2 * CDE - Common Desktop Environment
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23 /* $XConsortium: jidctint.c /main/2 1996/05/09 03:51:27 drk $ */
27 * Copyright (C) 1991-1996, Thomas G. Lane.
28 * This file is part of the Independent JPEG Group's software.
29 * For conditions of distribution and use, see the accompanying README file.
31 * This file contains a slow-but-accurate integer implementation of the
32 * inverse DCT (Discrete Cosine Transform). In the IJG code, this routine
33 * must also perform dequantization of the input coefficients.
35 * A 2-D IDCT can be done by 1-D IDCT on each column followed by 1-D IDCT
36 * on each row (or vice versa, but it's more convenient to emit a row at
37 * a time). Direct algorithms are also available, but they are much more
38 * complex and seem not to be any faster when reduced to code.
40 * This implementation is based on an algorithm described in
41 * C. Loeffler, A. Ligtenberg and G. Moschytz, "Practical Fast 1-D DCT
42 * Algorithms with 11 Multiplications", Proc. Int'l. Conf. on Acoustics,
43 * Speech, and Signal Processing 1989 (ICASSP '89), pp. 988-991.
44 * The primary algorithm described there uses 11 multiplies and 29 adds.
45 * We use their alternate method with 12 multiplies and 32 adds.
46 * The advantage of this method is that no data path contains more than one
47 * multiplication; this allows a very simple and accurate implementation in
48 * scaled fixed-point arithmetic, with a minimal number of shifts.
51 #define JPEG_INTERNALS
54 #include "jdct.h" /* Private declarations for DCT subsystem */
56 #ifdef DCT_ISLOW_SUPPORTED
60 * This module is specialized to the case DCTSIZE = 8.
64 Sorry, this code only copes with 8x8 DCTs. /* deliberate syntax err */
69 * The poop on this scaling stuff is as follows:
71 * Each 1-D IDCT step produces outputs which are a factor of sqrt(N)
72 * larger than the true IDCT outputs. The final outputs are therefore
73 * a factor of N larger than desired; since N=8 this can be cured by
74 * a simple right shift at the end of the algorithm. The advantage of
75 * this arrangement is that we save two multiplications per 1-D IDCT,
76 * because the y0 and y4 inputs need not be divided by sqrt(N).
78 * We have to do addition and subtraction of the integer inputs, which
79 * is no problem, and multiplication by fractional constants, which is
80 * a problem to do in integer arithmetic. We multiply all the constants
81 * by CONST_SCALE and convert them to integer constants (thus retaining
82 * CONST_BITS bits of precision in the constants). After doing a
83 * multiplication we have to divide the product by CONST_SCALE, with proper
84 * rounding, to produce the correct output. This division can be done
85 * cheaply as a right shift of CONST_BITS bits. We postpone shifting
86 * as long as possible so that partial sums can be added together with
87 * full fractional precision.
89 * The outputs of the first pass are scaled up by PASS1_BITS bits so that
90 * they are represented to better-than-integral precision. These outputs
91 * require BITS_IN_JSAMPLE + PASS1_BITS + 3 bits; this fits in a 16-bit word
92 * with the recommended scaling. (To scale up 12-bit sample data further, an
93 * intermediate INT32 array would be needed.)
95 * To avoid overflow of the 32-bit intermediate results in pass 2, we must
96 * have BITS_IN_JSAMPLE + CONST_BITS + PASS1_BITS <= 26. Error analysis
97 * shows that the values given below are the most effective.
100 #if BITS_IN_JSAMPLE == 8
101 #define CONST_BITS 13
104 #define CONST_BITS 13
105 #define PASS1_BITS 1 /* lose a little precision to avoid overflow */
108 /* Some C compilers fail to reduce "FIX(constant)" at compile time, thus
109 * causing a lot of useless floating-point operations at run time.
110 * To get around this we use the following pre-calculated constants.
111 * If you change CONST_BITS you may want to add appropriate values.
112 * (With a reasonable C compiler, you can just rely on the FIX() macro...)
116 #define FIX_0_298631336 ((INT32) 2446) /* FIX(0.298631336) */
117 #define FIX_0_390180644 ((INT32) 3196) /* FIX(0.390180644) */
118 #define FIX_0_541196100 ((INT32) 4433) /* FIX(0.541196100) */
119 #define FIX_0_765366865 ((INT32) 6270) /* FIX(0.765366865) */
120 #define FIX_0_899976223 ((INT32) 7373) /* FIX(0.899976223) */
121 #define FIX_1_175875602 ((INT32) 9633) /* FIX(1.175875602) */
122 #define FIX_1_501321110 ((INT32) 12299) /* FIX(1.501321110) */
123 #define FIX_1_847759065 ((INT32) 15137) /* FIX(1.847759065) */
124 #define FIX_1_961570560 ((INT32) 16069) /* FIX(1.961570560) */
125 #define FIX_2_053119869 ((INT32) 16819) /* FIX(2.053119869) */
126 #define FIX_2_562915447 ((INT32) 20995) /* FIX(2.562915447) */
127 #define FIX_3_072711026 ((INT32) 25172) /* FIX(3.072711026) */
129 #define FIX_0_298631336 FIX(0.298631336)
130 #define FIX_0_390180644 FIX(0.390180644)
131 #define FIX_0_541196100 FIX(0.541196100)
132 #define FIX_0_765366865 FIX(0.765366865)
133 #define FIX_0_899976223 FIX(0.899976223)
134 #define FIX_1_175875602 FIX(1.175875602)
135 #define FIX_1_501321110 FIX(1.501321110)
136 #define FIX_1_847759065 FIX(1.847759065)
137 #define FIX_1_961570560 FIX(1.961570560)
138 #define FIX_2_053119869 FIX(2.053119869)
139 #define FIX_2_562915447 FIX(2.562915447)
140 #define FIX_3_072711026 FIX(3.072711026)
144 /* Multiply an INT32 variable by an INT32 constant to yield an INT32 result.
145 * For 8-bit samples with the recommended scaling, all the variable
146 * and constant values involved are no more than 16 bits wide, so a
147 * 16x16->32 bit multiply can be used instead of a full 32x32 multiply.
148 * For 12-bit samples, a full 32-bit multiplication will be needed.
151 #if BITS_IN_JSAMPLE == 8
152 #define MULTIPLY(var,const) MULTIPLY16C16(var,const)
154 #define MULTIPLY(var,const) ((var) * (const))
158 /* Dequantize a coefficient by multiplying it by the multiplier-table
159 * entry; produce an int result. In this module, both inputs and result
160 * are 16 bits or less, so either int or short multiply will work.
163 #define DEQUANTIZE(coef,quantval) (((ISLOW_MULT_TYPE) (coef)) * (quantval))
167 * Perform dequantization and inverse DCT on one block of coefficients.
171 jpeg_idct_islow (j_decompress_ptr cinfo, jpeg_component_info * compptr,
173 JSAMPARRAY output_buf, JDIMENSION output_col)
175 INT32 tmp0, tmp1, tmp2, tmp3;
176 INT32 tmp10, tmp11, tmp12, tmp13;
177 INT32 z1, z2, z3, z4, z5;
179 ISLOW_MULT_TYPE * quantptr;
182 JSAMPLE *range_limit = IDCT_range_limit(cinfo);
184 int workspace[DCTSIZE2]; /* buffers data between passes */
187 /* Pass 1: process columns from input, store into work array. */
188 /* Note results are scaled up by sqrt(8) compared to a true IDCT; */
189 /* furthermore, we scale the results by 2**PASS1_BITS. */
192 quantptr = (ISLOW_MULT_TYPE *) compptr->dct_table;
194 for (ctr = DCTSIZE; ctr > 0; ctr--) {
195 /* Due to quantization, we will usually find that many of the input
196 * coefficients are zero, especially the AC terms. We can exploit this
197 * by short-circuiting the IDCT calculation for any column in which all
198 * the AC terms are zero. In that case each output is equal to the
199 * DC coefficient (with scale factor as needed).
200 * With typical images and quantization tables, half or more of the
201 * column DCT calculations can be simplified this way.
204 if ((inptr[DCTSIZE*1] | inptr[DCTSIZE*2] | inptr[DCTSIZE*3] |
205 inptr[DCTSIZE*4] | inptr[DCTSIZE*5] | inptr[DCTSIZE*6] |
206 inptr[DCTSIZE*7]) == 0) {
207 /* AC terms all zero */
208 int dcval = DEQUANTIZE(inptr[DCTSIZE*0], quantptr[DCTSIZE*0]) << PASS1_BITS;
210 wsptr[DCTSIZE*0] = dcval;
211 wsptr[DCTSIZE*1] = dcval;
212 wsptr[DCTSIZE*2] = dcval;
213 wsptr[DCTSIZE*3] = dcval;
214 wsptr[DCTSIZE*4] = dcval;
215 wsptr[DCTSIZE*5] = dcval;
216 wsptr[DCTSIZE*6] = dcval;
217 wsptr[DCTSIZE*7] = dcval;
219 inptr++; /* advance pointers to next column */
225 /* Even part: reverse the even part of the forward DCT. */
226 /* The rotator is sqrt(2)*c(-6). */
228 z2 = DEQUANTIZE(inptr[DCTSIZE*2], quantptr[DCTSIZE*2]);
229 z3 = DEQUANTIZE(inptr[DCTSIZE*6], quantptr[DCTSIZE*6]);
231 z1 = MULTIPLY(z2 + z3, FIX_0_541196100);
232 tmp2 = z1 + MULTIPLY(z3, - FIX_1_847759065);
233 tmp3 = z1 + MULTIPLY(z2, FIX_0_765366865);
235 z2 = DEQUANTIZE(inptr[DCTSIZE*0], quantptr[DCTSIZE*0]);
236 z3 = DEQUANTIZE(inptr[DCTSIZE*4], quantptr[DCTSIZE*4]);
238 tmp0 = (z2 + z3) << CONST_BITS;
239 tmp1 = (z2 - z3) << CONST_BITS;
246 /* Odd part per figure 8; the matrix is unitary and hence its
247 * transpose is its inverse. i0..i3 are y7,y5,y3,y1 respectively.
250 tmp0 = DEQUANTIZE(inptr[DCTSIZE*7], quantptr[DCTSIZE*7]);
251 tmp1 = DEQUANTIZE(inptr[DCTSIZE*5], quantptr[DCTSIZE*5]);
252 tmp2 = DEQUANTIZE(inptr[DCTSIZE*3], quantptr[DCTSIZE*3]);
253 tmp3 = DEQUANTIZE(inptr[DCTSIZE*1], quantptr[DCTSIZE*1]);
259 z5 = MULTIPLY(z3 + z4, FIX_1_175875602); /* sqrt(2) * c3 */
261 tmp0 = MULTIPLY(tmp0, FIX_0_298631336); /* sqrt(2) * (-c1+c3+c5-c7) */
262 tmp1 = MULTIPLY(tmp1, FIX_2_053119869); /* sqrt(2) * ( c1+c3-c5+c7) */
263 tmp2 = MULTIPLY(tmp2, FIX_3_072711026); /* sqrt(2) * ( c1+c3+c5-c7) */
264 tmp3 = MULTIPLY(tmp3, FIX_1_501321110); /* sqrt(2) * ( c1+c3-c5-c7) */
265 z1 = MULTIPLY(z1, - FIX_0_899976223); /* sqrt(2) * (c7-c3) */
266 z2 = MULTIPLY(z2, - FIX_2_562915447); /* sqrt(2) * (-c1-c3) */
267 z3 = MULTIPLY(z3, - FIX_1_961570560); /* sqrt(2) * (-c3-c5) */
268 z4 = MULTIPLY(z4, - FIX_0_390180644); /* sqrt(2) * (c5-c3) */
278 /* Final output stage: inputs are tmp10..tmp13, tmp0..tmp3 */
280 wsptr[DCTSIZE*0] = (int) DESCALE(tmp10 + tmp3, CONST_BITS-PASS1_BITS);
281 wsptr[DCTSIZE*7] = (int) DESCALE(tmp10 - tmp3, CONST_BITS-PASS1_BITS);
282 wsptr[DCTSIZE*1] = (int) DESCALE(tmp11 + tmp2, CONST_BITS-PASS1_BITS);
283 wsptr[DCTSIZE*6] = (int) DESCALE(tmp11 - tmp2, CONST_BITS-PASS1_BITS);
284 wsptr[DCTSIZE*2] = (int) DESCALE(tmp12 + tmp1, CONST_BITS-PASS1_BITS);
285 wsptr[DCTSIZE*5] = (int) DESCALE(tmp12 - tmp1, CONST_BITS-PASS1_BITS);
286 wsptr[DCTSIZE*3] = (int) DESCALE(tmp13 + tmp0, CONST_BITS-PASS1_BITS);
287 wsptr[DCTSIZE*4] = (int) DESCALE(tmp13 - tmp0, CONST_BITS-PASS1_BITS);
289 inptr++; /* advance pointers to next column */
294 /* Pass 2: process rows from work array, store into output array. */
295 /* Note that we must descale the results by a factor of 8 == 2**3, */
296 /* and also undo the PASS1_BITS scaling. */
299 for (ctr = 0; ctr < DCTSIZE; ctr++) {
300 outptr = output_buf[ctr] + output_col;
301 /* Rows of zeroes can be exploited in the same way as we did with columns.
302 * However, the column calculation has created many nonzero AC terms, so
303 * the simplification applies less often (typically 5% to 10% of the time).
304 * On machines with very fast multiplication, it's possible that the
305 * test takes more time than it's worth. In that case this section
306 * may be commented out.
309 #ifndef NO_ZERO_ROW_TEST
310 if ((wsptr[1] | wsptr[2] | wsptr[3] | wsptr[4] | wsptr[5] | wsptr[6] |
312 /* AC terms all zero */
313 JSAMPLE dcval = range_limit[(int) DESCALE((INT32) wsptr[0], PASS1_BITS+3)
325 wsptr += DCTSIZE; /* advance pointer to next row */
330 /* Even part: reverse the even part of the forward DCT. */
331 /* The rotator is sqrt(2)*c(-6). */
333 z2 = (INT32) wsptr[2];
334 z3 = (INT32) wsptr[6];
336 z1 = MULTIPLY(z2 + z3, FIX_0_541196100);
337 tmp2 = z1 + MULTIPLY(z3, - FIX_1_847759065);
338 tmp3 = z1 + MULTIPLY(z2, FIX_0_765366865);
340 tmp0 = ((INT32) wsptr[0] + (INT32) wsptr[4]) << CONST_BITS;
341 tmp1 = ((INT32) wsptr[0] - (INT32) wsptr[4]) << CONST_BITS;
348 /* Odd part per figure 8; the matrix is unitary and hence its
349 * transpose is its inverse. i0..i3 are y7,y5,y3,y1 respectively.
352 tmp0 = (INT32) wsptr[7];
353 tmp1 = (INT32) wsptr[5];
354 tmp2 = (INT32) wsptr[3];
355 tmp3 = (INT32) wsptr[1];
361 z5 = MULTIPLY(z3 + z4, FIX_1_175875602); /* sqrt(2) * c3 */
363 tmp0 = MULTIPLY(tmp0, FIX_0_298631336); /* sqrt(2) * (-c1+c3+c5-c7) */
364 tmp1 = MULTIPLY(tmp1, FIX_2_053119869); /* sqrt(2) * ( c1+c3-c5+c7) */
365 tmp2 = MULTIPLY(tmp2, FIX_3_072711026); /* sqrt(2) * ( c1+c3+c5-c7) */
366 tmp3 = MULTIPLY(tmp3, FIX_1_501321110); /* sqrt(2) * ( c1+c3-c5-c7) */
367 z1 = MULTIPLY(z1, - FIX_0_899976223); /* sqrt(2) * (c7-c3) */
368 z2 = MULTIPLY(z2, - FIX_2_562915447); /* sqrt(2) * (-c1-c3) */
369 z3 = MULTIPLY(z3, - FIX_1_961570560); /* sqrt(2) * (-c3-c5) */
370 z4 = MULTIPLY(z4, - FIX_0_390180644); /* sqrt(2) * (c5-c3) */
380 /* Final output stage: inputs are tmp10..tmp13, tmp0..tmp3 */
382 outptr[0] = range_limit[(int) DESCALE(tmp10 + tmp3,
383 CONST_BITS+PASS1_BITS+3)
385 outptr[7] = range_limit[(int) DESCALE(tmp10 - tmp3,
386 CONST_BITS+PASS1_BITS+3)
388 outptr[1] = range_limit[(int) DESCALE(tmp11 + tmp2,
389 CONST_BITS+PASS1_BITS+3)
391 outptr[6] = range_limit[(int) DESCALE(tmp11 - tmp2,
392 CONST_BITS+PASS1_BITS+3)
394 outptr[2] = range_limit[(int) DESCALE(tmp12 + tmp1,
395 CONST_BITS+PASS1_BITS+3)
397 outptr[5] = range_limit[(int) DESCALE(tmp12 - tmp1,
398 CONST_BITS+PASS1_BITS+3)
400 outptr[3] = range_limit[(int) DESCALE(tmp13 + tmp0,
401 CONST_BITS+PASS1_BITS+3)
403 outptr[4] = range_limit[(int) DESCALE(tmp13 - tmp0,
404 CONST_BITS+PASS1_BITS+3)
407 wsptr += DCTSIZE; /* advance pointer to next row */
411 #endif /* DCT_ISLOW_SUPPORTED */