幾何不變矩--Hu矩

【圖像算法】圖像特征：

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一 原理

    幾何矩是由Hu(Visual pattern recognition by moment invariants)在1962年提出的，具有平移、旋轉和尺度不變性。 定義如下：

① (p+q)階不變矩定義：

② 對於數字圖像，離散化，定義為：

 

③ 歸一化中心矩定義：

 

④Hu矩定義

 

 

 

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二 實現(源碼)

①自編函數模塊C++

//#################################################################################//
double M[7] = {0}; //HU不變矩
bool HuMoment(IplImage* img)
{
int bmpWidth = img->width;
int bmpHeight = img->height;
int bmpStep = img->widthStep; 
int bmpChannels = img->nChannels;
uchar*pBmpBuf = (uchar*)img->imageData;

double m00=0,m11=0,m20=0,m02=0,m30=0,m03=0,m12=0,m21=0; //中心矩 
double x0=0,y0=0; //計算中心距時所使用的臨時變量（x-x'） 
double u20=0,u02=0,u11=0,u30=0,u03=0,u12=0,u21=0;//規范化后的中心矩
//double M[7]; //HU不變矩 
double t1=0,t2=0,t3=0,t4=0,t5=0;//臨時變量， 
//double Center_x=0,Center_y=0;//重心 
int Center_x=0,Center_y=0;//重心 
int i,j; //循環變量

// 獲得圖像的區域重心(普通矩)
double s10=0,s01=0,s00=0; //0階矩和1階矩 
for(j=0;j<bmpHeight;j++)//y
{
for(i=0;i<bmpWidth;i++)//x
{
s10+=i*pBmpBuf[j*bmpStep+i];
s01+=j*pBmpBuf[j*bmpStep+i];
s00+=pBmpBuf[j*bmpStep+i];
}
}
Center_x=(int)(s10/s00+0.5);
Center_y=(int)(s01/s00+0.5);

// 計算二階、三階矩(中心矩)
m00=s00; 
for(j=0;j<bmpHeight;j++) 
{
for(i=0;i<bmpWidth;i++)//x 
{ 
x0=(i-Center_x); 
y0=(j-Center_y); 
m11+=x0*y0*pBmpBuf[j*bmpStep+i]; 
m20+=x0*x0*pBmpBuf[j*bmpStep+i]; 
m02+=y0*y0*pBmpBuf[j*bmpStep+i]; 
m03+=y0*y0*y0*pBmpBuf[j*bmpStep+i];
m30+=x0*x0*x0*pBmpBuf[j*bmpStep+i]; 
m12+=x0*y0*y0*pBmpBuf[j*bmpStep+i]; 
m21+=x0*x0*y0*pBmpBuf[j*bmpStep+i]; 
} 
}

// 計算規范化后的中心矩: mij/pow(m00,((i+j+2)/2)
u20=m20/pow(m00,2); 
u02=m02/pow(m00,2); 
u11=m11/pow(m00,2);
u30=m30/pow(m00,2.5); 
u03=m03/pow(m00,2.5);
u12=m12/pow(m00,2.5); 
u21=m21/pow(m00,2.5);

// 計算中間變量
t1=(u20-u02); 
t2=(u30-3*u12); 
t3=(3*u21-u03); 
t4=(u30+u12);
t5=(u21+u03);

// 計算不變矩 
M[0]=u20+u02; 
M[1]=t1*t1+4*u11*u11; 
M[2]=t2*t2+t3*t3; 
M[3]=t4*t4+t5*t5;
M[4]=t2*t4*(t4*t4-3*t5*t5)+t3*t5*(3*t4*t4-t5*t5); 
M[5]=t1*(t4*t4-t5*t5)+4*u11*t4*t5;
M[6]=t3*t4*(t4*t4-3*t5*t5)-t2*t5*(3*t4*t4-t5*t5);

returntrue;
}

　　

②調用OpenCV方法

1 //  利用OpenCV函數求7個Hu矩
2     CvMoments moments;
3     CvHuMoments hu;
4     cvMoments(bkImgEdge,&moments,0); 
5     cvGetHuMoments(&moments, &hu);
6     cout<<hu.hu1<<"/"<<hu.hu2<<"/"<<hu.hu3<<"/"<<hu.hu4<<"/"<<hu.hu5<<"/"<<hu.hu6<<"/"<<hu.hu7<<"/"<<"/"<<endl;
7     cvMoments(testImgEdge,&moments,0); 
8     cvGetHuMoments(&moments, &hu);
9     cout<<hu.hu1<<"/"<<hu.hu2<<"/"<<hu.hu3<<"/"<<hu.hu4<<"/"<<hu.hu5<<"/"<<hu.hu6<<"/"<<hu.hu7<<"/"<<"/"<<endl;

Python調用OpenCV：

#-*-coding:utf-8-*-
import cv2
from datetime import datetime
import numpy as np

def test(img):
    moments = cv2.moments(img)
    humoments = cv2.HuMoments(moments)
    # humoments = no.log(np.abs(humoments)) # 同樣建議取對數
    print(humoments)

if __name__ == '__main__':
    t1 = datetime.now()    
    fp = '/home/mamq/images/3.jpg'
    img = cv2.imread(fp)
    img_gray = cv2.cvtColor(img, cv2.COLOR_BGR2GRAY)
    test(img_gray)
    print datetime.now() - t1

　　

 

Python方法：

#-*-coding:utf-8-*-
import cv2
from datetime import datetime
import numpy as np
np.set_printoptions(suppress=True)

def humoments(img_gray):
    '''
    由於7個不變矩的變化范圍很大,為了便於比較,可利用取對數的方法進行數據壓縮;同時考慮到不變矩有可能出現負值的情況,因此,在取對數之前先取絕對值
    經修正后的不變矩特征具有平移 、旋轉和比例不變性
    '''
    # 標准矩定義為m_pq = sumsum(x^p * y^q * f(x, y))
    row, col = img_gray.shape
    #計算圖像的0階幾何矩
    m00 = img_gray.sum()
    m10 = m01 = 0
    #　計算圖像的二階、三階幾何矩
    m11 = m20 = m02 = m12 = m21 = m30 = m03 = 0
    for i in range(row):
        m10 += (i * img_gray[i]).sum()
        m20 += (i ** 2 * img_gray[i]).sum()
        m30 += (i ** 3 * img_gray[i]).sum()
        for j in range(col):
            m11 += i * j * img_gray[i][j]
            m12 += i * j ** 2 * img_gray[i][j]
            m21 += i ** 2 * j * img_gray[i][j]
    for j in range(col):
        m01 += (j * img_gray[:, j]).sum()
        m02 += (j ** 2 * img_gray[:, j]).sum()
        m30 += (j ** 3 * img_gray[:, j]).sum()
    # 由標准矩我們可以得到圖像的"重心"
    u10 = m10 / m00
    u01 = m01 / m00
    # 計算圖像的二階中心矩、三階中心矩
    y00 = m00
    y10 = y01 = 0
    y11 = m11 - u01 * m10
    y20 = m20 - u10 * m10
    y02 = m02 - u01 * m01
    y30 = m30 - 3 * u10 * m20 + 2 * u10 ** 2 * m10
    y12 = m12 - 2 * u01 * m11 - u10 * m02 + 2 * u01 ** 2 * m10
    y21 = m21 - 2 * u10 * m11 - u01 * m20 + 2 * u10 ** 2 * m01
    y03 = m03 - 3 * u01 * m02 + 2 * u01 ** 2 * m01
    # 計算圖像的歸格化中心矩
    n20 = y20 / m00 ** 2
    n02 = y02 / m00 ** 2
    n11 = y11 / m00 ** 2
    n30 = y30 / m00 ** 2.5
    n03 = y03 / m00 ** 2.5
    n12 = y12 / m00 ** 2.5
    n21 = y21 / m00 ** 2.5
    # 計算圖像的七個不變矩
    h1 = n20 + n02
    h2 = (n20 - n02) ** 2 + 4 * n11 ** 2
    h3 = (n30 - 3 * n12) ** 2 + (3 * n21 - n03) ** 2
    h4 = (n30 + n12) ** 2 + (n21 + n03) ** 2
    h5 = (n30 - 3 * n12) * (n30 + n12) * ((n30 + n12) ** 2 - 3 * (n21 + n03) ** 2) + (3 * n21 - n03) * (n21 + n03) \
        * (3 * (n30 + n12) ** 2 - (n21 + n03) ** 2)
    h6 = (n20 - n02) * ((n30 + n12) ** 2 - (n21 + n03) ** 2) + 4 * n11 * (n30 + n12) * (n21 + n03)
    h7 = (3 * n21 - n03) * (n30 + n12) * ((n30 + n12) ** 2 - 3 * (n21 + n03) ** 2) + (3 * n12 - n30) * (n21 + n03) \
        * (3 * (n30 + n12) ** 2 - (n21 + n03) ** 2)
    inv_m7 = [h1, h2, h3, h4, h5, h6, h7]
    inv_m7 = np.log(np.abs(inv_m7))
    return inv_m7

if __name__ == '__main__':
    t1 = datetime.now()
    fp = '/home/mamq/images/3.jpg'
    img = cv2.imread(fp)
    img_gray = cv2.cvtColor(img, cv2.COLOR_BGR2GRAY)
    print humoments(img_gray)
    print datetime.now() - t1

　　

　　

MATLAB方法：

invariable_moment(imread('lena.jpg'));

function inv_m7 = invariable_moment(in_image)
% 功能：計算圖像的Hu的七個不變矩
% 輸入：in_image-RGB圖像
% 輸出：inv_m7-七個不變矩
 
% 將輸入的RGB圖像轉換為灰度圖像   
image=rgb2gray(in_image);     
%將圖像矩陣的數據類型轉換成雙精度型
image=double(image);      
%%%=================計算 、 、 =========================
%計算灰度圖像的零階幾何矩 
m00=sum(sum(image));     
m10=0;
m01=0;
[row,col]=size(image);
for i=1:row
    for j=1:col
        m10=m10+i*image(i,j);
        m01=m01+j*image(i,j);
    end
end
%%%=================計算 、 ================================
u10=m10/m00;
u01=m01/m00;
%%%=================計算圖像的二階幾何矩、三階幾何矩============
m20 = 0;m02 = 0;m11 = 0;m30 = 0;m12 = 0;m21 = 0;m03 = 0;
for i=1:row
    for j=1:col
        m20=m20+i^2*image(i,j);
        m02=m02+j^2*image(i,j);
        m11=m11+i*j*image(i,j);
        m30=m30+i^3*image(i,j);
        m03=m03+j^3*image(i,j);
        m12=m12+i*j^2*image(i,j);
        m21=m21+i^2*j*image(i,j);
    end
end
%%%=================計算圖像的二階中心矩、三階中心矩============
y00=m00;
y10=0;
y01=0;
y11=m11-u01*m10;
y20=m20-u10*m10;
y02=m02-u01*m01;
y30=m30-3*u10*m20+2*u10^2*m10;
y12=m12-2*u01*m11-u10*m02+2*u01^2*m10;
y21=m21-2*u10*m11-u01*m20+2*u10^2*m01;
y03=m03-3*u01*m02+2*u01^2*m01;
%%%=================計算圖像的歸格化中心矩====================
        n20=y20/m00^2;
        n02=y02/m00^2;
        n11=y11/m00^2;
        n30=y30/m00^2.5;
        n03=y03/m00^2.5;
        n12=y12/m00^2.5;
        n21=y21/m00^2.5;
%%%=================計算圖像的七個不變矩======================
h1 = n20 + n02;                      
h2 = (n20-n02)^2 + 4*(n11)^2;
h3 = (n30-3*n12)^2 + (3*n21-n03)^2;  
h4 = (n30+n12)^2 + (n21+n03)^2;
h5 = (n30-3*n12)*(n30+n12)*((n30+n12)^2-3*(n21+n03)^2)+(3*n21-n03)*(n21+n03)*(3*(n30+n12)^2-(n21+n03)^2);
h6 = (n20-n02)*((n30+n12)^2-(n21+n03)^2)+4*n11*(n30+n12)*(n21+n03);
  h7 = (3*n21-n03)*(n30+n12)*((n30+n12)^2-3*(n21+n03)^2)+(3*n12-n30)*(n21+n03)*(3*(n30+n12)^2-(n21+n03)^2);
 
inv_m7= [h1 h2 h3 h4 h5 h6 h7];  

　　

 

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三 相似性准則

①法一

//  計算相似度1
double dbR =0; //相似度
double dSigmaST =0;
    double dSigmaS =0;
    double dSigmaT =0;
    double temp =0;  
    {for(int i=0;i<7;i++)
    {
        temp = fabs(Sa[i]*Ta[i]);
        dSigmaST+=temp;
        dSigmaS+=pow(Sa[i],2);
        dSigmaT+=pow(Ta[i],2);
    }}
    dbR = dSigmaST/(sqrt(dSigmaS)*sqrt(dSigmaT));

　　

②法二 

 1 //  計算相似度2
 2 double dbR2 =0; //相似度
 3 double temp2 =0;
 4 double temp3 =0;  
 5     {for(int i=0;i<7;i++)
 6     {
 7         temp2 += fabs(Sa[i]-Ta[i]);
 8         temp3 += fabs(Sa[i]+Ta[i]);
 9     }}
10     dbR2 =1- (temp2*1.0)/(temp3);

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Author:         SKySeraph

 

Email/GTalk: zgzhaobo@gmail.com    QQ：452728574

 

From：         http://www.cnblogs.com/skyseraph/

 

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