1/*
2 * jidctint.c
3 *
4 * Copyright (C) 1991-1998, 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 * inverse DCT (Discrete Cosine Transform). In the IJG code, this routine
10 * must also perform dequantization of the input coefficients.
11 *
12 * A 2-D IDCT can be done by 1-D IDCT on each column followed by 1-D IDCT
13 * on each row (or vice versa, but it's more convenient to emit a row at
14 * a time). Direct algorithms are also available, but they are much more
15 * complex and seem not to be any faster when reduced to code.
16 *
17 * This implementation is based on an algorithm described in
18 * C. Loeffler, A. Ligtenberg and G. Moschytz, "Practical Fast 1-D DCT
19 * Algorithms with 11 Multiplications", Proc. Int'l. Conf. on Acoustics,
20 * Speech, and Signal Processing 1989 (ICASSP '89), pp. 988-991.
21 * The primary algorithm described there uses 11 multiplies and 29 adds.
22 * We use their alternate method with 12 multiplies and 32 adds.
23 * The advantage of this method is that no data path contains more than one
24 * multiplication; this allows a very simple and accurate implementation in
25 * scaled fixed-point arithmetic, with a minimal number of shifts.
26 */
27
28#define JPEG_INTERNALS
29#include "jinclude.h"
30#include "jpeglib.h"
31#include "jdct.h" /* Private declarations for DCT subsystem */
32
33#ifdef DCT_ISLOW_SUPPORTED
34
35
36/*
37 * This module is specialized to the case DCTSIZE = 8.
38 */
39
40#if DCTSIZE != 8
41 Sorry, this code only copes with 8x8 DCTs. /* deliberate syntax err */
42#endif
43
44
45/*
46 * The poop on this scaling stuff is as follows:
47 *
48 * Each 1-D IDCT step produces outputs which are a factor of sqrt(N)
49 * larger than the true IDCT outputs. The final outputs are therefore
50 * a factor of N larger than desired; since N=8 this can be cured by
51 * a simple right shift at the end of the algorithm. The advantage of
52 * this arrangement is that we save two multiplications per 1-D IDCT,
53 * because the y0 and y4 inputs need not be divided by sqrt(N).
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. (To scale up 12-bit sample data further, an
70 * intermediate INT32 array would be needed.)
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/* Dequantize a coefficient by multiplying it by the multiplier-table
136 * entry; produce an int result. In this module, both inputs and result
137 * are 16 bits or less, so either int or short multiply will work.
138 */
139
140#define DEQUANTIZE(coef,quantval) (((ISLOW_MULT_TYPE) (coef)) * (quantval))
141
142
143/*
144 * Perform dequantization and inverse DCT on one block of coefficients.
145 */
146
147GLOBAL(void)
148jpeg_idct_islow (j_decompress_ptr cinfo, jpeg_component_info * compptr,
149 JCOEFPTR coef_block,
150 JSAMPARRAY output_buf, JDIMENSION output_col)
151{
152 INT32 tmp0, tmp1, tmp2, tmp3;
153 INT32 tmp10, tmp11, tmp12, tmp13;
154 INT32 z1, z2, z3, z4, z5;
155 JCOEFPTR inptr;
156 ISLOW_MULT_TYPE * quantptr;
157 int * wsptr;
158 JSAMPROW outptr;
159 JSAMPLE *range_limit = IDCT_range_limit(cinfo);
160 int ctr;
161 int workspace[DCTSIZE2]; /* buffers data between passes */
162 SHIFT_TEMPS
163
164 /* Pass 1: process columns from input, store into work array. */
165 /* Note results are scaled up by sqrt(8) compared to a true IDCT; */
166 /* furthermore, we scale the results by 2**PASS1_BITS. */
167
168 inptr = coef_block;
169 quantptr = (ISLOW_MULT_TYPE *) compptr->dct_table;
170 wsptr = workspace;
171 for (ctr = DCTSIZE; ctr > 0; ctr--) {
172 /* Due to quantization, we will usually find that many of the input
173 * coefficients are zero, especially the AC terms. We can exploit this
174 * by short-circuiting the IDCT calculation for any column in which all
175 * the AC terms are zero. In that case each output is equal to the
176 * DC coefficient (with scale factor as needed).
177 * With typical images and quantization tables, half or more of the
178 * column DCT calculations can be simplified this way.
179 */
180
181 if (inptr[DCTSIZE*1] == 0 && inptr[DCTSIZE*2] == 0 &&
182 inptr[DCTSIZE*3] == 0 && inptr[DCTSIZE*4] == 0 &&
183 inptr[DCTSIZE*5] == 0 && inptr[DCTSIZE*6] == 0 &&
184 inptr[DCTSIZE*7] == 0) {
185 /* AC terms all zero */
186 int dcval = DEQUANTIZE(inptr[DCTSIZE*0], quantptr[DCTSIZE*0]) << PASS1_BITS;
187
188 wsptr[DCTSIZE*0] = dcval;
189 wsptr[DCTSIZE*1] = dcval;
190 wsptr[DCTSIZE*2] = dcval;
191 wsptr[DCTSIZE*3] = dcval;
192 wsptr[DCTSIZE*4] = dcval;
193 wsptr[DCTSIZE*5] = dcval;
194 wsptr[DCTSIZE*6] = dcval;
195 wsptr[DCTSIZE*7] = dcval;
196
197 inptr++; /* advance pointers to next column */
198 quantptr++;
199 wsptr++;
200 continue;
201 }
202
203 /* Even part: reverse the even part of the forward DCT. */
204 /* The rotator is sqrt(2)*c(-6). */
205
206 z2 = DEQUANTIZE(inptr[DCTSIZE*2], quantptr[DCTSIZE*2]);
207 z3 = DEQUANTIZE(inptr[DCTSIZE*6], quantptr[DCTSIZE*6]);
208
209 z1 = MULTIPLY(z2 + z3, FIX_0_541196100);
210 tmp2 = z1 + MULTIPLY(z3, - FIX_1_847759065);
211 tmp3 = z1 + MULTIPLY(z2, FIX_0_765366865);
212
213 z2 = DEQUANTIZE(inptr[DCTSIZE*0], quantptr[DCTSIZE*0]);
214 z3 = DEQUANTIZE(inptr[DCTSIZE*4], quantptr[DCTSIZE*4]);
215
216 tmp0 = (z2 + z3) << CONST_BITS;
217 tmp1 = (z2 - z3) << CONST_BITS;
218
219 tmp10 = tmp0 + tmp3;
220 tmp13 = tmp0 - tmp3;
221 tmp11 = tmp1 + tmp2;
222 tmp12 = tmp1 - tmp2;
223
224 /* Odd part per figure 8; the matrix is unitary and hence its
225 * transpose is its inverse. i0..i3 are y7,y5,y3,y1 respectively.
226 */
227
228 tmp0 = DEQUANTIZE(inptr[DCTSIZE*7], quantptr[DCTSIZE*7]);
229 tmp1 = DEQUANTIZE(inptr[DCTSIZE*5], quantptr[DCTSIZE*5]);
230 tmp2 = DEQUANTIZE(inptr[DCTSIZE*3], quantptr[DCTSIZE*3]);
231 tmp3 = DEQUANTIZE(inptr[DCTSIZE*1], quantptr[DCTSIZE*1]);
232
233 z1 = tmp0 + tmp3;
234 z2 = tmp1 + tmp2;
235 z3 = tmp0 + tmp2;
236 z4 = tmp1 + tmp3;
237 z5 = MULTIPLY(z3 + z4, FIX_1_175875602); /* sqrt(2) * c3 */
238
239 tmp0 = MULTIPLY(tmp0, FIX_0_298631336); /* sqrt(2) * (-c1+c3+c5-c7) */
240 tmp1 = MULTIPLY(tmp1, FIX_2_053119869); /* sqrt(2) * ( c1+c3-c5+c7) */
241 tmp2 = MULTIPLY(tmp2, FIX_3_072711026); /* sqrt(2) * ( c1+c3+c5-c7) */
242 tmp3 = MULTIPLY(tmp3, FIX_1_501321110); /* sqrt(2) * ( c1+c3-c5-c7) */
243 z1 = MULTIPLY(z1, - FIX_0_899976223); /* sqrt(2) * (c7-c3) */
244 z2 = MULTIPLY(z2, - FIX_2_562915447); /* sqrt(2) * (-c1-c3) */
245 z3 = MULTIPLY(z3, - FIX_1_961570560); /* sqrt(2) * (-c3-c5) */
246 z4 = MULTIPLY(z4, - FIX_0_390180644); /* sqrt(2) * (c5-c3) */
247
248 z3 += z5;
249 z4 += z5;
250
251 tmp0 += z1 + z3;
252 tmp1 += z2 + z4;
253 tmp2 += z2 + z3;
254 tmp3 += z1 + z4;
255
256 /* Final output stage: inputs are tmp10..tmp13, tmp0..tmp3 */
257
258 wsptr[DCTSIZE*0] = (int) DESCALE(tmp10 + tmp3, CONST_BITS-PASS1_BITS);
259 wsptr[DCTSIZE*7] = (int) DESCALE(tmp10 - tmp3, CONST_BITS-PASS1_BITS);
260 wsptr[DCTSIZE*1] = (int) DESCALE(tmp11 + tmp2, CONST_BITS-PASS1_BITS);
261 wsptr[DCTSIZE*6] = (int) DESCALE(tmp11 - tmp2, CONST_BITS-PASS1_BITS);
262 wsptr[DCTSIZE*2] = (int) DESCALE(tmp12 + tmp1, CONST_BITS-PASS1_BITS);
263 wsptr[DCTSIZE*5] = (int) DESCALE(tmp12 - tmp1, CONST_BITS-PASS1_BITS);
264 wsptr[DCTSIZE*3] = (int) DESCALE(tmp13 + tmp0, CONST_BITS-PASS1_BITS);
265 wsptr[DCTSIZE*4] = (int) DESCALE(tmp13 - tmp0, CONST_BITS-PASS1_BITS);
266
267 inptr++; /* advance pointers to next column */
268 quantptr++;
269 wsptr++;
270 }
271
272 /* Pass 2: process rows from work array, store into output array. */
273 /* Note that we must descale the results by a factor of 8 == 2**3, */
274 /* and also undo the PASS1_BITS scaling. */
275
276 wsptr = workspace;
277 for (ctr = 0; ctr < DCTSIZE; ctr++) {
278 outptr = output_buf[ctr] + output_col;
279 /* Rows of zeroes can be exploited in the same way as we did with columns.
280 * However, the column calculation has created many nonzero AC terms, so
281 * the simplification applies less often (typically 5% to 10% of the time).
282 * On machines with very fast multiplication, it's possible that the
283 * test takes more time than it's worth. In that case this section
284 * may be commented out.
285 */
286
287#ifndef NO_ZERO_ROW_TEST
288 if (wsptr[1] == 0 && wsptr[2] == 0 && wsptr[3] == 0 && wsptr[4] == 0 &&
289 wsptr[5] == 0 && wsptr[6] == 0 && wsptr[7] == 0) {
290 /* AC terms all zero */
291 JSAMPLE dcval = range_limit[(int) DESCALE((INT32) wsptr[0], PASS1_BITS+3)
292 & RANGE_MASK];
293
294 outptr[0] = dcval;
295 outptr[1] = dcval;
296 outptr[2] = dcval;
297 outptr[3] = dcval;
298 outptr[4] = dcval;
299 outptr[5] = dcval;
300 outptr[6] = dcval;
301 outptr[7] = dcval;
302
303 wsptr += DCTSIZE; /* advance pointer to next row */
304 continue;
305 }
306#endif
307
308 /* Even part: reverse the even part of the forward DCT. */
309 /* The rotator is sqrt(2)*c(-6). */
310
311 z2 = (INT32) wsptr[2];
312 z3 = (INT32) wsptr[6];
313
314 z1 = MULTIPLY(z2 + z3, FIX_0_541196100);
315 tmp2 = z1 + MULTIPLY(z3, - FIX_1_847759065);
316 tmp3 = z1 + MULTIPLY(z2, FIX_0_765366865);
317
318 tmp0 = ((INT32) wsptr[0] + (INT32) wsptr[4]) << CONST_BITS;
319 tmp1 = ((INT32) wsptr[0] - (INT32) wsptr[4]) << CONST_BITS;
320
321 tmp10 = tmp0 + tmp3;
322 tmp13 = tmp0 - tmp3;
323 tmp11 = tmp1 + tmp2;
324 tmp12 = tmp1 - tmp2;
325
326 /* Odd part per figure 8; the matrix is unitary and hence its
327 * transpose is its inverse. i0..i3 are y7,y5,y3,y1 respectively.
328 */
329
330 tmp0 = (INT32) wsptr[7];
331 tmp1 = (INT32) wsptr[5];
332 tmp2 = (INT32) wsptr[3];
333 tmp3 = (INT32) wsptr[1];
334
335 z1 = tmp0 + tmp3;
336 z2 = tmp1 + tmp2;
337 z3 = tmp0 + tmp2;
338 z4 = tmp1 + tmp3;
339 z5 = MULTIPLY(z3 + z4, FIX_1_175875602); /* sqrt(2) * c3 */
340
341 tmp0 = MULTIPLY(tmp0, FIX_0_298631336); /* sqrt(2) * (-c1+c3+c5-c7) */
342 tmp1 = MULTIPLY(tmp1, FIX_2_053119869); /* sqrt(2) * ( c1+c3-c5+c7) */
343 tmp2 = MULTIPLY(tmp2, FIX_3_072711026); /* sqrt(2) * ( c1+c3+c5-c7) */
344 tmp3 = MULTIPLY(tmp3, FIX_1_501321110); /* sqrt(2) * ( c1+c3-c5-c7) */
345 z1 = MULTIPLY(z1, - FIX_0_899976223); /* sqrt(2) * (c7-c3) */
346 z2 = MULTIPLY(z2, - FIX_2_562915447); /* sqrt(2) * (-c1-c3) */
347 z3 = MULTIPLY(z3, - FIX_1_961570560); /* sqrt(2) * (-c3-c5) */
348 z4 = MULTIPLY(z4, - FIX_0_390180644); /* sqrt(2) * (c5-c3) */
349
350 z3 += z5;
351 z4 += z5;
352
353 tmp0 += z1 + z3;
354 tmp1 += z2 + z4;
355 tmp2 += z2 + z3;
356 tmp3 += z1 + z4;
357
358 /* Final output stage: inputs are tmp10..tmp13, tmp0..tmp3 */
359
360 outptr[0] = range_limit[(int) DESCALE(tmp10 + tmp3,
361 CONST_BITS+PASS1_BITS+3)
362 & RANGE_MASK];
363 outptr[7] = range_limit[(int) DESCALE(tmp10 - tmp3,
364 CONST_BITS+PASS1_BITS+3)
365 & RANGE_MASK];
366 outptr[1] = range_limit[(int) DESCALE(tmp11 + tmp2,
367 CONST_BITS+PASS1_BITS+3)
368 & RANGE_MASK];
369 outptr[6] = range_limit[(int) DESCALE(tmp11 - tmp2,
370 CONST_BITS+PASS1_BITS+3)
371 & RANGE_MASK];
372 outptr[2] = range_limit[(int) DESCALE(tmp12 + tmp1,
373 CONST_BITS+PASS1_BITS+3)
374 & RANGE_MASK];
375 outptr[5] = range_limit[(int) DESCALE(tmp12 - tmp1,
376 CONST_BITS+PASS1_BITS+3)
377 & RANGE_MASK];
378 outptr[3] = range_limit[(int) DESCALE(tmp13 + tmp0,
379 CONST_BITS+PASS1_BITS+3)
380 & RANGE_MASK];
381 outptr[4] = range_limit[(int) DESCALE(tmp13 - tmp0,
382 CONST_BITS+PASS1_BITS+3)
383 & RANGE_MASK];
384
385 wsptr += DCTSIZE; /* advance pointer to next row */
386 }
387}
388
389#endif /* DCT_ISLOW_SUPPORTED */
390