2 # Fast discrete cosine transform algorithm (Python)
4 # Modifications made to create an in-place iterative DCT:
5 # Copyright (c) 2021 Luke Kenneth Casson Leighton <lkcl@lkcl.net>
7 # License for modifications - SPDX: LGPLv3+
9 # Original fastdctlee.py by Nayuki:
10 # Copyright (c) 2020 Project Nayuki. (MIT License)
11 # https://www.nayuki.io/page/fast-discrete-cosine-transform-algorithms
13 # License for original fastdctlee.py by Nayuki:
15 # Permission is hereby granted, free of charge, to any person obtaining
16 # a copy of this software and associated documentation files (the
17 # "Software"), to deal in the Software without restriction, including
18 # without limitation the rights to use, copy, modify, merge, publish,
19 # distribute, sublicense, and/or sell copies of the Software, and to
20 # permit persons to whom the Software is furnished to do so, subject to
21 # the following conditions:
22 # - The above copyright notice and this permission notice shall be included in
23 # all copies or substantial portions of the Software.
24 # - The Software is provided "as is", without warranty of any kind, express or
25 # implied, including but not limited to the warranties of merchantability,
26 # fitness for a particular purpose and noninfringement. In no event shall the
27 # authors or copyright holders be liable for any claim, damages or other
28 # liability, whether in an action of contract, tort or otherwise,
29 # arising from, out of or in connection with the Software or the use
30 # or other dealings in the Software.
33 # The modifications made are firstly to create an iterative schedule,
34 # rather than the more normal recursive algorithm. Secondly, the
35 # two butterflys are also separated out: inner butterfly does COS +/-
36 # whilst outer butterfly does the iterative summing.
38 # However, to avoid data copying some additional tricks are played:
39 # - firstly, the data is LOADed in bit-reversed order (which is normally
40 # covered by the recursive algorithm due to the odd-even reconstruction)
41 # but then to reference the data in the correct order an array of
42 # bit-reversed indices is created, as a level of indirection.
43 # the data is bit-reversed but so are the indices, making it all A-Ok.
44 # - secondly, normally in DCT a 2nd target (copy) array is used where
45 # the top half is read in reverse order (7 6 5 4) and written out
46 # to the target 4 5 6 7. the plan was to do this in two stages:
47 # write in-place in order 4 5 6 7 then swap afterwards (7-4), (6-5).
48 # however by leaving the data *in-place* and having subsequent
49 # loops refer to the data *where it now is*, the swap is avoided
50 # - thirdly, arrange for the data to be *pre-swapped* (in an inverse
51 # order of how it would have got there, if that makes sense), such
52 # that *when* it gets swapped, it ends up in the right order.
53 # given that that will be a LD operation it's no big deal.
55 # End result is that once the first butterfly is done - bear in mind
56 # it's in-place - the data is in the right order so that a second
57 # dead-straightforward iterative sum can be done: again, in-place.
61 from copy
import deepcopy
63 # bits of the integer 'val'.
64 def reverse_bits(val
, width
):
66 for _
in range(width
):
67 result
= (result
<< 1) |
(val
& 1)
72 # reverse top half of a list, recursively. the recursion can be
73 # applied *after* or *before* the reversal of the top half. these
74 # are inverses of each other.
75 # this function is unused except to test the iterative version (halfrev2)
76 def halfrev(l
, pre_rev
=True):
80 ll
, lh
= l
[:n
//2], l
[n
//2:]
82 ll
, lh
= halfrev(ll
, pre_rev
), halfrev(lh
, pre_rev
)
85 ll
, lh
= halfrev(ll
, pre_rev
), halfrev(lh
, pre_rev
)
89 # iterative version of [recursively-applied] half-rev.
90 # relies on the list lengths being power-of-two and the fact
91 # that bit-inversion of a list of binary numbers is the same
92 # as reversing the order of the list
93 # this version is dead easy to implement in hardware.
94 # a big surprise is that the half-reversal can be done with
95 # such a simple XOR. the inverse operation is slightly trickier
96 def halfrev2(vec
, pre_rev
=True):
98 for i
in range(len(vec
)):
100 res
.append(i ^
(i
>>1))
104 for ji
in range(1, bl
):
110 # DCT type II, unscaled. Algorithm by Byeong Gi Lee, 1984.
111 # See: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.118.3056&rep=rep1&type=pdf#page=34
112 # original (recursive) algorithm by Nayuki
113 def transform(vector
, indent
=0):
118 elif n
== 0 or n
% 2 != 0:
122 alpha
= [(vector
[i
] + vector
[-(i
+ 1)]) for i
in range(half
)]
123 beta
= [(vector
[i
] - vector
[-(i
+ 1)]) /
124 (math
.cos((i
+ 0.5) * math
.pi
/ n
) * 2.0)
125 for i
in range(half
)]
126 alpha
= transform(alpha
)
127 beta
= transform(beta
)
129 for i
in range(half
- 1):
130 result
.append(alpha
[i
])
131 result
.append(beta
[i
] + beta
[i
+ 1])
132 result
.append(alpha
[-1])
133 result
.append(beta
[-1])
137 # modified recursive algorithm, based on Nayuki original, which simply
138 # prints out an awful lot of debug data. used to work out the ordering
139 # for the iterative version by analysing the indices printed out
140 def transform(vector
, indent
=0):
145 elif n
== 0 or n
% 2 != 0:
151 print (idt
, "xf", vector
)
152 print (idt
, "coeff", n
, "->", end
=" ")
153 for i
in range(half
):
154 t1
, t2
= vector
[i
], vector
[n
-i
-1]
155 k
= (math
.cos((i
+ 0.5) * math
.pi
/ n
) * 2.0)
156 print (i
, n
-i
-1, "i/n", (i
+0.5)/n
, ":", k
, end
= " ")
158 beta
[i
] = (t1
- t2
) * (1/k
)
160 print (idt
, "n", n
, "alpha", end
=" ")
161 for i
in range(0, n
, 2):
162 print (i
, i
//2, alpha
[i
//2], end
=" ")
164 print (idt
, "n", n
, "beta", end
=" ")
165 for i
in range(0, n
, 2):
166 print (i
, beta
[i
//2], end
=" ")
168 alpha
= transform(alpha
, indent
+1)
169 beta
= transform(beta
, indent
+1)
171 for i
in range(half
):
172 result
[i
*2] = alpha
[i
]
173 result
[i
*2+1] = beta
[i
]
174 print(idt
, "merge", result
)
175 for i
in range(half
- 1):
176 result
[i
*2+1] += result
[i
*2+3]
177 print(idt
, "result", result
)
181 # totally cool *in-place* DCT algorithm
187 print ("transform2", n
)
188 levels
= n
.bit_length() - 1
190 # reference (read/write) the in-place data in *reverse-bit-order*
192 ri
= [ri
[reverse_bits(i
, levels
)] for i
in range(n
)]
194 # reference list for not needing to do data-swaps, just swap what
195 # *indices* are referenced (two levels of indirection at the moment)
196 # pre-reverse the data-swap list so that it *ends up* in the order 0123..
198 ji
= halfrev2(ji
, True)
200 # and pretend we LDed data in half-swapped *and* bit-reversed order as well
201 # TODO: merge these two
202 vec
= halfrev2(vec
, False)
203 vec
= [vec
[ri
[i
]] for i
in range(n
)]
208 # create a cos table: not strictly necessary but here for illustrative
209 # purposes, to demonstrate the point that it really *is* iterative.
210 # this table could be cached and used multiple times rather than
211 # computed every time.
216 for i
in range(n
//size
):
217 for ci
in range(halfsize
):
218 ctable
.append((math
.cos((ci
+ 0.5) * math
.pi
/ size
) * 2.0))
221 # start the inner butterfly
226 tablestep
= n
// size
227 ir
= list(range(0, n
, size
))
228 print (" xform", size
, ir
)
230 # two lists of half-range indices, e.g. j 0123, jr 7654
231 j
= list(range(i
, i
+ halfsize
))
232 jr
= list(range(i
+halfsize
, i
+ size
))
234 print (" xform jr", j
, jr
)
235 for ci
, (jl
, jh
) in enumerate(zip(j
, jr
)):
236 t1
, t2
= vec
[ri
[ji
[jl
]]], vec
[ri
[ji
[jh
]]]
237 #coeff = (math.cos((ci + 0.5) * math.pi / size) * 2.0)
240 # normally DCT would use jl+halfsize not jh, here.
241 # to be able to work in-place, the idea is to perform a
243 vec
[ri
[ji
[jl
]]] = t1
+ t2
244 vec
[ri
[ji
[jh
]]] = (t1
- t2
) * (1/coeff
)
245 print ("coeff", size
, i
, "ci", ci
,
246 "jl", ri
[ji
[jl
]], "jh", ri
[ji
[jh
]],
247 "i/n", (ci
+0.5)/size
, coeff
, vec
[ri
[ji
[jl
]]],
249 # instead of using jl+halfsize, perform a swap here.
250 # use half of j/jr because actually jl+halfsize = reverse(j)
251 hz2
= halfsize
// 2 # can be zero which stops reversing 1-item lists
252 for ci
, (jl
, jh
) in enumerate(zip(j
[:hz2
], jr
[:hz2
])):
254 # swap indices, NOT the data
255 tmp1
, tmp2
= ji
[jlh
], ji
[jh
]
256 ji
[jlh
], ji
[jh
] = tmp2
, tmp1
257 print (" swap", size
, i
, ji
[jlh
], ji
[jh
])
260 print("post-swapped", ri
)
261 print("ji-swapped", ji
)
262 print("transform2 pre-itersum", vec
)
264 # now things are in the right order for the outer butterfly.
269 ir
= list(range(0, halfsize
))
270 print ("itersum", halfsize
, size
, ir
)
272 jr
= list(range(i
+halfsize
, i
+n
-halfsize
, size
))
273 print ("itersum jr", i
+halfsize
, i
+size
, jr
)
275 vec
[jh
] += vec
[jh
+size
]
276 print (" itersum", size
, i
, jh
, jh
+size
)
279 print("transform2 result", vec
)
284 # DCT type III, unscaled. Algorithm by Byeong Gi Lee, 1984.
285 # See: https://www.nayuki.io/res/fast-discrete-cosine-transform-algorithms/lee-new-algo-discrete-cosine-transform.pdf
286 def inverse_transform(vector
, root
=True, indent
=0):
289 vector
= list(vector
)
294 elif n
== 0 or n
% 2 != 0:
300 for i
in range(2, n
, 2):
301 alpha
.append(vector
[i
])
302 beta
.append(vector
[i
- 1] + vector
[i
+ 1])
303 print (idt
, "n", n
, "alpha 0", end
=" ")
304 for i
in range(2, n
, 2):
306 print ("beta 1", end
=" ")
307 for i
in range(2, n
, 2):
308 print ("%d+%d" % (i
-1, i
+1), end
=" ")
310 inverse_transform(alpha
, False, indent
+1)
311 inverse_transform(beta
, False, indent
+1)
312 for i
in range(half
):
314 y
= beta
[i
] / (math
.cos((i
+ 0.5) * math
.pi
/ n
) * 2)
316 vector
[-(i
+ 1)] = x
- y
317 print (idt
, " v[%d] = alpha[%d]+beta[%d]" % (i
, i
, i
))
318 print (idt
, " v[%d] = alpha[%d]-beta[%d]" % (n
-i
-1, i
, i
))
322 def inverse_transform2(vector
, root
=True):
325 vector
= list(vector
)
328 elif n
== 0 or n
% 2 != 0:
334 for i
in range(2, n
, 2):
336 beta
.append(("add", i
- 1, i
+ 1))
337 inverse_transform2(alpha
, False)
338 inverse_transform2(beta
, False)
339 for i
in range(half
):
341 y
= ("cos", beta
[i
], i
)
342 vector
[i
] = ("add", x
, y
)
343 vector
[-(i
+ 1)] = ("sub", x
, y
)
347 # does the outer butterfly in a recursive fashion, used in an
348 # intermediary development of the in-place DCT.
349 def transform_itersum(vector
, indent
=0):
354 elif n
== 0 or n
% 2 != 0:
360 for i
in range(half
):
361 t1
, t2
= vector
[i
], vector
[i
+half
]
364 alpha
= transform_itersum(alpha
, indent
+1)
365 beta
= transform_itersum(beta
, indent
+1)
367 for i
in range(half
):
368 result
[i
*2] = alpha
[i
]
369 result
[i
*2+1] = beta
[i
]
370 print(idt
, "iter-merge", result
)
371 for i
in range(half
- 1):
372 result
[i
*2+1] += result
[i
*2+3]
373 print(idt
, "iter-result", result
)
377 # prints out an "add" schedule for the outer butterfly, recursively,
378 # matching what transform_itersum does.
379 def itersum_explore(vector
, indent
=0):
384 elif n
== 0 or n
% 2 != 0:
390 for i
in range(half
):
391 t1
, t2
= vector
[i
], vector
[i
+half
]
394 alpha
= itersum_explore(alpha
, indent
+1)
395 beta
= itersum_explore(beta
, indent
+1)
397 for i
in range(half
):
398 result
[i
*2] = alpha
[i
]
399 result
[i
*2+1] = beta
[i
]
400 print(idt
, "iter-merge", result
)
401 for i
in range(half
- 1):
402 result
[i
*2+1] = ("add", result
[i
*2+1], result
[i
*2+3])
403 print(idt
, "iter-result", result
)
407 # prints out the exact same outer butterfly but does so iteratively.
408 # by comparing the output from itersum_explore and itersum_explore2
409 # and by drawing out the resultant ADDs as a graph it was possible
410 # to deduce what the heck was going on.
411 def itersum_explore2(vec
, indent
=0):
416 ir
= list(range(0, halfsize
))
417 print ("itersum", halfsize
, size
, ir
)
419 jr
= list(range(i
+halfsize
, i
+n
-halfsize
, size
))
420 print ("itersum jr", i
+halfsize
, i
+size
, jr
)
422 vec
[jh
] = ("add", vec
[jh
], vec
[jh
+size
])
423 print (" itersum", size
, i
, jh
, jh
+size
)
428 if __name__
== '__main__':
431 levels
= n
.bit_length() - 1
432 vec
= [vec
[reverse_bits(i
, levels
)] for i
in range(n
)]
433 ops
= itersum_explore(vec
)
434 for i
, x
in enumerate(ops
):
439 levels
= n
.bit_length() - 1
440 ops
= itersum_explore2(vec
)
441 for i
, x
in enumerate(ops
):
445 vec
= list(range(16))
446 print ("orig vec", vec
)
447 vecr
= halfrev(vec
, True)
448 print ("reversed", vecr
)
449 for i
, v
in enumerate(vecr
):
450 print ("%2d %2d %04s %04s %04s" % (i
, v
,
451 bin(i
)[2:], bin(v ^ i
)[2:], bin(v
)[2:]))
452 vecrr
= halfrev(vecr
, False)
454 vecrr
= halfrev(vec
, False)
455 print ("pre-reversed", vecrr
)
456 for i
, v
in enumerate(vecrr
):
457 print ("%2d %2d %04s %04s %04s" % (i
, v
,
458 bin(i
)[2:], bin(v ^ i
)[2:], bin(v
)[2:]))
459 il
= halfrev2(vec
, False)
460 print ("iterative rev", il
)
461 il
= halfrev2(vec
, True)
462 print ("iterative rev-true", il
)