模式识别是一种通过对数据进行分析和学习,从中提取模式并做出决策的技术。这一领域涵盖了多种技术和方法,可用于处理各种类型的数据,包括图像、语音、文本等。以下是一些常见的模式识别技术:
这些技术通常不是孤立存在的,而是相互交叉和融合的,以解决更复杂的问题。在实际应用中,根据具体的问题和数据特点选择合适的模式识别技术是至关重要的。
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Fisher线性判别(Fisher Linear Discriminant,简称FLD)是一种用于进行模式识别和分类的线性判别方法。它旨在找到一个投影方向,将不同类别的数据在该方向上的投影尽可能分开,从而使得类间方差最大,而类内方差最小。FLD通常用于二分类问题,但也可以扩展到多类别情况。 以下是Fisher线性判别的算法原理: 1. 定义问题: 假设有两个类别,分别为类别1和类别2,目标是找到一个投影方向,使得两个类别在这个方向上的投影差异尽可能大。 2. 计算类内散度矩阵: 对于每个类别,计算其数据点的协方差矩阵,然后将这些协方差矩阵加和,得到类内散度矩阵(Sw)。 Sw=∑i=1c∑j=1ni(xij−μi)⋅(xij−μi)T 其中:
3. 计算类间散度矩阵: 计算两个类别的均值向量之差的外积,得到类间散度矩阵(Sb)。 Sb=(μ1−μ2)⋅(μ1−μ2)T 4. 计算广义特征值问题: 通过解决广义特征值问题,得到投影方向的系数。广义特征值问题的解决将产生一个特征值和对应的特征向量,其中特征向量表示投影方向。 Sw−1⋅Sb⋅w=λ⋅w 其中:
5. 选择投影方向: 选择前 C−1 个最大广义特征值对应的特征向量,其中 C 是类别的数量。这些特征向量构成了投影方向的基,也称为Fisher判别子空间。 6. 进行投影: 将数据点投影到所选择的判别子空间,然后通过选择一个阈值对投影结果进行分类。 Fisher线性判别的优势在于在选择投影方向时,不仅考虑了类别之间的差异(类间散度),还考虑了类别内部的差异(类内散度),因此更加适用于具有类别内部差异的情况。
C语言程序:
// fisher.cpp : Defines the entry point for the console application.
//
// rmbdis.cpp : Defines the entry point for the console application.
//
#include "stdafx.h"
#include "math.h"
#include "conio.h"
#include <fstream>
#include <iomanip>
using namespace std;
#define PNUM 60
unsigned char dat[10][4][8][8][60]={
//0--样本1,1--样本1,......,8--样本9,9--样本10
//0--100,1--50,2--20,3--10
//0--A向,1--B向,2--C向,3--D向,4--新A向,5--新B向,6--新C向,7--新D向
//0--传感1,1--传感2,2--传感3,3--传感4,4--传感5,5--传感6,6--传感7,7--传感8
//
{
#include "样本\\rmb00.txt"
},
{
#include "样本\\rmb01.txt"
},
{
#include "样本\\rmb02.txt"
},
{
#include "样本\\rmb03.txt"
},
{
#include "样本\\rmb04.txt"
},
{
#include "样本\\rmb05.txt"
},
{
#include "样本\\rmb06.txt"
},
{
#include "样本\\rmb07.txt"
},
{
#include "样本\\rmb08.txt"
},
{
#include "样本\\rmb09.txt"
}
};
#define NUM 8
double Eucliden(double x[],double y[],int n)
{
double d;
d=0.0;
for (int i=0;i<n;i++) {
d+=(x[i]-y[i])*(x[i]-y[i]);
}
d=sqrt(d);
return d;
}
double Manhattan(double x[],double y[],int n)
{
double d;
d=0.0;
for (int i=0;i<n;i++) {
d+=fabs(x[i]-y[i]);
}
return d;
}
double Chebyshev(double x[],double y[],int n)
{
double d;
d=0.0;
for (int i=0;i<n;i++) {
if(fabs(x[i]-y[i])>d) d=fabs(x[i]-y[i]);
}
return d;
}
double Minkowski(double x[],double y[],int n,int m)
{
double d;
d=0.0;
for (int i=0;i<n;i++) {
d+=(double)powf((float)(x[i]-y[i]),(float)m);
}
d=(double)powf((float)d,1.0f/m);
return d;
}
double Mahalanobis(double x[],double y[],double matv1[8][8])
{
double dx,dy;
int i,j;
dx=0.0;
for (i=0;i<8;i++) {
dy=0.0;
for (j=0;j<8;j++) {
dy+=matv1[i][j]*(x[j]-y[j]);
}
dx+=dy*(x[i]-y[i]);
}
return dx;
}
void GetMatV(double V[8][8],int k)
{
int i,j,m,n1,n2,n3;
double xm[8],d,x,y;
m=4*8*PNUM;
for (i=0;i<8;i++) {
d=0;
for (n1=0;n1<4;n1++) {
for (n2=0;n2<8;n2++) {
for (n3=0;n3<PNUM;n3++) {
d+=(double)dat[k][n1][n2][i][n3];
}
}
}
d/=m;
xm[i]=d;
}
for (i=0;i<8;i++) {
for (j=0;j<8;j++) {
d=0;
for (n1=0;n1<4;n1++) {
for (n2=0;n2<8;n2++) {
for (n3=0;n3<PNUM;n3++) {
x=(double)dat[k][n1][n2][i][n3]-xm[i];
y=(double)dat[k][n1][n2][j][n3]-xm[j];
d+=x*y;
}
}
}
d/=m-1.0;
V[i][j]=d;
}
}
}
void Gauss_Jordan(double matv[8][8],double matv1[8][8])
{
int n=8;
double mat[8][16],d;
int i,j,l,k;
for (i=0;i<n;i++) {
for (j=0;j<2*n;j++) {
if (j<n)
mat[i][j]=matv[i][j];
else
mat[i][j]=0.0;
}
}
for (i=0;i<n;i++) mat[i][n+i]=1.0;
for (k=0;k<n;k++) {
d=fabs(mat[k][k]);
j=k;
for (i=k+1;i<n;i++) {//选主元
if (fabs(mat[i][k])>d) {
d=fabs(mat[i][k]);
j=i;
}
}
if (j!=k) { //交换
for (l=0;l<2*n;l++) {
d=mat[j][l];
mat[j][l]=mat[k][l];
mat[k][l]=d;
}
}
for (j=k+1;j<2*n;j++) {
mat[k][j]/=mat[k][k];
}
for (i=0;i<n;i++) {
if (i==k) continue;
for (j=k+1;j<2*n;j++) {
mat[i][j]-=mat[i][k]*mat[k][j];
}
}
}
for (i=0;i<n;i++) {
for (j=0;j<n;j++) {
matv1[i][j]=mat[i][j+n];
}
}
}
void getswj(double mats[8][8],double mj[8],unsigned char data[8][60])
{
int i,j,k;
for (i=0;i<8;i++)
{
mj[i]=0.0;
for (k=0;k<PNUM;k++)
{
mj[i]+=(double)data[i][k];
}
mj[i]/=60.0;
}
for (i=0;i<8;i++)
{
for (j=0;j<8;j++)
{
mats[i][j]=0;
for (k=0;k<PNUM;k++)
{
mats[i][j]+=(data[i][k]-mj[i])*(data[j][k]-mj[j]);
}
mats[i][j]/=59.0;
}
}
}
void get4sw(double mats[8][8],double mj[8],unsigned char data[8][8][60])
{
int i,j,k,m;
for (i=0;i<NUM;i++) {
mj[i]=0.0;
for (j=0;j<8;j++) {
for (k=0;k<PNUM;k++) {
mj[i]+=(double)data[j][i][k];
}
}
mj[i]/=8.0*PNUM;
}
for (i=0;i<NUM;i++) {
for (j=0;j<NUM;j++) {
mats[i][j]=0;
for (m=0;m<8;m++) {
for (k=0;k<PNUM;k++) {
mats[i][j]+=(data[m][i][k]-mj[i])*(data[m][j][k]-mj[j]);
}
}
mats[i][j]/=8*PNUM-1;
}
}
}
void getsb(double sb[8][8],double mj[32][8],unsigned char data[4][8][8][60])
{
int i,j,k;
double m[8];
for (i=0;i<8;i++) {
m[i]=0;
for (j=0;j<32;j++) {
for (k=0;k<60;k++) {
m[i]+=data[j/8][j%8][i][k];
}
}
m[i]/=60.0*32.0;
}
for (i=0;i<8;i++) {
for (j=0;j<8;j++) {
sb[i][j]=0;
for (k=0;k<32;k++) {
sb[i][j]+=(mj[k][i]-m[i])*(mj[k][j]-m[j]);
}
sb[i][j]/=32;
}
}
}
void getsw(double swj[32][8][8],double sw[8][8])
{
int i,j,k;
for (i=0;i<8;i++) {
for (j=0;j<8;j++) {
sw[i][j]=0;
for (k=0;k<32;k++) {
sw[i][j]+=swj[k][i][j];
}
sw[i][j]/=32.0;
}
}
}
void MatMul(double mata[8][8],double matb[8][8],double matc[8][8])
{
int i,j,k;
for (i=0;i<NUM;i++) {
for (j=0;j<NUM;j++) {
matc[i][j]=0;
for (k=0;k<NUM;k++) {
matc[i][j]+=mata[i][k]*matb[k][j];
}
}
}
}
void MatAdd(double mata[8][8],double matb[8][8],double matc[8][8])
{
int i,j;
for (i=0;i<NUM;i++) {
for (j=0;j<NUM;j++) {
matc[i][j]=mata[i][j]+matb[i][j];
}
}
}
void MatDec(double mata[8][8],double matb[8][8],double matc[8][8])
{
int i,j;
for (i=0;i<NUM;i++) {
for (j=0;j<NUM;j++) {
matc[i][j]=mata[i][j]-matb[i][j];
}
}
}
void getst(double sw[8][8],double sb[8][8],double st[8][8])
{
MatAdd(sw,sb,st);
}
double MatTrace(double mat[8][8])
{
int i;
double d=0.0;
for(i=0;i<NUM;i++) {
d+=mat[i][i];
}
return d;
}
void OutSw(ofstream outfile,double sw[NUM][NUM])
{
int i,j;
for (i=0;i<NUM;i++) {
for (j=0;j<NUM;j++) {
outfile<<setprecision(5)<<sw[i][j];
if (j<NUM-1) outfile<<",";
else outfile<<endl;
}
}
}
double MulVector(double x[NUM],double y[NUM])
{
int i;
double d;
d=0.0;
for (i=0;i<NUM;i++) {
d+=x[i]*y[i];
}
return d;
}
int main(int argc, char* argv[])
{
double sw[32][8][8];
double mj[32][8];
double sww[8][8];
double sww1[8][8];
int i,j;
// char name[20]="sw100aa.h";
/* for (i=0;i<32;i++) {
getswj(sw[i],mj[i],dat[0][i/8][i%8]);
}
MatAdd(sw[0],sw[8],sww);
Gauss_Jordan(sww,sww1);
ofstream outfile;
outfile.open("sw100ab.txt");
outfile<<"//100A m1: \n";
for (i=0;i<NUM;i++) {
outfile<<setw(5)<<setprecision(3)<<mj[0][i]<<",";
}
outfile<<endl;
outfile<<"//100b m2: \n";
for (i=0;i<NUM;i++) {
outfile<<setw(5)<<setprecision(3)<<mj[8][i]<<",";
}
outfile<<endl;
outfile<<"//100A SW1: \n";
for (i=0;i<NUM;i++) {
for (j=0;j<NUM;j++) {
outfile<<setw(5)<<setprecision(3)<<sw[0][i][j];
if (j<NUM-1) outfile<<",";
else outfile<<endl;
}
}
outfile<<"//100b SW2: \n";
for (i=0;i<NUM;i++) {
for (j=0;j<NUM;j++) {
outfile<<setw(5)<<setprecision(3)<<sw[8][i][j];
if (j<NUM-1) outfile<<",";
else outfile<<endl;
}
}
outfile<<"//SW=SW1+SW2: \n";
for (i=0;i<NUM;i++) {
for (j=0;j<NUM;j++) {
outfile<<setw(5)<<setprecision(3)<<sww[i][j];
if (j<NUM-1) outfile<<",";
else outfile<<endl;
}
}
outfile<<"//SW-1: \n";
for (i=0;i<NUM;i++) {
for (j=0;j<NUM;j++) {
outfile<<setw(5)<<setprecision(3)<<sww1[i][j];
if (j<NUM-1) outfile<<",";
else outfile<<endl;
}
}
double d,u[NUM];
for (i=0;i<NUM;i++) {
d=0.0;
for (j=0;j<NUM;j++) {
d+=sww1[i][j]*(mj[0][j]-mj[8][j]);
}
u[i]=d;
}
outfile<<"//u=sw-1(m1-m2): \n";
for (i=0;i<NUM;i++) {
outfile<<setw(5)<<setprecision(3)<<u[i]<<",";
}
outfile<<endl;
d=MulVector(u,mj[0]);
outfile<<"u*m1="<<d<<endl;
double d2;
d2=MulVector(u,mj[8]);
outfile<<"u*m2="<<d2<<endl;
d=(d+d2)/2.0;
outfile<<"yt="<<d<<endl;
double pt[NUM];
outfile<<"100AResult:\n";
for (i=0;i<PNUM;i++)
{
for (j=0;j<NUM;j++)
{
pt[j]=(double)dat[0][0][0][j][i];
}
d=MulVector(u,pt);
outfile<<setw(5)<<setprecision(3)<<d;
if ((i+1)%8==0) outfile<<endl;
else outfile<<",";
}
outfile<<endl;
outfile<<"100bResult:\n";
for (i=0;i<PNUM;i++) {
for (j=0;j<NUM;j++) {
pt[j]=dat[0][1][0][j][i];
}
d=MulVector(u,pt);
outfile<<setw(5)<<setprecision(3)<<d;
if ((i+1)%8==0) outfile<<endl;
else outfile<<",";
}
outfile<<endl;
outfile.close();
*/
for (i=0;i<32;i++) {
getswj(sw[i],mj[i],dat[0][i/8][i%8]);
}
// MatAdd(sw[0],sw[8],sww);
MatAdd(sw[1],sw[8],sww);
Gauss_Jordan(sww,sww1);
ofstream outfile;
outfile.open("sw100b50a.h");
// outfile<<"//100A m1: \n";
outfile<<"//100b m1: \n";
for (i=0;i<NUM;i++) {
// outfile<<setw(5)<<setprecision(3)<<mj[0][i]<<",";
outfile<<setw(5)<<setprecision(3)<<mj[1][i]<<",";
}
outfile<<endl;
outfile<<"//50A m2: \n";
for (i=0;i<NUM;i++) {
outfile<<setw(5)<<setprecision(3)<<mj[8][i]<<",";
}
outfile<<endl;
// outfile<<"//100A SW1: \n";
outfile<<"//100b SW1: \n";
for (i=0;i<NUM;i++) {
for (j=0;j<NUM;j++) {
// outfile<<setw(5)<<setprecision(3)<<sw[0][i][j];
outfile<<setw(5)<<setprecision(3)<<sw[1][i][j];
if (j<NUM-1) outfile<<",";
else outfile<<endl;
}
}
outfile<<"//50A SW2: \n";
for (i=0;i<NUM;i++) {
for (j=0;j<NUM;j++) {
outfile<<setw(5)<<setprecision(3)<<sw[8][i][j];
if (j<NUM-1) outfile<<",";
else outfile<<endl;
}
}
outfile<<"//SW=SW1+SW2: \n";
for (i=0;i<NUM;i++) {
for (j=0;j<NUM;j++) {
outfile<<setw(5)<<setprecision(3)<<sww[i][j];
if (j<NUM-1) outfile<<",";
else outfile<<endl;
}
}
outfile<<"//SW-1: \n";
for (i=0;i<NUM;i++) {
for (j=0;j<NUM;j++) {
outfile<<setw(5)<<setprecision(3)<<sww1[i][j];
if (j<NUM-1) outfile<<",";
else outfile<<endl;
}
}
double d,u[NUM];
for (i=0;i<NUM;i++) {
d=0.0;
for (j=0;j<NUM;j++) {
// d+=sww1[i][j]*(mj[0][j]-mj[8][j]);
d+=sww1[i][j]*(mj[1][j]-mj[8][j]);
}
u[i]=d;
}
outfile<<"//u=sw-1(m1-m2): \n";
for (i=0;i<NUM;i++) {
outfile<<setw(5)<<setprecision(3)<<u[i]<<",";
}
outfile<<endl;
// d=MulVector(u,mj[0]);
d=MulVector(u,mj[1]);
outfile<<"u*m1="<<d<<endl;
double d2;
d2=MulVector(u,mj[8]);
outfile<<"u*m2="<<d2<<endl;
d=(d+d2)/2.0;
outfile<<"yt="<<d<<endl;
double pt[NUM];
// outfile<<"100AResult:\n";
outfile<<"100bResult:\n";
for (i=0;i<PNUM;i++) {
for (j=0;j<NUM;j++) {
// pt[j]=(double)dat[0][0][0][j][i];
pt[j]=(double)dat[0][0][1][j][i];
}
d=MulVector(u,pt);
outfile<<setw(5)<<setprecision(3)<<d;
if ((i+1)%8==0) outfile<<endl;
else outfile<<",";
}
outfile<<endl;
outfile<<"50AResult:\n";
for (i=0;i<PNUM;i++) {
for (j=0;j<NUM;j++) {
pt[j]=dat[0][1][0][j][i];
}
d=MulVector(u,pt);
outfile<<setw(5)<<setprecision(3)<<d;
if ((i+1)%8==0) outfile<<endl;
else outfile<<",";
}
outfile<<endl;
outfile.close();
return 0;
}
分析实验结果,比较PCA与Fisher线性判别前后的人脸识别性能差异,将结果输出到sw100b50a.h文件,结果如下:
//100b m1:
57, 67.2, 72.8, 67, 94.4, 93.9, 83, 84.4,
//50A m2:
59.2, 55.5, 81.9, 63.9, 95.1, 91, 91.1, 86.5,
//100b SW1:
3.95,-2.11, 1.74,-2.51, 3.09, 1.85, 1.49,0.804
-2.11, 11.7,-17.7, 2.45,-1.32, 0.31,-6.01,-0.041
1.74,-17.7, 67.9, 16.7, 3.62, 2.19, 24.8, 12.3
-2.51, 2.45, 16.7, 32.9,-2.14, 1.17, 6.49, 14.1
3.09,-1.32, 3.62,-2.14, 8.55, 2.94, 1.98, 2.31
1.85, 0.31, 2.19, 1.17, 2.94, 5.28, 3.67, 3.86
1.49,-6.01, 24.8, 6.49, 1.98, 3.67, 15.1, 6.28
0.804,-0.041, 12.3, 14.1, 2.31, 3.86, 6.28, 9.13
//50A SW2:
28.3, 45.7, 49.6, 48.6, 1.59, 17.4, 14, 15.2
45.7, 124, 63.2, 88.6, 6.21, 44.1, 15.8, 17.4
49.6, 63.2, 119, 82.6, 3.31, 25.9, 38, 28.8
48.6, 88.6, 82.6, 109,-2.86, 33, 14.1, 28.6
1.59, 6.21, 3.31,-2.86, 4.93, 3.69, 5.21,-1.53
17.4, 44.1, 25.9, 33, 3.69, 17.7, 7.79, 6.71
14, 15.8, 38, 14.1, 5.21, 7.79, 23, 9.07
15.2, 17.4, 28.8, 28.6,-1.53, 6.71, 9.07, 13.9
//SW=SW1+SW2:
32.2, 43.6, 51.4, 46.1, 4.69, 19.2, 15.5, 16
43.6, 136, 45.5, 91, 4.9, 44.4, 9.84, 17.3
51.4, 45.5, 187, 99.3, 6.93, 28, 62.8, 41
46.1, 91, 99.3, 142, -5, 34.2, 20.6, 42.7
4.69, 4.9, 6.93, -5, 13.5, 6.63, 7.19,0.775
19.2, 44.4, 28, 34.2, 6.63, 23, 11.5, 10.6
15.5, 9.84, 62.8, 20.6, 7.19, 11.5, 38.1, 15.4
16, 17.3, 41, 42.7,0.775, 10.6, 15.4, 23
//SW-1:
0.112,-0.0211,-0.0242,0.00562,-0.0137,-0.0229,0.0187,-0.0307
-0.0211,0.0369,0.00917,-0.0212,0.00628,-0.0489,-0.00547,0.0356
-0.0242,0.00917,0.0266,-0.0185,-0.00154,0.00792,-0.0352,0.0167
0.00562,-0.0212,-0.0185,0.0492,0.0245,-0.00716,0.0305,-0.0639
-0.0137,0.00628,-0.00154,0.0245,0.122,-0.0629,-0.00764,-0.00795
-0.0229,-0.0489,0.00792,-0.00716,-0.0629,0.198,-0.0275,-0.0187
0.0187,-0.00547,-0.0352,0.0305,-0.00764,-0.0275,0.0918,-0.0508
-0.0307,0.0356,0.0167,-0.0639,-0.00795,-0.0187,-0.0508,0.169
//u=sw-1(m1-m2):
-0.401,0.158,0.135,-0.0915,0.0103,0.253,-0.399,0.148,
u*m1=-4.4
u*m2=-9.25
yt=-6.83
100bResult:
-8.94,-8.94,-7.42,-6.08,-4.08,-4.02,-4.44,-4.54
-4.8,-4.76,-5.03,-5.03,-5.11,-4.81,-4.81,-4.31
-3.8,-3.67, -3.5,-3.59,-3.49,-2.94,-3.12,-2.97
-4.25,-3.98,-3.54, -3,-3.21,-3.18,-3.36,-3.84
-4.24,-3.75,-3.92,-3.98,-4.97, -4.5,-4.21,-3.21
-2.68,-3.02,-2.97,-3.35, -3.3,-3.83,-4.63,-4.41
-4.97, -4.2,-4.69,-5.22,-5.77,-6.28,-6.16, -5.9
-5.03,-4.78,-3.78, -3.9,
50AResult:
-7.69,-7.91,-7.68,-7.34,-6.79,-7.32,-7.95,-8.68
-8.24,-8.03,-7.78,-8.27,-8.87,-9.26,-9.72, -9.4
-8.77,-8.75,-8.55,-9.28, -8.1,-8.43,-7.86,-8.78
-9.11,-9.55,-9.15,-9.19,-9.64,-10.1,-9.96,-9.61
-8.91,-8.28,-8.14,-8.14,-8.93,-9.14,-10.3,-10.1
-9.79, -10,-10.5,-10.6,-10.5,-10.7,-11.7,-12.5
-13.2,-13.8,-13.7,-13.6,-12.4,-11.2,-9.61,-8.41
-6.74, -6.4,-5.85,-6.27,
模式匹配领域就像一片未被勘探的信息大海,引领你勇敢踏入数据科学的神秘领域。这是一场独特的学习冒险,从基本概念到算法实现,逐步揭示更深层次的模式分析、匹配算法和智能模式识别的奥秘。