【OpenCV】SIFT原理与源码分析:关键点搜索与定位

《SIFT原理与源码分析》系列文章索引：/article/1357504.html

由前一步《DoG尺度空间构造》，我们得到了DoG高斯差分金字塔：



如上图的金字塔，高斯尺度空间金字塔中每组有五层不同尺度图像，相邻两层相减得到四层DoG结果。关键点搜索就在这四层DoG图像上寻找局部极值点。

DoG局部极值点

寻找DoG极值点时，每一个像素点和它所有的相邻点比较，当其大于（或小于）它的图像域和尺度域的所有相邻点时，即为极值点。如下图所示，比较的范围是个3×3的立方体：中间的检测点和它同尺度的8个相邻点，以及和上下相邻尺度对应的9×2个点——共26个点比较，以确保在尺度空间和二维图像空间都检测到极值点。



在一组中，搜索从每组的第二层开始，以第二层为当前层，第一层和第三层分别作为立方体的的上下层；搜索完成后再以第三层为当前层做同样的搜索。所以每层的点搜索两次。通常我们将组Octaves索引以-1开始，则在比较时牺牲了-1组的第0层和第N组的最高层



高斯金字塔，DoG图像及极值计算的相互关系如上图所示。

关键点精确定位

以上极值点的搜索是在离散空间进行搜索的，由下图可以看到，在离散空间找到的极值点不一定是真正意义上的极值点。可以通过对尺度空间DoG函数进行曲线拟合寻找极值点来减小这种误差。



利用DoG函数在尺度空间的Taylor展开式：



则极值点为：



程序中还除去了极值小于0.04的点。如下所示：

[cpp] view
plaincopy

// Detects features at extrema in DoG scale space. Bad features are discarded

// based on contrast and ratio of principal curvatures.

// 在DoG尺度空间寻特征点（极值点）

void SIFT::findScaleSpaceExtrema( const vector<Mat>& gauss_pyr, const vector<Mat>& dog_pyr,

vector<KeyPoint>& keypoints ) const

{

int nOctaves = (int)gauss_pyr.size()/(nOctaveLayers + 3);

// The contrast threshold used to filter out weak features in semi-uniform

// (low-contrast) regions. The larger the threshold, the less features are produced by the detector.

// 过滤掉弱特征的阈值 contrastThreshold默认为0.04

int threshold = cvFloor(0.5 * contrastThreshold / nOctaveLayers * 255 * SIFT_FIXPT_SCALE);

const int n = SIFT_ORI_HIST_BINS; //36

float hist
;

KeyPoint kpt;

keypoints.clear();

for( int o = 0; o < nOctaves; o++ )

for( int i = 1; i <= nOctaveLayers; i++ )

{

int idx = o*(nOctaveLayers+2)+i;

const Mat& img = dog_pyr[idx];

const Mat& prev = dog_pyr[idx-1];

const Mat& next = dog_pyr[idx+1];

int step = (int)img.step1();

int rows = img.rows, cols = img.cols;

for( int r = SIFT_IMG_BORDER; r < rows-SIFT_IMG_BORDER; r++)

{

const short* currptr = img.ptr<short>(r);

const short* prevptr = prev.ptr<short>(r);

const short* nextptr = next.ptr<short>(r);

for( int c = SIFT_IMG_BORDER; c < cols-SIFT_IMG_BORDER; c++)

{

int val = currptr[c];

// find local extrema with pixel accuracy

// 寻找局部极值点，DoG中每个点与其所在的立方体周围的26个点比较

// if （val比所有都大 或者 val比所有都小）

if( std::abs(val) > threshold &&

((val > 0 && val >= currptr[c-1] && val >= currptr[c+1] &&

val >= currptr[c-step-1] && val >= currptr[c-step] &&

val >= currptr[c-step+1] && val >= currptr[c+step-1] &&

val >= currptr[c+step] && val >= currptr[c+step+1] &&

val >= nextptr[c] && val >= nextptr[c-1] &&

val >= nextptr[c+1] && val >= nextptr[c-step-1] &&

val >= nextptr[c-step] && val >= nextptr[c-step+1] &&

val >= nextptr[c+step-1] && val >= nextptr[c+step] &&

val >= nextptr[c+step+1] && val >= prevptr[c] &&

val >= prevptr[c-1] && val >= prevptr[c+1] &&

val >= prevptr[c-step-1] && val >= prevptr[c-step] &&

val >= prevptr[c-step+1] && val >= prevptr[c+step-1] &&

val >= prevptr[c+step] && val >= prevptr[c+step+1]) ||

(val < 0 && val <= currptr[c-1] && val <= currptr[c+1] &&

val <= currptr[c-step-1] && val <= currptr[c-step] &&

val <= currptr[c-step+1] && val <= currptr[c+step-1] &&

val <= currptr[c+step] && val <= currptr[c+step+1] &&

val <= nextptr[c] && val <= nextptr[c-1] &&

val <= nextptr[c+1] && val <= nextptr[c-step-1] &&

val <= nextptr[c-step] && val <= nextptr[c-step+1] &&

val <= nextptr[c+step-1] && val <= nextptr[c+step] &&

val <= nextptr[c+step+1] && val <= prevptr[c] &&

val <= prevptr[c-1] && val <= prevptr[c+1] &&

val <= prevptr[c-step-1] && val <= prevptr[c-step] &&

val <= prevptr[c-step+1] && val <= prevptr[c+step-1] &&

val <= prevptr[c+step] && val <= prevptr[c+step+1])))

{

int r1 = r, c1 = c, layer = i;

// 关键点精确定位

if( !adjustLocalExtrema(dog_pyr, kpt, o, layer, r1, c1,

nOctaveLayers, (float)contrastThreshold,

(float)edgeThreshold, (float)sigma) )

continue;

float scl_octv = kpt.size*0.5f/(1 << o);

// 计算梯度直方图

float omax = calcOrientationHist(

gauss_pyr[o*(nOctaveLayers+3) + layer],

Point(c1, r1),

cvRound(SIFT_ORI_RADIUS * scl_octv),

SIFT_ORI_SIG_FCTR * scl_octv,

hist, n);

float mag_thr = (float)(omax * SIFT_ORI_PEAK_RATIO);

for( int j = 0; j < n; j++ )

{

int l = j > 0 ? j - 1 : n - 1;

int r2 = j < n-1 ? j + 1 : 0;

if( hist[j] > hist[l] && hist[j] > hist[r2] && hist[j] >= mag_thr )

{

float bin = j + 0.5f * (hist[l]-hist[r2]) /

(hist[l] - 2*hist[j] + hist[r2]);

bin = bin < 0 ? n + bin : bin >= n ? bin - n : bin;

kpt.angle = (float)((360.f/n) * bin);

keypoints.push_back(kpt);

}

}

}

}

}

}

}

删除边缘效应

除了DoG响应较低的点，还有一些响应较强的点也不是稳定的特征点。DoG对图像中的边缘有较强的响应值，所以落在图像边缘的点也不是稳定的特征点。

一个平坦的DoG响应峰值在横跨边缘的地方有较大的主曲率，而在垂直边缘的地方有较小的主曲率。主曲率可以通过2×2的Hessian矩阵H求出：



D值可以通过求临近点差分得到。H的特征值与D的主曲率成正比，具体可参见Harris角点检测算法。

为了避免求具体的值，我们可以通过H将特征值的比例表示出来。令

为最大特征值，

为最小特征值，那么：



Tr(H)表示矩阵H的迹，Det(H)表示H的行列式。

令

表示最大特征值与最小特征值的比值，则有：



上式与两个特征值的比例有关。随着主曲率比值的增加，

也会增加。我们只需要去掉比率大于一定值的特征点。Lowe论文中去掉r=10的点。

[cpp] view
plaincopy

// Interpolates a scale-space extremum's location and scale to subpixel

// accuracy to form an image feature. Rejects features with low contrast.

// Based on Section 4 of Lowe's paper.

// 特征点精确定位

static bool adjustLocalExtrema( const vector<Mat>& dog_pyr, KeyPoint& kpt, int octv,

int& layer, int& r, int& c, int nOctaveLayers,

float contrastThreshold, float edgeThreshold, float sigma )

{

const float img_scale = 1.f/(255*SIFT_FIXPT_SCALE);

const float deriv_scale = img_scale*0.5f;

const float second_deriv_scale = img_scale;

const float cross_deriv_scale = img_scale*0.25f;

float xi=0, xr=0, xc=0, contr;

int i = 0;

//三维子像元插值

for( ; i < SIFT_MAX_INTERP_STEPS; i++ )

{

int idx = octv*(nOctaveLayers+2) + layer;

const Mat& img = dog_pyr[idx];

const Mat& prev = dog_pyr[idx-1];

const Mat& next = dog_pyr[idx+1];

Vec3f dD((img.at<short>(r, c+1) - img.at<short>(r, c-1))*deriv_scale,

(img.at<short>(r+1, c) - img.at<short>(r-1, c))*deriv_scale,

(next.at<short>(r, c) - prev.at<short>(r, c))*deriv_scale);

float v2 = (float)img.at<short>(r, c)*2;

float dxx = (img.at<short>(r, c+1) +

img.at<short>(r, c-1) - v2)*second_deriv_scale;

float dyy = (img.at<short>(r+1, c) +

img.at<short>(r-1, c) - v2)*second_deriv_scale;

float dss = (next.at<short>(r, c) +

prev.at<short>(r, c) - v2)*second_deriv_scale;

float dxy = (img.at<short>(r+1, c+1) -

img.at<short>(r+1, c-1) - img.at<short>(r-1, c+1) +

img.at<short>(r-1, c-1))*cross_deriv_scale;

float dxs = (next.at<short>(r, c+1) -

next.at<short>(r, c-1) - prev.at<short>(r, c+1) +

prev.at<short>(r, c-1))*cross_deriv_scale;

float dys = (next.at<short>(r+1, c) -

next.at<short>(r-1, c) - prev.at<short>(r+1, c) +

prev.at<short>(r-1, c))*cross_deriv_scale;

Matx33f H(dxx, dxy, dxs,

dxy, dyy, dys,

dxs, dys, dss);

Vec3f X = H.solve(dD, DECOMP_LU);

xi = -X[2];

xr = -X[1];

xc = -X[0];

if( std::abs( xi ) < 0.5f && std::abs( xr ) < 0.5f && std::abs( xc ) < 0.5f )

break;

//将找到的极值点对应成像素（整数）

c += cvRound( xc );

r += cvRound( xr );

layer += cvRound( xi );

if( layer < 1 || layer > nOctaveLayers ||

c < SIFT_IMG_BORDER || c >= img.cols - SIFT_IMG_BORDER ||

r < SIFT_IMG_BORDER || r >= img.rows - SIFT_IMG_BORDER )

return false;

}

/* ensure convergence of interpolation */

// SIFT_MAX_INTERP_STEPS:插值最大步数，避免插值不收敛，程序中默认为5

if( i >= SIFT_MAX_INTERP_STEPS )

return false;

{

int idx = octv*(nOctaveLayers+2) + layer;

const Mat& img = dog_pyr[idx];

const Mat& prev = dog_pyr[idx-1];

const Mat& next = dog_pyr[idx+1];

Matx31f dD((img.at<short>(r, c+1) - img.at<short>(r, c-1))*deriv_scale,

(img.at<short>(r+1, c) - img.at<short>(r-1, c))*deriv_scale,

(next.at<short>(r, c) - prev.at<short>(r, c))*deriv_scale);

float t = dD.dot(Matx31f(xc, xr, xi));

contr = img.at<short>(r, c)*img_scale + t * 0.5f;

if( std::abs( contr ) * nOctaveLayers < contrastThreshold )

return false;

/* principal curvatures are computed using the trace and det of Hessian */

//利用Hessian矩阵的迹和行列式计算主曲率的比值

float v2 = img.at<short>(r, c)*2.f;

float dxx = (img.at<short>(r, c+1) +

img.at<short>(r, c-1) - v2)*second_deriv_scale;

float dyy = (img.at<short>(r+1, c) +

img.at<short>(r-1, c) - v2)*second_deriv_scale;

float dxy = (img.at<short>(r+1, c+1) -

img.at<short>(r+1, c-1) - img.at<short>(r-1, c+1) +

img.at<short>(r-1, c-1)) * cross_deriv_scale;

float tr = dxx + dyy;

float det = dxx * dyy - dxy * dxy;

//这里edgeThreshold可以在调用SIFT()时输入；

//其实代码中定义了 static const float SIFT_CURV_THR = 10.f 可以直接使用

if( det <= 0 || tr*tr*edgeThreshold >= (edgeThreshold + 1)*(edgeThreshold + 1)*det )

return false;

}

kpt.pt.x = (c + xc) * (1 << octv);

kpt.pt.y = (r + xr) * (1 << octv);

kpt.octave = octv + (layer << 8) + (cvRound((xi + 0.5)*255) << 16);

kpt.size = sigma*powf(2.f, (layer + xi) / nOctaveLayers)*(1 << octv)*2;

return true;

}

至此，SIFT第二步就完成了。参见《SIFT原理与源码分析》

（转载请注明作者和出处：http://blog.csdn.net/xiaowei_cqu 未经允许请勿用于商业用途）