1/*
2 * jfdctint.c
3 *
4 * Copyright (C) 1991-1996, Thomas G. Lane.
5 * This file is part of the Independent JPEG Group's software.
6 * For conditions of distribution and use, see the accompanying README file.
7 *
8 * This file contains a slow-but-accurate integer implementation of the
9 * forward DCT (Discrete Cosine Transform).
10 *
11 * A 2-D DCT can be done by 1-D DCT on each row followed by 1-D DCT
12 * on each column. Direct algorithms are also available, but they are
13 * much more complex and seem not to be any faster when reduced to code.
14 *
15 * This implementation is based on an algorithm described in
16 * C. Loeffler, A. Ligtenberg and G. Moschytz, "Practical Fast 1-D DCT
17 * Algorithms with 11 Multiplications", Proc. Int'l. Conf. on Acoustics,
18 * Speech, and Signal Processing 1989 (ICASSP '89), pp. 988-991.
19 * The primary algorithm described there uses 11 multiplies and 29 adds.
20 * We use their alternate method with 12 multiplies and 32 adds.
21 * The advantage of this method is that no data path contains more than one
22 * multiplication; this allows a very simple and accurate implementation in
23 * scaled fixed-point arithmetic, with a minimal number of shifts.
24 */
25
26#define JPEG_INTERNALS
27#include "jinclude.h"
28#include "jpeglib.h"
29#include "jdct.h" /* Private declarations for DCT subsystem */
30
31#ifdef DCT_ISLOW_SUPPORTED
32
33
34/*
35 * This module is specialized to the case DCTSIZE = 8.
36 */
37
38#if DCTSIZE != 8
39 Sorry, this code only copes with 8x8 DCTs. /* deliberate syntax err */
40#endif
41
42
43/*
44 * The poop on this scaling stuff is as follows:
45 *
46 * Each 1-D DCT step produces outputs which are a factor of sqrt(N)
47 * larger than the true DCT outputs. The final outputs are therefore
48 * a factor of N larger than desired; since N=8 this can be cured by
49 * a simple right shift at the end of the algorithm. The advantage of
50 * this arrangement is that we save two multiplications per 1-D DCT,
51 * because the y0 and y4 outputs need not be divided by sqrt(N).
52 * In the IJG code, this factor of 8 is removed by the quantization step
53 * (in jcdctmgr.c), NOT in this module.
54 *
55 * We have to do addition and subtraction of the integer inputs, which
56 * is no problem, and multiplication by fractional constants, which is
57 * a problem to do in integer arithmetic. We multiply all the constants
58 * by CONST_SCALE and convert them to integer constants (thus retaining
59 * CONST_BITS bits of precision in the constants). After doing a
60 * multiplication we have to divide the product by CONST_SCALE, with proper
61 * rounding, to produce the correct output. This division can be done
62 * cheaply as a right shift of CONST_BITS bits. We postpone shifting
63 * as long as possible so that partial sums can be added together with
64 * full fractional precision.
65 *
66 * The outputs of the first pass are scaled up by PASS1_BITS bits so that
67 * they are represented to better-than-integral precision. These outputs
68 * require BITS_IN_JSAMPLE + PASS1_BITS + 3 bits; this fits in a 16-bit word
69 * with the recommended scaling. (For 12-bit sample data, the intermediate
70 * array is INT32 anyway.)
71 *
72 * To avoid overflow of the 32-bit intermediate results in pass 2, we must
73 * have BITS_IN_JSAMPLE + CONST_BITS + PASS1_BITS <= 26. Error analysis
74 * shows that the values given below are the most effective.
75 */
76
77#if BITS_IN_JSAMPLE == 8
78#define CONST_BITS 13
79#define PASS1_BITS 2
80#else
81#define CONST_BITS 13
82#define PASS1_BITS 1 /* lose a little precision to avoid overflow */
83#endif
84
85/* Some C compilers fail to reduce "FIX(constant)" at compile time, thus
86 * causing a lot of useless floating-point operations at run time.
87 * To get around this we use the following pre-calculated constants.
88 * If you change CONST_BITS you may want to add appropriate values.
89 * (With a reasonable C compiler, you can just rely on the FIX() macro...)
90 */
91
92#if CONST_BITS == 13
93#define FIX_0_298631336 ((INT32) 2446) /* FIX(0.298631336) */
94#define FIX_0_390180644 ((INT32) 3196) /* FIX(0.390180644) */
95#define FIX_0_541196100 ((INT32) 4433) /* FIX(0.541196100) */
96#define FIX_0_765366865 ((INT32) 6270) /* FIX(0.765366865) */
97#define FIX_0_899976223 ((INT32) 7373) /* FIX(0.899976223) */
98#define FIX_1_175875602 ((INT32) 9633) /* FIX(1.175875602) */
99#define FIX_1_501321110 ((INT32) 12299) /* FIX(1.501321110) */
100#define FIX_1_847759065 ((INT32) 15137) /* FIX(1.847759065) */
101#define FIX_1_961570560 ((INT32) 16069) /* FIX(1.961570560) */
102#define FIX_2_053119869 ((INT32) 16819) /* FIX(2.053119869) */
103#define FIX_2_562915447 ((INT32) 20995) /* FIX(2.562915447) */
104#define FIX_3_072711026 ((INT32) 25172) /* FIX(3.072711026) */
105#else
106#define FIX_0_298631336 FIX(0.298631336)
107#define FIX_0_390180644 FIX(0.390180644)
108#define FIX_0_541196100 FIX(0.541196100)
109#define FIX_0_765366865 FIX(0.765366865)
110#define FIX_0_899976223 FIX(0.899976223)
111#define FIX_1_175875602 FIX(1.175875602)
112#define FIX_1_501321110 FIX(1.501321110)
113#define FIX_1_847759065 FIX(1.847759065)
114#define FIX_1_961570560 FIX(1.961570560)
115#define FIX_2_053119869 FIX(2.053119869)
116#define FIX_2_562915447 FIX(2.562915447)
117#define FIX_3_072711026 FIX(3.072711026)
118#endif
119
120
121/* Multiply an INT32 variable by an INT32 constant to yield an INT32 result.
122 * For 8-bit samples with the recommended scaling, all the variable
123 * and constant values involved are no more than 16 bits wide, so a
124 * 16x16->32 bit multiply can be used instead of a full 32x32 multiply.
125 * For 12-bit samples, a full 32-bit multiplication will be needed.
126 */
127
128#if BITS_IN_JSAMPLE == 8
129#define MULTIPLY(var,const) MULTIPLY16C16(var,const)
130#else
131#define MULTIPLY(var,const) ((var) * (const))
132#endif
133
134
135/*
136 * Perform the forward DCT on one block of samples.
137 */
138
139GLOBAL(void)
140jpeg_fdct_islow (DCTELEM * data)
141{
142 INT32 tmp0, tmp1, tmp2, tmp3, tmp4, tmp5, tmp6, tmp7;
143 INT32 tmp10, tmp11, tmp12, tmp13;
144 INT32 z1, z2, z3, z4, z5;
145 DCTELEM *dataptr;
146 int ctr;
147 SHIFT_TEMPS
148
149 /* Pass 1: process rows. */
150 /* Note results are scaled up by sqrt(8) compared to a true DCT; */
151 /* furthermore, we scale the results by 2**PASS1_BITS. */
152
153 dataptr = data;
154 for (ctr = DCTSIZE-1; ctr >= 0; ctr--) {
155 tmp0 = dataptr[0] + dataptr[7];
156 tmp7 = dataptr[0] - dataptr[7];
157 tmp1 = dataptr[1] + dataptr[6];
158 tmp6 = dataptr[1] - dataptr[6];
159 tmp2 = dataptr[2] + dataptr[5];
160 tmp5 = dataptr[2] - dataptr[5];
161 tmp3 = dataptr[3] + dataptr[4];
162 tmp4 = dataptr[3] - dataptr[4];
163
164 /* Even part per LL&M figure 1 --- note that published figure is faulty;
165 * rotator "sqrt(2)*c1" should be "sqrt(2)*c6".
166 */
167
168 tmp10 = tmp0 + tmp3;
169 tmp13 = tmp0 - tmp3;
170 tmp11 = tmp1 + tmp2;
171 tmp12 = tmp1 - tmp2;
172
173 dataptr[0] = (DCTELEM) ((tmp10 + tmp11) << PASS1_BITS);
174 dataptr[4] = (DCTELEM) ((tmp10 - tmp11) << PASS1_BITS);
175
176 z1 = MULTIPLY(tmp12 + tmp13, FIX_0_541196100);
177 dataptr[2] = (DCTELEM) DESCALE(z1 + MULTIPLY(tmp13, FIX_0_765366865),
178 CONST_BITS-PASS1_BITS);
179 dataptr[6] = (DCTELEM) DESCALE(z1 + MULTIPLY(tmp12, - FIX_1_847759065),
180 CONST_BITS-PASS1_BITS);
181
182 /* Odd part per figure 8 --- note paper omits factor of sqrt(2).
183 * cK represents cos(K*pi/16).
184 * i0..i3 in the paper are tmp4..tmp7 here.
185 */
186
187 z1 = tmp4 + tmp7;
188 z2 = tmp5 + tmp6;
189 z3 = tmp4 + tmp6;
190 z4 = tmp5 + tmp7;
191 z5 = MULTIPLY(z3 + z4, FIX_1_175875602); /* sqrt(2) * c3 */
192
193 tmp4 = MULTIPLY(tmp4, FIX_0_298631336); /* sqrt(2) * (-c1+c3+c5-c7) */
194 tmp5 = MULTIPLY(tmp5, FIX_2_053119869); /* sqrt(2) * ( c1+c3-c5+c7) */
195 tmp6 = MULTIPLY(tmp6, FIX_3_072711026); /* sqrt(2) * ( c1+c3+c5-c7) */
196 tmp7 = MULTIPLY(tmp7, FIX_1_501321110); /* sqrt(2) * ( c1+c3-c5-c7) */
197 z1 = MULTIPLY(z1, - FIX_0_899976223); /* sqrt(2) * (c7-c3) */
198 z2 = MULTIPLY(z2, - FIX_2_562915447); /* sqrt(2) * (-c1-c3) */
199 z3 = MULTIPLY(z3, - FIX_1_961570560); /* sqrt(2) * (-c3-c5) */
200 z4 = MULTIPLY(z4, - FIX_0_390180644); /* sqrt(2) * (c5-c3) */
201
202 z3 += z5;
203 z4 += z5;
204
205 dataptr[7] = (DCTELEM) DESCALE(tmp4 + z1 + z3, CONST_BITS-PASS1_BITS);
206 dataptr[5] = (DCTELEM) DESCALE(tmp5 + z2 + z4, CONST_BITS-PASS1_BITS);
207 dataptr[3] = (DCTELEM) DESCALE(tmp6 + z2 + z3, CONST_BITS-PASS1_BITS);
208 dataptr[1] = (DCTELEM) DESCALE(tmp7 + z1 + z4, CONST_BITS-PASS1_BITS);
209
210 dataptr += DCTSIZE; /* advance pointer to next row */
211 }
212
213 /* Pass 2: process columns.
214 * We remove the PASS1_BITS scaling, but leave the results scaled up
215 * by an overall factor of 8.
216 */
217
218 dataptr = data;
219 for (ctr = DCTSIZE-1; ctr >= 0; ctr--) {
220 tmp0 = dataptr[DCTSIZE*0] + dataptr[DCTSIZE*7];
221 tmp7 = dataptr[DCTSIZE*0] - dataptr[DCTSIZE*7];
222 tmp1 = dataptr[DCTSIZE*1] + dataptr[DCTSIZE*6];
223 tmp6 = dataptr[DCTSIZE*1] - dataptr[DCTSIZE*6];
224 tmp2 = dataptr[DCTSIZE*2] + dataptr[DCTSIZE*5];
225 tmp5 = dataptr[DCTSIZE*2] - dataptr[DCTSIZE*5];
226 tmp3 = dataptr[DCTSIZE*3] + dataptr[DCTSIZE*4];
227 tmp4 = dataptr[DCTSIZE*3] - dataptr[DCTSIZE*4];
228
229 /* Even part per LL&M figure 1 --- note that published figure is faulty;
230 * rotator "sqrt(2)*c1" should be "sqrt(2)*c6".
231 */
232
233 tmp10 = tmp0 + tmp3;
234 tmp13 = tmp0 - tmp3;
235 tmp11 = tmp1 + tmp2;
236 tmp12 = tmp1 - tmp2;
237
238 dataptr[DCTSIZE*0] = (DCTELEM) DESCALE(tmp10 + tmp11, PASS1_BITS);
239 dataptr[DCTSIZE*4] = (DCTELEM) DESCALE(tmp10 - tmp11, PASS1_BITS);
240
241 z1 = MULTIPLY(tmp12 + tmp13, FIX_0_541196100);
242 dataptr[DCTSIZE*2] = (DCTELEM) DESCALE(z1 + MULTIPLY(tmp13, FIX_0_765366865),
243 CONST_BITS+PASS1_BITS);
244 dataptr[DCTSIZE*6] = (DCTELEM) DESCALE(z1 + MULTIPLY(tmp12, - FIX_1_847759065),
245 CONST_BITS+PASS1_BITS);
246
247 /* Odd part per figure 8 --- note paper omits factor of sqrt(2).
248 * cK represents cos(K*pi/16).
249 * i0..i3 in the paper are tmp4..tmp7 here.
250 */
251
252 z1 = tmp4 + tmp7;
253 z2 = tmp5 + tmp6;
254 z3 = tmp4 + tmp6;
255 z4 = tmp5 + tmp7;
256 z5 = MULTIPLY(z3 + z4, FIX_1_175875602); /* sqrt(2) * c3 */
257
258 tmp4 = MULTIPLY(tmp4, FIX_0_298631336); /* sqrt(2) * (-c1+c3+c5-c7) */
259 tmp5 = MULTIPLY(tmp5, FIX_2_053119869); /* sqrt(2) * ( c1+c3-c5+c7) */
260 tmp6 = MULTIPLY(tmp6, FIX_3_072711026); /* sqrt(2) * ( c1+c3+c5-c7) */
261 tmp7 = MULTIPLY(tmp7, FIX_1_501321110); /* sqrt(2) * ( c1+c3-c5-c7) */
262 z1 = MULTIPLY(z1, - FIX_0_899976223); /* sqrt(2) * (c7-c3) */
263 z2 = MULTIPLY(z2, - FIX_2_562915447); /* sqrt(2) * (-c1-c3) */
264 z3 = MULTIPLY(z3, - FIX_1_961570560); /* sqrt(2) * (-c3-c5) */
265 z4 = MULTIPLY(z4, - FIX_0_390180644); /* sqrt(2) * (c5-c3) */
266
267 z3 += z5;
268 z4 += z5;
269
270 dataptr[DCTSIZE*7] = (DCTELEM) DESCALE(tmp4 + z1 + z3,
271 CONST_BITS+PASS1_BITS);
272 dataptr[DCTSIZE*5] = (DCTELEM) DESCALE(tmp5 + z2 + z4,
273 CONST_BITS+PASS1_BITS);
274 dataptr[DCTSIZE*3] = (DCTELEM) DESCALE(tmp6 + z2 + z3,
275 CONST_BITS+PASS1_BITS);
276 dataptr[DCTSIZE*1] = (DCTELEM) DESCALE(tmp7 + z1 + z4,
277 CONST_BITS+PASS1_BITS);
278
279 dataptr++; /* advance pointer to next column */
280 }
281}
282
283#endif /* DCT_ISLOW_SUPPORTED */
284