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 
139 GLOBAL(void)
140 jpeg_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