
 306     idt = "   " * indent
 307     n = len(vector)
 308     if n == 1:
 309         return list(vector)
 310     elif n == 0 or n % 2 != 0:
 311         raise ValueError()
 312     else:
 313         half = n // 2
 314         alpha = [0] * half
 315         beta = [0] * half
 316         for i in range(half):
 317             t1, t2 = vector[i], vector[i+half]
 318             alpha[i] = t1
 319             beta[i] = t2
 320         alpha = transform_itersum(alpha, indent+1)
 321         beta  = transform_itersum(beta , indent+1)
 322         result = [0] * n
 323         for i in range(half):
 324             result[i*2] = alpha[i]
 325             result[i*2+1] = beta[i]
 326         print(idt, "iter-merge", result)
 327         for i in range(half - 1):
 328             result[i*2+1] += result[i*2+3]
 329         print(idt, "iter-result", result)
 330         return result
 331
 332
 333 # prints out an "add" schedule for the outer butterfly, recursively,
 334 # matching what transform_itersum does.
 335 def itersum_explore(vector, indent=0):
 336     idt = "   " * indent
 337     n = len(vector)
 338     if n == 1:
 339         return list(vector)
 340     elif n == 0 or n % 2 != 0:
 341         raise ValueError()
 342     else:
 343         half = n // 2
 344         alpha = [0] * half
 345         beta = [0] * half
 346         for i in range(half):
 347             t1, t2 = vector[i], vector[i+half]
 348             alpha[i] = t1
 349             beta[i] = t2
 350         alpha = itersum_explore(alpha, indent+1)
 351         beta  = itersum_explore(beta , indent+1)
 352         result = [0] * n
 353         for i in range(half):
 354             result[i*2] = alpha[i]
 355             result[i*2+1] = beta[i]
 356         print(idt, "iter-merge", result)
 357         for i in range(half - 1):
 358             result[i*2+1] = ("add", result[i*2+1], result[i*2+3])
 359         print(idt, "iter-result", result)
 360         return result
 361
 362
 363 # prints out the exact same outer butterfly but does so iteratively.
 364 # by comparing the output from itersum_explore and itersum_explore2
 365 # and by drawing out the resultant ADDs as a graph it was possible
 366 # to deduce what the heck was going on.
 367 def itersum_explore2(vec, indent=0):
 368     n = len(vec)
 369     size = n // 2
 370     while size >= 2:
 371         halfsize = size // 2
 372         ir = list(range(0, halfsize))
 373         print ("itersum", halfsize, size, ir)
 374         for i in ir:
 375             jr = list(range(i+halfsize, i+n-halfsize, size))
 376             print ("itersum    jr", i+halfsize, i+size, jr)
 377             for jh in jr:
 378                 vec[jh] = ("add", vec[jh], vec[jh+size])
 379                 print ("    itersum", size, i, jh, jh+size)
 380         size //= 2
 381
 382     return vec
 383
 384 if __name__ == '__main__':
 385     n = 16
 386     vec = list(range(n))
 387     levels = n.bit_length() - 1
 388     vec = [vec[reverse_bits(i, levels)] for i in range(n)]
 389     ops = itersum_explore(vec)
 390     for i, x in enumerate(ops):
 391         print (i, x)
 392
 393     n = 16
 394     vec = list(range(n))
 395     levels = n.bit_length() - 1
 396     ops = itersum_explore2(vec)
 397     for i, x in enumerate(ops):
 398         print (i, x)