【ORB_SLAM2源码解读】计算具有旋转不变性的特征点描述子（11）
原理讲解
BEIFE描述子计算步骤,首先以特征点为圆心, 半径位15个像素的的圆形范围,计算灰度质心和描述子
在这个图像块内按照一定的方法选择N对点,N一般取256(一定的方法会生成256个点对,就是前面转换的那个pattern)
256个随机点对,生成256位的描述子信息,每个位bit的值是0或者1,每一位来自与点对的比较结果
8, -3, 9, 5 这四个值就是 相对于特征点的坐标,选取的点对,所以后面我们会将这个相对位置进行旋转来达到旋转不变性
同时根据图像的矩阵存储结构来推算这两个点对位置对应的像素值,进而生成0或者1的比较结果, 所以描述子由256位的二进制数字组成
视频讲解见哔哩哔哩
代码讲解
const float factorPI = (float) (CV_PI / 180.f);
static void computeOrbDescriptor(const KeyPoint &kpt, const Mat &img, const Point *pattern, uchar *desc) {
// 角度制转弧度制
float angle = (float) kpt.angle * factorPI;
// 所谓宏定义，就是用一个标识符来表示一个字符串，后面的代码中出现该标识符，全部替换成指定的字符串。
// 宏定义,无参宏定义的一般形式为：#define 宏名 字符串
// 带参宏定义的一般形式为：#define 宏名（参数表） 字符串
// pattern 点对 旋转前坐标为(x,y), 旋转后坐标(x',y')
// [x'] = cos(θ) -sin(θ) [x]
// [y'] = sin(θ) cos(θ) [y]
// x'= cos(θ)x - sin(θ)y
// y'= sin(θ)x + cos(θ)y
// y'* step + x' = 相对与特征点坐标的行列偏移量
// cvRound ( x * sin(θ) + y * cos(θ) ) * step
// 提取FAST特征点,特征点点角度,这个角度可以用来计算描述子,保证每次计算的描述子,都具有可比性。实现特征点的旋转不变性 Oriented FAST
// cos(angle) 角度的余弦值 sin(angle) 角度的正弦值
// float a = (float) cos(angle), b = (float) sin(angle);
float a = (float) cos(angle), b = (float) sin(angle);
// 特征点所在位置的一个指针，通过它，以及偏移量，可以获取特征点周围一定范围内像素值
const uchar *center = &img.at<uchar>(cvRound(kpt.pt.y), cvRound(kpt.pt.x));
// 图像的每行的字节数 也就是是原始图像两行像素之间的偏移量
const int step = (int) img.step;// 752
#define GET_VALUE(idx) \
center[cvRound(pattern[idx].x * b + pattern[idx].y * a) * step + \
cvRound(pattern[idx].x * a - pattern[idx].y * b)]
// "static int bit_pattern_31_[256 * 4] = {" 是一个1024维的数组,数组数据类型是int,是特征点keypoint为中心周围256对点的坐标,强制转换成 cv::Point 类型的 512个点.
// 一次比较特征点周围16个点=八个点对所在位置的像素值,比较32次, pattern中一共512个点。
for (int i = 0; i < 32; ++i, pattern += 16) {
std::cout << "idx " << i << std::endl;// idx 0
std::cout << "pattern[idx] " << pattern[i] << std::endl;// pattern[idx] [8, -3]
std::cout << "pattern[idx].x " << pattern[i].x << std::endl;// pattern[idx].x 8
std::cout << "b " << b << std::endl;// b -0.160526
std::cout << "pattern[idx].x * b " << pattern[i].x * b << std::endl;// pattern[idx].x * b -1.28421
std::cout << "pattern[idx].y " << pattern[i].y << std::endl;// pattern[idx].y -3
std::cout << "a " << a << std::endl;// a 0.987032
std::cout << "pattern[idx].y * a " << pattern[i].y * a << std::endl;// pattern[idx].y * a -2.96109
std::cout << "cvRound(pattern[idx].x * b + pattern[idx].y * a) " << cvRound(pattern[i].x * b + pattern[i].y * a) << std::endl;// -4
std::cout << "step " << step << std::endl;// 752
std::cout << "cvRound(pattern[idx].x * b + pattern[idx].y * a) * step " << cvRound(pattern[i].x * b + pattern[i].y * a) * step << std::endl;// -3008
std::cout << "y'* step + x' " << int(GET_VALUE(0)) << std::endl;// 183
int t0, t1, val;
// 0和1编码了关键点附近连个随机像素点的大小关系, 如果第一个点大于第二个点则取1, 如果第一个点小于第二个点则取0,
t0 = GET_VALUE(0);// 获取参与比较的第1个像素点的灰度值
t1 = GET_VALUE(1);// 获取参与比较的第2个像素点的灰度值
val = t0 < t1;
std::cout << "t0=" << t0 << " t1=" << t1 << " val " << val << std::endl;
// > 检查左操作数的值是否大于右操作数的值,如果是则条件为真val=1, 否则val=0
// t0=124 t1=128 -> 1
// val = 0000 0001 = 1
// 0 0 0 0 0 0 0 0
// 128 64 32 16 8 4 2 1
t0 = GET_VALUE(2);// 157
t1 = GET_VALUE(3);
// > 大于运算符
// < 小于运算符
// << 二进制左移运算符,将一个运算对象的各二进制位全部左移若干位(左边的二进制位丢弃右边补0), A << 2 将得到 240, 即为 1111 0000
val |= (t0 < t1) << 1;
std::cout << "t0=" << t0 << " t1=" << t1 << " val " << val << std::endl;
// t0=157 t1=190 -> 1 = 0000 0001
// << 二进制左移运算符 0000 0010
// |= 位或运算后赋值 0000 0011 = val 3
t0 = GET_VALUE(4);
t1 = GET_VALUE(5);
val |= (t0 < t1) << 2;
//std::cout << "t0=" << t0 << " t1=" << t1 << " val " << val << std::endl;
// t0=454 t1=243 -> 0
// val = 0000 0011
// 0000 0000
// 0000 0011 = val 3
t0 = GET_VALUE(6);
t1 = GET_VALUE(7);
val |= (t0 < t1) << 3;
//std::cout << "t0=" << t0 << " t1=" << t1 << " val " << val << std::endl;
// t0=190 t1=134 -> 0
// val = 0000 0111
// 0000 1000
// 0000 0111 = val 7
t0 = GET_VALUE(8);
t1 = GET_VALUE(9);
val |= (t0 < t1) << 4;
//std::cout << "t0=" << t0 << " t1=" << t1 << " val " << val << std::endl;
// t0=92 t1=135 -> 1
// val = 0000 0111
// 0001 0000
// 0001 0111 = val 23
t0 = GET_VALUE(10);
t1 = GET_VALUE(11);
val |= (t0 < t1) << 5;
//std::cout << "t0=" << t0 << " t1=" << t1 << " val " << val << std::endl;
// t0=92 t1=135 -> 1
// val = 0000 0111
// 0001 0000
// 0001 0111 = val 23
t0 = GET_VALUE(12);
t1 = GET_VALUE(13);
val |= (t0 < t1) << 6;
//std::cout << "t0=" << t0 << " t1=" << t1 << " val " << val << std::endl;
// t0=73 t1=48 -> 0
// val = 0000 0111
// 0001 0000
// 0001 0111 = val 23
t0 = GET_VALUE(14);
t1 = GET_VALUE(15);
val |= (t0 < t1) << 7;
//std::cout << "t0=" << t0 << " t1=" << t1 << " val " << val << std::endl;
// t0=26 t1=38 -> 1
// val = 0001 0111
// 1000 0000
// 1001 0111 = val 151
desc[i] = (uchar)val;
cout << "desc[" << i << "] = " << int(desc[i]) << std::endl;
// desc[1] = 173
// desc[2] = 186
// desc[3] = 122
// desc[4] = 159
// desc[5] = 209
// desc[6] = 150
// desc[7] = 233
// desc[8] = 223
// desc[9] = 19
// desc[10] = 245
// desc[11] = 26
// desc[12] = 46
// desc[13] = 216
// desc[14] = 110
// desc[15] = 37
// desc[16] = 233
// desc[17] = 251
// desc[18] = 58
// desc[19] = 83
// desc[20] = 126
// desc[21] = 111
// desc[22] = 138
// desc[23] = 247
// desc[24] = 187
// desc[25] = 94
// desc[26] = 23
// desc[27] = 123
// desc[28] = 81
// desc[29] = 79
// desc[30] = 169
// desc[31] = 142
desc[i] = (uchar) val;
}
#undef GET_VALUE
}
// BEIFE描述子计算步骤,首先以特征点为圆心, 半径位15个像素的的圆形范围,计算灰度质心和描述子.
// 在这个图像块内按照一定的方法选择N对点,N一般取256(一定的方法会生成256个点对,就是前面转换的那个pattern)
// 256个随机点对,生成256位的描述子信息,每个位bit的值是0或者1,每一位来自与点对的比较结果
// 8, -3, 9, 5 这四个值就是 相对于特征点的坐标,选取的点对,所以后面我们会将这个相对位置进行旋转来达到旋转不变性,
// 同时根据图像的矩阵存储结构来推算这两个点对位置对应的像素值,进而生成0或者1的比较结果, 所以描述子由256位的二进制数字组成.
static int bit_pattern_31_[256 * 4] = {
8, -3, 9, 5 /*mean (0), correlation (0)*/,
4, 2, 7, -12 /*mean (1.12461e-05), correlation (0.0437584)*/,
-11, 9, -8, 2 /*mean (3.37382e-05), correlation (0.0617409)*/,
7, -12, 12, -13 /*mean (5.62303e-05), correlation (0.0636977)*/,
2, -13, 2, 12 /*mean (0.000134953), correlation (0.085099)*/,
1, -7, 1, 6 /*mean (0.000528565), correlation (0.0857175)*/,
-2, -10, -2, -4 /*mean (0.0188821), correlation (0.0985774)*/,
-13, -13, -11, -8 /*mean (0.0363135), correlation (0.0899616)*/,
-13, -3, -12, -9 /*mean (0.121806), correlation (0.099849)*/,
10, 4, 11, 9 /*mean (0.122065), correlation (0.093285)*/,
-13, -8, -8, -9 /*mean (0.162787), correlation (0.0942748)*/,
-11, 7, -9, 12 /*mean (0.21561), correlation (0.0974438)*/,
7, 7, 12, 6 /*mean (0.160583), correlation (0.130064)*/,
-4, -5, -3, 0 /*mean (0.228171), correlation (0.132998)*/,
-13, 2, -12, -3 /*mean (0.00997526), correlation (0.145926)*/,
-9, 0, -7, 5 /*mean (0.198234), correlation (0.143636)*/,
12, -6, 12, -1 /*mean (0.0676226), correlation (0.16689)*/,
-3, 6, -2, 12 /*mean (0.166847), correlation (0.171682)*/,
-6, -13, -4, -8 /*mean (0.101215), correlation (0.179716)*/,
11, -13, 12, -8 /*mean (0.200641), correlation (0.192279)*/,
4, 7, 5, 1 /*mean (0.205106), correlation (0.186848)*/,
5, -3, 10, -3 /*mean (0.234908), correlation (0.192319)*/,
3, -7, 6, 12 /*mean (0.0709964), correlation (0.210872)*/,
-8, -7, -6, -2 /*mean (0.0939834), correlation (0.212589)*/,
-2, 11, -1, -10 /*mean (0.127778), correlation (0.20866)*/,
-13, 12, -8, 10 /*mean (0.14783), correlation (0.206356)*/,
-7, 3, -5, -3 /*mean (0.182141), correlation (0.198942)*/,
-4, 2, -3, 7 /*mean (0.188237), correlation (0.21384)*/,
-10, -12, -6, 11 /*mean (0.14865), correlation (0.23571)*/,
5, -12, 6, -7 /*mean (0.222312), correlation (0.23324)*/,
5, -6, 7, -1 /*mean (0.229082), correlation (0.23389)*/,
1, 0, 4, -5 /*mean (0.241577), correlation (0.215286)*/,
9, 11, 11, -13 /*mean (0.00338507), correlation (0.251373)*/,
4, 7, 4, 12 /*mean (0.131005), correlation (0.257622)*/,
2, -1, 4, 4 /*mean (0.152755), correlation (0.255205)*/,
-4, -12, -2, 7 /*mean (0.182771), correlation (0.244867)*/,
-8, -5, -7, -10 /*mean (0.186898), correlation (0.23901)*/,
4, 11, 9, 12 /*mean (0.226226), correlation (0.258255)*/,
0, -8, 1, -13 /*mean (0.0897886), correlation (0.274827)*/,
-13, -2, -8, 2 /*mean (0.148774), correlation (0.28065)*/,
-3, -2, -2, 3 /*mean (0.153048), correlation (0.283063)*/,
-6, 9, -4, -9 /*mean (0.169523), correlation (0.278248)*/,
8, 12, 10, 7 /*mean (0.225337), correlation (0.282851)*/,
0, 9, 1, 3 /*mean (0.226687), correlation (0.278734)*/,
7, -5, 11, -10 /*mean (0.00693882), correlation (0.305161)*/,
-13, -6, -11, 0 /*mean (0.0227283), correlation (0.300181)*/,
10, 7, 12, 1 /*mean (0.125517), correlation (0.31089)*/,
-6, -3, -6, 12 /*mean (0.131748), correlation (0.312779)*/,
10, -9, 12, -4 /*mean (0.144827), correlation (0.292797)*/,
-13, 8, -8, -12 /*mean (0.149202), correlation (0.308918)*/,
-13, 0, -8, -4 /*mean (0.160909), correlation (0.310013)*/,
3, 3, 7, 8 /*mean (0.177755), correlation (0.309394)*/,
5, 7, 10, -7 /*mean (0.212337), correlation (0.310315)*/,
-1, 7, 1, -12 /*mean (0.214429), correlation (0.311933)*/,
3, -10, 5, 6 /*mean (0.235807), correlation (0.313104)*/,
2, -4, 3, -10 /*mean (0.00494827), correlation (0.344948)*/,
-13, 0, -13, 5 /*mean (0.0549145), correlation (0.344675)*/,
-13, -7, -12, 12 /*mean (0.103385), correlation (0.342715)*/,
-13, 3, -11, 8 /*mean (0.134222), correlation (0.322922)*/,
-7, 12, -4, 7 /*mean (0.153284), correlation (0.337061)*/,
6, -10, 12, 8 /*mean (0.154881), correlation (0.329257)*/,
-9, -1, -7, -6 /*mean (0.200967), correlation (0.33312)*/,
-2, -5, 0, 12 /*mean (0.201518), correlation (0.340635)*/,
-12, 5, -7, 5 /*mean (0.207805), correlation (0.335631)*/,
3, -10, 8, -13 /*mean (0.224438), correlation (0.34504)*/,
-7, -7, -4, 5 /*mean (0.239361), correlation (0.338053)*/,
-3, -2, -1, -7 /*mean (0.240744), correlation (0.344322)*/,
2, 9, 5, -11 /*mean (0.242949), correlation (0.34145)*/,
-11, -13, -5, -13 /*mean (0.244028), correlation (0.336861)*/,
-1, 6, 0, -1 /*mean (0.247571), correlation (0.343684)*/,
5, -3, 5, 2 /*mean (0.000697256), correlation (0.357265)*/,
-4, -13, -4, 12 /*mean (0.00213675), correlation (0.373827)*/,
-9, -6, -9, 6 /*mean (0.0126856), correlation (0.373938)*/,
-12, -10, -8, -4 /*mean (0.0152497), correlation (0.364237)*/,
10, 2, 12, -3 /*mean (0.0299933), correlation (0.345292)*/,
7, 12, 12, 12 /*mean (0.0307242), correlation (0.366299)*/,
-7, -13, -6, 5 /*mean (0.0534975), correlation (0.368357)*/,
-4, 9, -3, 4 /*mean (0.099865), correlation (0.372276)*/,
7, -1, 12, 2 /*mean (0.117083), correlation (0.364529)*/,
-7, 6, -5, 1 /*mean (0.126125), correlation (0.369606)*/,
-13, 11, -12, 5 /*mean (0.130364), correlation (0.358502)*/,
-3, 7, -2, -6 /*mean (0.131691), correlation (0.375531)*/,
7, -8, 12, -7 /*mean (0.160166), correlation (0.379508)*/,
-13, -7, -11, -12 /*mean (0.167848), correlation (0.353343)*/,
1, -3, 12, 12 /*mean (0.183378), correlation (0.371916)*/,
2, -6, 3, 0 /*mean (0.228711), correlation (0.371761)*/,
-4, 3, -2, -13 /*mean (0.247211), correlation (0.364063)*/,
-1, -13, 1, 9 /*mean (0.249325), correlation (0.378139)*/,
7, 1, 8, -6 /*mean (0.000652272), correlation (0.411682)*/,
1, -1, 3, 12 /*mean (0.00248538), correlation (0.392988)*/,
9, 1, 12, 6 /*mean (0.0206815), correlation (0.386106)*/,
-1, -9, -1, 3 /*mean (0.0364485), correlation (0.410752)*/,
-13, -13, -10, 5 /*mean (0.0376068), correlation (0.398374)*/,
7, 7, 10, 12 /*mean (0.0424202), correlation (0.405663)*/,
12, -5, 12, 9 /*mean (0.0942645), correlation (0.410422)*/,
6, 3, 7, 11 /*mean (0.1074), correlation (0.413224)*/,
5, -13, 6, 10 /*mean (0.109256), correlation (0.408646)*/,
2, -12, 2, 3 /*mean (0.131691), correlation (0.416076)*/,
3, 8, 4, -6 /*mean (0.165081), correlation (0.417569)*/,
2, 6, 12, -13 /*mean (0.171874), correlation (0.408471)*/,
9, -12, 10, 3 /*mean (0.175146), correlation (0.41296)*/,
-8, 4, -7, 9 /*mean (0.183682), correlation (0.402956)*/,
-11, 12, -4, -6 /*mean (0.184672), correlation (0.416125)*/,
1, 12, 2, -8 /*mean (0.191487), correlation (0.386696)*/,
6, -9, 7, -4 /*mean (0.192668), correlation (0.394771)*/,
2, 3, 3, -2 /*mean (0.200157), correlation (0.408303)*/,
6, 3, 11, 0 /*mean (0.204588), correlation (0.411762)*/,
3, -3, 8, -8 /*mean (0.205904), correlation (0.416294)*/,
7, 8, 9, 3 /*mean (0.213237), correlation (0.409306)*/,
-11, -5, -6, -4 /*mean (0.243444), correlation (0.395069)*/,
-10, 11, -5, 10 /*mean (0.247672), correlation (0.413392)*/,
-5, -8, -3, 12 /*mean (0.24774), correlation (0.411416)*/,
-10, 5, -9, 0 /*mean (0.00213675), correlation (0.454003)*/,
8, -1, 12, -6 /*mean (0.0293635), correlation (0.455368)*/,
4, -6, 6, -11 /*mean (0.0404971), correlation (0.457393)*/,
-10, 12, -8, 7 /*mean (0.0481107), correlation (0.448364)*/,
4, -2, 6, 7 /*mean (0.050641), correlation (0.455019)*/,
-2, 0, -2, 12 /*mean (0.0525978), correlation (0.44338)*/,
-5, -8, -5, 2 /*mean (0.0629667), correlation (0.457096)*/,
7, -6, 10, 12 /*mean (0.0653846), correlation (0.445623)*/,
-9, -13, -8, -8 /*mean (0.0858749), correlation (0.449789)*/,
-5, -13, -5, -2 /*mean (0.122402), correlation (0.450201)*/,
8, -8, 9, -13 /*mean (0.125416), correlation (0.453224)*/,
-9, -11, -9, 0 /*mean (0.130128), correlation (0.458724)*/,
1, -8, 1, -2 /*mean (0.132467), correlation (0.440133)*/,
7, -4, 9, 1 /*mean (0.132692), correlation (0.454)*/,
-2, 1, -1, -4 /*mean (0.135695), correlation (0.455739)*/,
11, -6, 12, -11 /*mean (0.142904), correlation (0.446114)*/,
-12, -9, -6, 4 /*mean (0.146165), correlation (0.451473)*/,
3, 7, 7, 12 /*mean (0.147627), correlation (0.456643)*/,
5, 5, 10, 8 /*mean (0.152901), correlation (0.455036)*/,
0, -4, 2, 8 /*mean (0.167083), correlation (0.459315)*/,
-9, 12, -5, -13 /*mean (0.173234), correlation (0.454706)*/,
0, 7, 2, 12 /*mean (0.18312), correlation (0.433855)*/,
-1, 2, 1, 7 /*mean (0.185504), correlation (0.443838)*/,
5, 11, 7, -9 /*mean (0.185706), correlation (0.451123)*/,
3, 5, 6, -8 /*mean (0.188968), correlation (0.455808)*/,
-13, -4, -8, 9 /*mean (0.191667), correlation (0.459128)*/,
-5, 9, -3, -3 /*mean (0.193196), correlation (0.458364)*/,
-4, -7, -3, -12 /*mean (0.196536), correlation (0.455782)*/,
6, 5, 8, 0 /*mean (0.1972), correlation (0.450481)*/,
-7, 6, -6, 12 /*mean (0.199438), correlation (0.458156)*/,
-13, 6, -5, -2 /*mean (0.211224), correlation (0.449548)*/,
1, -10, 3, 10 /*mean (0.211718), correlation (0.440606)*/,
4, 1, 8, -4 /*mean (0.213034), correlation (0.443177)*/,
-2, -2, 2, -13 /*mean (0.234334), correlation (0.455304)*/,
2, -12, 12, 12 /*mean (0.235684), correlation (0.443436)*/,
-2, -13, 0, -6 /*mean (0.237674), correlation (0.452525)*/,
4, 1, 9, 3 /*mean (0.23962), correlation (0.444824)*/,
-6, -10, -3, -5 /*mean (0.248459), correlation (0.439621)*/,
-3, -13, -1, 1 /*mean (0.249505), correlation (0.456666)*/,
7, 5, 12, -11 /*mean (0.00119208), correlation (0.495466)*/,
4, -2, 5, -7 /*mean (0.00372245), correlation (0.484214)*/,
-13, 9, -9, -5 /*mean (0.00741116), correlation (0.499854)*/,
7, 1, 8, 6 /*mean (0.0208952), correlation (0.499773)*/,
7, -8, 7, 6 /*mean (0.0220085), correlation (0.501609)*/,
-7, -4, -7, 1 /*mean (0.0233806), correlation (0.496568)*/,
-8, 11, -7, -8 /*mean (0.0236505), correlation (0.489719)*/,
-13, 6, -12, -8 /*mean (0.0268781), correlation (0.503487)*/,
2, 4, 3, 9 /*mean (0.0323324), correlation (0.501938)*/,
10, -5, 12, 3 /*mean (0.0399235), correlation (0.494029)*/,
-6, -5, -6, 7 /*mean (0.0420153), correlation (0.486579)*/,
8, -3, 9, -8 /*mean (0.0548021), correlation (0.484237)*/,
2, -12, 2, 8 /*mean (0.0616622), correlation (0.496642)*/,
-11, -2, -10, 3 /*mean (0.0627755), correlation (0.498563)*/,
-12, -13, -7, -9 /*mean (0.0829622), correlation (0.495491)*/,
-11, 0, -10, -5 /*mean (0.0843342), correlation (0.487146)*/,
5, -3, 11, 8 /*mean (0.0929937), correlation (0.502315)*/,
-2, -13, -1, 12 /*mean (0.113327), correlation (0.48941)*/,
-1, -8, 0, 9 /*mean (0.132119), correlation (0.467268)*/,
-13, -11, -12, -5 /*mean (0.136269), correlation (0.498771)*/,
-10, -2, -10, 11 /*mean (0.142173), correlation (0.498714)*/,
-3, 9, -2, -13 /*mean (0.144141), correlation (0.491973)*/,
2, -3, 3, 2 /*mean (0.14892), correlation (0.500782)*/,
-9, -13, -4, 0 /*mean (0.150371), correlation (0.498211)*/,
-4, 6, -3, -10 /*mean (0.152159), correlation (0.495547)*/,
-4, 12, -2, -7 /*mean (0.156152), correlation (0.496925)*/,
-6, -11, -4, 9 /*mean (0.15749), correlation (0.499222)*/,
6, -3, 6, 11 /*mean (0.159211), correlation (0.503821)*/,
-13, 11, -5, 5 /*mean (0.162427), correlation (0.501907)*/,
11, 11, 12, 6 /*mean (0.16652), correlation (0.497632)*/,
7, -5, 12, -2 /*mean (0.169141), correlation (0.484474)*/,
-1, 12, 0, 7 /*mean (0.169456), correlation (0.495339)*/,
-4, -8, -3, -2 /*mean (0.171457), correlation (0.487251)*/,
-7, 1, -6, 7 /*mean (0.175), correlation (0.500024)*/,
-13, -12, -8, -13 /*mean (0.175866), correlation (0.497523)*/,
-7, -2, -6, -8 /*mean (0.178273), correlation (0.501854)*/,
-8, 5, -6, -9 /*mean (0.181107), correlation (0.494888)*/,
-5, -1, -4, 5 /*mean (0.190227), correlation (0.482557)*/,
-13, 7, -8, 10 /*mean (0.196739), correlation (0.496503)*/,
1, 5, 5, -13 /*mean (0.19973), correlation (0.499759)*/,
1, 0, 10, -13 /*mean (0.204465), correlation (0.49873)*/,
9, 12, 10, -1 /*mean (0.209334), correlation (0.49063)*/,
5, -8, 10, -9 /*mean (0.211134), correlation (0.503011)*/,
-1, 11, 1, -13 /*mean (0.212), correlation (0.499414)*/,
-9, -3, -6, 2 /*mean (0.212168), correlation (0.480739)*/,
-1, -10, 1, 12 /*mean (0.212731), correlation (0.502523)*/,
-13, 1, -8, -10 /*mean (0.21327), correlation (0.489786)*/,
8, -11, 10, -6 /*mean (0.214159), correlation (0.488246)*/,
2, -13, 3, -6 /*mean (0.216993), correlation (0.50287)*/,
7, -13, 12, -9 /*mean (0.223639), correlation (0.470502)*/,
-10, -10, -5, -7 /*mean (0.224089), correlation (0.500852)*/,
-10, -8, -8, -13 /*mean (0.228666), correlation (0.502629)*/,
4, -6, 8, 5 /*mean (0.22906), correlation (0.498305)*/,
3, 12, 8, -13 /*mean (0.233378), correlation (0.503825)*/,
-4, 2, -3, -3 /*mean (0.234323), correlation (0.476692)*/,
5, -13, 10, -12 /*mean (0.236392), correlation (0.475462)*/,
4, -13, 5, -1 /*mean (0.236842), correlation (0.504132)*/,
-9, 9, -4, 3 /*mean (0.236977), correlation (0.497739)*/,
0, 3, 3, -9 /*mean (0.24314), correlation (0.499398)*/,
-12, 1, -6, 1 /*mean (0.243297), correlation (0.489447)*/,
3, 2, 4, -8 /*mean (0.00155196), correlation (0.553496)*/,
-10, -10, -10, 9 /*mean (0.00239541), correlation (0.54297)*/,
8, -13, 12, 12 /*mean (0.0034413), correlation (0.544361)*/,
-8, -12, -6, -5 /*mean (0.003565), correlation (0.551225)*/,
2, 2, 3, 7 /*mean (0.00835583), correlation (0.55285)*/,
10, 6, 11, -8 /*mean (0.00885065), correlation (0.540913)*/,
6, 8, 8, -12 /*mean (0.0101552), correlation (0.551085)*/,
-7, 10, -6, 5 /*mean (0.0102227), correlation (0.533635)*/,
-3, -9, -3, 9 /*mean (0.0110211), correlation (0.543121)*/,
-1, -13, -1, 5 /*mean (0.0113473), correlation (0.550173)*/,
-3, -7, -3, 4 /*mean (0.0140913), correlation (0.554774)*/,
-8, -2, -8, 3 /*mean (0.017049), correlation (0.55461)*/,
4, 2, 12, 12 /*mean (0.01778), correlation (0.546921)*/,
2, -5, 3, 11 /*mean (0.0224022), correlation (0.549667)*/,
6, -9, 11, -13 /*mean (0.029161), correlation (0.546295)*/,
3, -1, 7, 12 /*mean (0.0303081), correlation (0.548599)*/,
11, -1, 12, 4 /*mean (0.0355151), correlation (0.523943)*/,
-3, 0, -3, 6 /*mean (0.0417904), correlation (0.543395)*/,
4, -11, 4, 12 /*mean (0.0487292), correlation (0.542818)*/,
2, -4, 2, 1 /*mean (0.0575124), correlation (0.554888)*/,
-10, -6, -8, 1 /*mean (0.0594242), correlation (0.544026)*/,
-13, 7, -11, 1 /*mean (0.0597391), correlation (0.550524)*/,
-13, 12, -11, -13 /*mean (0.0608974), correlation (0.55383)*/,
6, 0, 11, -13 /*mean (0.065126), correlation (0.552006)*/,
0, -1, 1, 4 /*mean (0.074224), correlation (0.546372)*/,
-13, 3, -9, -2 /*mean (0.0808592), correlation (0.554875)*/,
-9, 8, -6, -3 /*mean (0.0883378), correlation (0.551178)*/,
-13, -6, -8, -2 /*mean (0.0901035), correlation (0.548446)*/,
5, -9, 8, 10 /*mean (0.0949843), correlation (0.554694)*/,
2, 7, 3, -9 /*mean (0.0994152), correlation (0.550979)*/,
-1, -6, -1, -1 /*mean (0.10045), correlation (0.552714)*/,
9, 5, 11, -2 /*mean (0.100686), correlation (0.552594)*/,
11, -3, 12, -8 /*mean (0.101091), correlation (0.532394)*/,
3, 0, 3, 5 /*mean (0.101147), correlation (0.525576)*/,
-1, 4, 0, 10 /*mean (0.105263), correlation (0.531498)*/,
3, -6, 4, 5 /*mean (0.110785), correlation (0.540491)*/,
-13, 0, -10, 5 /*mean (0.112798), correlation (0.536582)*/,
5, 8, 12, 11 /*mean (0.114181), correlation (0.555793)*/,
8, 9, 9, -6 /*mean (0.117431), correlation (0.553763)*/,
7, -4, 8, -12 /*mean (0.118522), correlation (0.553452)*/,
-10, 4, -10, 9 /*mean (0.12094), correlation (0.554785)*/,
7, 3, 12, 4 /*mean (0.122582), correlation (0.555825)*/,
9, -7, 10, -2 /*mean (0.124978), correlation (0.549846)*/,
7, 0, 12, -2 /*mean (0.127002), correlation (0.537452)*/,
-1, -6, 0, -11 /*mean (0.127148), correlation (0.547401)*/
};