《SIFT原理与源码分析》系列文章索引：http://www.cnblogs.com/tianyalu/p/5467813.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：

// 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[n];
    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。

// 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》

本文转自：http://blog.csdn.net/xiaowei_cqu/article/details/8087239
原文链接: https://www.cnblogs.com/tianyalu/p/5467854.html

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