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, 2022, D. R. Commander.
8 * For conditions of distribution and use, see the accompanying README.ijg
11 * This file contains a slower but more accurate integer implementation of the
12 * forward DCT (Discrete Cosine Transform).
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.
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.
29 #define JPEG_INTERNALS
32 #include "jdct.h" /* Private declarations for DCT subsystem */
34 #ifdef DCT_ISLOW_SUPPORTED
38 * This module is specialized to the case DCTSIZE = 8.
42 Sorry
, this code only copes with
8x8 DCTs
. /* deliberate syntax err */
47 * The poop on this scaling stuff is as follows:
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.
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.
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.)
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.
80 #if BITS_IN_JSAMPLE == 8
85 #define PASS1_BITS 1 /* lose a little precision to avoid overflow */
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...)
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) */
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)
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.
131 #if BITS_IN_JSAMPLE == 8
132 #define MULTIPLY(var, const) MULTIPLY16C16(var, const)
134 #define MULTIPLY(var, const) ((var) * (const))
139 * Perform the forward DCT on one block of samples.
143 _jpeg_fdct_islow(DCTELEM
*data
)
145 JLONG tmp0
, tmp1
, tmp2
, tmp3
, tmp4
, tmp5
, tmp6
, tmp7
;
146 JLONG tmp10
, tmp11
, tmp12
, tmp13
;
147 JLONG z1
, z2
, z3
, z4
, z5
;
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. */
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];
167 /* Even part per LL&M figure 1 --- note that published figure is faulty;
168 * rotator "sqrt(2)*c1" should be "sqrt(2)*c6".
176 dataptr
[0] = (DCTELEM
)LEFT_SHIFT(tmp10
+ tmp11
, PASS1_BITS
);
177 dataptr
[4] = (DCTELEM
)LEFT_SHIFT(tmp10
- tmp11
, PASS1_BITS
);
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
);
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.
194 z5
= MULTIPLY(z3
+ z4
, FIX_1_175875602
); /* sqrt(2) * c3 */
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) */
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
);
213 dataptr
+= DCTSIZE
; /* advance pointer to next row */
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.
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];
232 /* Even part per LL&M figure 1 --- note that published figure is faulty;
233 * rotator "sqrt(2)*c1" should be "sqrt(2)*c6".
241 dataptr
[DCTSIZE
* 0] = (DCTELEM
)DESCALE(tmp10
+ tmp11
, PASS1_BITS
);
242 dataptr
[DCTSIZE
* 4] = (DCTELEM
)DESCALE(tmp10
- tmp11
, PASS1_BITS
);
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
);
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.
261 z5
= MULTIPLY(z3
+ z4
, FIX_1_175875602
); /* sqrt(2) * c3 */
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) */
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
);
284 dataptr
++; /* advance pointer to next column */
288 #endif /* DCT_ISLOW_SUPPORTED */