1/*
2 * jfdctint.c
3 *
4 * This file was part of the Independent JPEG Group's software:
5 * Copyright (C) 1991-1996, Thomas G. Lane.
6 * libjpeg-turbo Modifications:
7 * Copyright (C) 2015, 2020, D. R. Commander.
8 * For conditions of distribution and use, see the accompanying README.ijg
9 * file.
10 *
11 * This file contains a slower but more accurate integer implementation of the
12 * forward DCT (Discrete Cosine Transform).
13 *
14 * A 2-D DCT can be done by 1-D DCT on each row followed by 1-D DCT
15 * on each column. Direct algorithms are also available, but they are
16 * much more complex and seem not to be any faster when reduced to code.
17 *
18 * This implementation is based on an algorithm described in
19 * C. Loeffler, A. Ligtenberg and G. Moschytz, "Practical Fast 1-D DCT
20 * Algorithms with 11 Multiplications", Proc. Int'l. Conf. on Acoustics,
21 * Speech, and Signal Processing 1989 (ICASSP '89), pp. 988-991.
22 * The primary algorithm described there uses 11 multiplies and 29 adds.
23 * We use their alternate method with 12 multiplies and 32 adds.
24 * The advantage of this method is that no data path contains more than one
25 * multiplication; this allows a very simple and accurate implementation in
26 * scaled fixed-point arithmetic, with a minimal number of shifts.
27 */
28
29#define JPEG_INTERNALS
30#include "jinclude.h"
31#include "jpeglib.h"
32#include "jdct.h" /* Private declarations for DCT subsystem */
33
34#ifdef DCT_ISLOW_SUPPORTED
35
36
37/*
38 * This module is specialized to the case DCTSIZE = 8.
39 */
40
41#if DCTSIZE != 8
42 Sorry, this code only copes with 8x8 DCTs. /* deliberate syntax err */
43#endif
44
45
46/*
47 * The poop on this scaling stuff is as follows:
48 *
49 * Each 1-D DCT step produces outputs which are a factor of sqrt(N)
50 * larger than the true DCT outputs. The final outputs are therefore
51 * a factor of N larger than desired; since N=8 this can be cured by
52 * a simple right shift at the end of the algorithm. The advantage of
53 * this arrangement is that we save two multiplications per 1-D DCT,
54 * because the y0 and y4 outputs need not be divided by sqrt(N).
55 * In the IJG code, this factor of 8 is removed by the quantization step
56 * (in jcdctmgr.c), NOT in this module.
57 *
58 * We have to do addition and subtraction of the integer inputs, which
59 * is no problem, and multiplication by fractional constants, which is
60 * a problem to do in integer arithmetic. We multiply all the constants
61 * by CONST_SCALE and convert them to integer constants (thus retaining
62 * CONST_BITS bits of precision in the constants). After doing a
63 * multiplication we have to divide the product by CONST_SCALE, with proper
64 * rounding, to produce the correct output. This division can be done
65 * cheaply as a right shift of CONST_BITS bits. We postpone shifting
66 * as long as possible so that partial sums can be added together with
67 * full fractional precision.
68 *
69 * The outputs of the first pass are scaled up by PASS1_BITS bits so that
70 * they are represented to better-than-integral precision. These outputs
71 * require BITS_IN_JSAMPLE + PASS1_BITS + 3 bits; this fits in a 16-bit word
72 * with the recommended scaling. (For 12-bit sample data, the intermediate
73 * array is JLONG anyway.)
74 *
75 * To avoid overflow of the 32-bit intermediate results in pass 2, we must
76 * have BITS_IN_JSAMPLE + CONST_BITS + PASS1_BITS <= 26. Error analysis
77 * shows that the values given below are the most effective.
78 */
79
80#if BITS_IN_JSAMPLE == 8
81#define CONST_BITS 13
82#define PASS1_BITS 2
83#else
84#define CONST_BITS 13
85#define PASS1_BITS 1 /* lose a little precision to avoid overflow */
86#endif
87
88/* Some C compilers fail to reduce "FIX(constant)" at compile time, thus
89 * causing a lot of useless floating-point operations at run time.
90 * To get around this we use the following pre-calculated constants.
91 * If you change CONST_BITS you may want to add appropriate values.
92 * (With a reasonable C compiler, you can just rely on the FIX() macro...)
93 */
94
95#if CONST_BITS == 13
96#define FIX_0_298631336 ((JLONG)2446) /* FIX(0.298631336) */
97#define FIX_0_390180644 ((JLONG)3196) /* FIX(0.390180644) */
98#define FIX_0_541196100 ((JLONG)4433) /* FIX(0.541196100) */
99#define FIX_0_765366865 ((JLONG)6270) /* FIX(0.765366865) */
100#define FIX_0_899976223 ((JLONG)7373) /* FIX(0.899976223) */
101#define FIX_1_175875602 ((JLONG)9633) /* FIX(1.175875602) */
102#define FIX_1_501321110 ((JLONG)12299) /* FIX(1.501321110) */
103#define FIX_1_847759065 ((JLONG)15137) /* FIX(1.847759065) */
104#define FIX_1_961570560 ((JLONG)16069) /* FIX(1.961570560) */
105#define FIX_2_053119869 ((JLONG)16819) /* FIX(2.053119869) */
106#define FIX_2_562915447 ((JLONG)20995) /* FIX(2.562915447) */
107#define FIX_3_072711026 ((JLONG)25172) /* FIX(3.072711026) */
108#else
109#define FIX_0_298631336 FIX(0.298631336)
110#define FIX_0_390180644 FIX(0.390180644)
111#define FIX_0_541196100 FIX(0.541196100)
112#define FIX_0_765366865 FIX(0.765366865)
113#define FIX_0_899976223 FIX(0.899976223)
114#define FIX_1_175875602 FIX(1.175875602)
115#define FIX_1_501321110 FIX(1.501321110)
116#define FIX_1_847759065 FIX(1.847759065)
117#define FIX_1_961570560 FIX(1.961570560)
118#define FIX_2_053119869 FIX(2.053119869)
119#define FIX_2_562915447 FIX(2.562915447)
120#define FIX_3_072711026 FIX(3.072711026)
121#endif
122
123
124/* Multiply an JLONG variable by an JLONG constant to yield an JLONG result.
125 * For 8-bit samples with the recommended scaling, all the variable
126 * and constant values involved are no more than 16 bits wide, so a
127 * 16x16->32 bit multiply can be used instead of a full 32x32 multiply.
128 * For 12-bit samples, a full 32-bit multiplication will be needed.
129 */
130
131#if BITS_IN_JSAMPLE == 8
132#define MULTIPLY(var, const) MULTIPLY16C16(var, const)
133#else
134#define MULTIPLY(var, const) ((var) * (const))
135#endif
136
137
138/*
139 * Perform the forward DCT on one block of samples.
140 */
141
142GLOBAL(void)
143jpeg_fdct_islow(DCTELEM *data)
144{
145 JLONG tmp0, tmp1, tmp2, tmp3, tmp4, tmp5, tmp6, tmp7;
146 JLONG tmp10, tmp11, tmp12, tmp13;
147 JLONG z1, z2, z3, z4, z5;
148 DCTELEM *dataptr;
149 int ctr;
150 SHIFT_TEMPS
151
152 /* Pass 1: process rows. */
153 /* Note results are scaled up by sqrt(8) compared to a true DCT; */
154 /* furthermore, we scale the results by 2**PASS1_BITS. */
155
156 dataptr = data;
157 for (ctr = DCTSIZE - 1; ctr >= 0; ctr--) {
158 tmp0 = dataptr[0] + dataptr[7];
159 tmp7 = dataptr[0] - dataptr[7];
160 tmp1 = dataptr[1] + dataptr[6];
161 tmp6 = dataptr[1] - dataptr[6];
162 tmp2 = dataptr[2] + dataptr[5];
163 tmp5 = dataptr[2] - dataptr[5];
164 tmp3 = dataptr[3] + dataptr[4];
165 tmp4 = dataptr[3] - dataptr[4];
166
167 /* Even part per LL&M figure 1 --- note that published figure is faulty;
168 * rotator "sqrt(2)*c1" should be "sqrt(2)*c6".
169 */
170
171 tmp10 = tmp0 + tmp3;
172 tmp13 = tmp0 - tmp3;
173 tmp11 = tmp1 + tmp2;
174 tmp12 = tmp1 - tmp2;
175
176 dataptr[0] = (DCTELEM)LEFT_SHIFT(tmp10 + tmp11, PASS1_BITS);
177 dataptr[4] = (DCTELEM)LEFT_SHIFT(tmp10 - tmp11, PASS1_BITS);
178
179 z1 = MULTIPLY(tmp12 + tmp13, FIX_0_541196100);
180 dataptr[2] = (DCTELEM)DESCALE(z1 + MULTIPLY(tmp13, FIX_0_765366865),
181 CONST_BITS - PASS1_BITS);
182 dataptr[6] = (DCTELEM)DESCALE(z1 + MULTIPLY(tmp12, -FIX_1_847759065),
183 CONST_BITS - PASS1_BITS);
184
185 /* Odd part per figure 8 --- note paper omits factor of sqrt(2).
186 * cK represents cos(K*pi/16).
187 * i0..i3 in the paper are tmp4..tmp7 here.
188 */
189
190 z1 = tmp4 + tmp7;
191 z2 = tmp5 + tmp6;
192 z3 = tmp4 + tmp6;
193 z4 = tmp5 + tmp7;
194 z5 = MULTIPLY(z3 + z4, FIX_1_175875602); /* sqrt(2) * c3 */
195
196 tmp4 = MULTIPLY(tmp4, FIX_0_298631336); /* sqrt(2) * (-c1+c3+c5-c7) */
197 tmp5 = MULTIPLY(tmp5, FIX_2_053119869); /* sqrt(2) * ( c1+c3-c5+c7) */
198 tmp6 = MULTIPLY(tmp6, FIX_3_072711026); /* sqrt(2) * ( c1+c3+c5-c7) */
199 tmp7 = MULTIPLY(tmp7, FIX_1_501321110); /* sqrt(2) * ( c1+c3-c5-c7) */
200 z1 = MULTIPLY(z1, -FIX_0_899976223); /* sqrt(2) * ( c7-c3) */
201 z2 = MULTIPLY(z2, -FIX_2_562915447); /* sqrt(2) * (-c1-c3) */
202 z3 = MULTIPLY(z3, -FIX_1_961570560); /* sqrt(2) * (-c3-c5) */
203 z4 = MULTIPLY(z4, -FIX_0_390180644); /* sqrt(2) * ( c5-c3) */
204
205 z3 += z5;
206 z4 += z5;
207
208 dataptr[7] = (DCTELEM)DESCALE(tmp4 + z1 + z3, CONST_BITS - PASS1_BITS);
209 dataptr[5] = (DCTELEM)DESCALE(tmp5 + z2 + z4, CONST_BITS - PASS1_BITS);
210 dataptr[3] = (DCTELEM)DESCALE(tmp6 + z2 + z3, CONST_BITS - PASS1_BITS);
211 dataptr[1] = (DCTELEM)DESCALE(tmp7 + z1 + z4, CONST_BITS - PASS1_BITS);
212
213 dataptr += DCTSIZE; /* advance pointer to next row */
214 }
215
216 /* Pass 2: process columns.
217 * We remove the PASS1_BITS scaling, but leave the results scaled up
218 * by an overall factor of 8.
219 */
220
221 dataptr = data;
222 for (ctr = DCTSIZE - 1; ctr >= 0; ctr--) {
223 tmp0 = dataptr[DCTSIZE * 0] + dataptr[DCTSIZE * 7];
224 tmp7 = dataptr[DCTSIZE * 0] - dataptr[DCTSIZE * 7];
225 tmp1 = dataptr[DCTSIZE * 1] + dataptr[DCTSIZE * 6];
226 tmp6 = dataptr[DCTSIZE * 1] - dataptr[DCTSIZE * 6];
227 tmp2 = dataptr[DCTSIZE * 2] + dataptr[DCTSIZE * 5];
228 tmp5 = dataptr[DCTSIZE * 2] - dataptr[DCTSIZE * 5];
229 tmp3 = dataptr[DCTSIZE * 3] + dataptr[DCTSIZE * 4];
230 tmp4 = dataptr[DCTSIZE * 3] - dataptr[DCTSIZE * 4];
231
232 /* Even part per LL&M figure 1 --- note that published figure is faulty;
233 * rotator "sqrt(2)*c1" should be "sqrt(2)*c6".
234 */
235
236 tmp10 = tmp0 + tmp3;
237 tmp13 = tmp0 - tmp3;
238 tmp11 = tmp1 + tmp2;
239 tmp12 = tmp1 - tmp2;
240
241 dataptr[DCTSIZE * 0] = (DCTELEM)DESCALE(tmp10 + tmp11, PASS1_BITS);
242 dataptr[DCTSIZE * 4] = (DCTELEM)DESCALE(tmp10 - tmp11, PASS1_BITS);
243
244 z1 = MULTIPLY(tmp12 + tmp13, FIX_0_541196100);
245 dataptr[DCTSIZE * 2] =
246 (DCTELEM)DESCALE(z1 + MULTIPLY(tmp13, FIX_0_765366865),
247 CONST_BITS + PASS1_BITS);
248 dataptr[DCTSIZE * 6] =
249 (DCTELEM)DESCALE(z1 + MULTIPLY(tmp12, -FIX_1_847759065),
250 CONST_BITS + PASS1_BITS);
251
252 /* Odd part per figure 8 --- note paper omits factor of sqrt(2).
253 * cK represents cos(K*pi/16).
254 * i0..i3 in the paper are tmp4..tmp7 here.
255 */
256
257 z1 = tmp4 + tmp7;
258 z2 = tmp5 + tmp6;
259 z3 = tmp4 + tmp6;
260 z4 = tmp5 + tmp7;
261 z5 = MULTIPLY(z3 + z4, FIX_1_175875602); /* sqrt(2) * c3 */
262
263 tmp4 = MULTIPLY(tmp4, FIX_0_298631336); /* sqrt(2) * (-c1+c3+c5-c7) */
264 tmp5 = MULTIPLY(tmp5, FIX_2_053119869); /* sqrt(2) * ( c1+c3-c5+c7) */
265 tmp6 = MULTIPLY(tmp6, FIX_3_072711026); /* sqrt(2) * ( c1+c3+c5-c7) */
266 tmp7 = MULTIPLY(tmp7, FIX_1_501321110); /* sqrt(2) * ( c1+c3-c5-c7) */
267 z1 = MULTIPLY(z1, -FIX_0_899976223); /* sqrt(2) * ( c7-c3) */
268 z2 = MULTIPLY(z2, -FIX_2_562915447); /* sqrt(2) * (-c1-c3) */
269 z3 = MULTIPLY(z3, -FIX_1_961570560); /* sqrt(2) * (-c3-c5) */
270 z4 = MULTIPLY(z4, -FIX_0_390180644); /* sqrt(2) * ( c5-c3) */
271
272 z3 += z5;
273 z4 += z5;
274
275 dataptr[DCTSIZE * 7] = (DCTELEM)DESCALE(tmp4 + z1 + z3,
276 CONST_BITS + PASS1_BITS);
277 dataptr[DCTSIZE * 5] = (DCTELEM)DESCALE(tmp5 + z2 + z4,
278 CONST_BITS + PASS1_BITS);
279 dataptr[DCTSIZE * 3] = (DCTELEM)DESCALE(tmp6 + z2 + z3,
280 CONST_BITS + PASS1_BITS);
281 dataptr[DCTSIZE * 1] = (DCTELEM)DESCALE(tmp7 + z1 + z4,
282 CONST_BITS + PASS1_BITS);
283
284 dataptr++; /* advance pointer to next column */
285 }
286}
287
288#endif /* DCT_ISLOW_SUPPORTED */
289