SIFT特征的原理学习资料开始学习

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1. SIFT特征的性质

SIFT特征不只具有尺度不变性,即使改变图像的旋转角度,亮度或拍摄视角,仍然能够得到好的检测效果。

2. SIFT算法的流程

2.1 构建尺度空间

这是一个初始化操作,尺度空间理论目的是模拟图像数据的多尺度特征。高斯卷积核是实现尺度变换的唯一线性核,于是一副二维图像的尺度空间定义为:

其中 G(x,y,σ) 是尺度可变高斯函数

(x,y)是空间坐标,是尺度坐标。σ大小决定图像的平滑程度,大尺度对应图像的概貌特征,小尺度对应图像的细节特征。大的σ值对应粗糙尺度(低分辨率),反之,对应精细尺度(高分辨率)。为了有效的在尺度空间检测到稳定的关键点,提出了高斯差分尺度空间(DOG scale-space)。利用不同尺度的高斯差分核与图像卷积生成。

下图所示不同σ下图像尺度空间:

关于尺度空间的理解说明:2kσ中的2是必须的,尺度空间是连续的。在 Lowe的论文中 ,将第0层的初始尺度定为1.6(最模糊),图片的初始尺度定为0.5(最清晰). 在检测极值点前对原始图像的高斯平滑以致图像丢失高频信息,所以 Lowe 建议在建立尺度空间前首先对原始图像长宽扩展一倍,以保留原始图像信息,增加特征点数量。尺度越大图像越模糊。

图像金字塔的建立:对于一幅图像I,建立其在不同尺度(scale)的图像,也成为子八度(octave),这是为了scale-invariant,也就是在任何尺度都能够有对应的特征点,第一个子八度的scale为原图大小,后面每个octave为上一个octave降采样的结果,即原图的1/4(长宽分别减半),构成下一个子八度(高一层金字塔)。

尺度空间的所有取值,i为octave的塔数(第几个塔),s为每塔层数 由图片size决定建几个塔,每塔几(S)层图像(S一般为3-5层)。0塔的第0层是原始图像(或你double后的图像),往上每一层是对其下一层进行Laplacian变换(高斯卷积,其中σ值渐大,例如可以是σ, kσ, kk*σ…),直观上看来越往上图片越模糊。塔间的图片是降采样关系,例如1塔的第0层可以由0塔的第3层down sample得到,然后进行与0塔类似的高斯卷积操作。

2.2 LoG近似DoG找到关键点(检测DOG尺度空间极值点)

注意:使用Difference of Gaussian图像的极大极小值近似寻找特征点计算简单,是尺度归一化的LoG算子的近似

为了寻找尺度空间的极值点,每一个采样点要和它所有的相邻点比较,看其是否比它的图像域和尺度域的相邻点大或者小。 一个点如果在DOG尺度空间本层以及上下两层的26个领域中是最大或最小值时,就认为该点是图像在该尺度下的一个特征点,如图所示。

X点和尺度空间中相邻26个点作比较

整个高斯金字塔如下图所示,其中每个Octave代表一个金字塔,同一个金字塔内图像尺寸一样,同一个金字塔内每张图通过不同的高斯卷积核产生。

SIFT特征提取过程中建立的所有金字塔

在极值比较的过程中,每一组图像的首末两层是无法进行极值比较的,为了满足尺度变化的连续性,我们在每一组图像的顶层继续用高斯模糊生成了 3 幅图像,高斯金字塔有每组S+3层图像。DOG金字塔每组有S+2层图像。

解释下什么叫“为了满足尺度变化的连续性”: 假设s=3,也就是每个塔里有3层,则k=21/s=21/3,那么按照上图可得Gauss Space和DoG space 分别有3个(s个) 和2个(s-1个)分量,在DoG space中,1st-octave两项分别是σ,kσ; 2nd-octave两项分别是2σ,2kσ; 由于无法比较极值,我们必须在高斯空间继续添加高斯模糊项,使得形成σ,kσ,k2σ,k3σ,k4σ(?)这样就可以选择 DoG space中的中间三项kσ,k2σ,k3σ(只有左右都有才能有极值),那么下一octave中(由上一层降采样获得) 所得三项即为2kσ,2k2σ,2k3σ,其首项2kσ=24/3。刚好与上一octave末项k3σ=23/3尺度变化连续起来,所以每次要在 Gaussian space添加3项,每组(塔)共S+3层图像,相应的DoG金字塔有S+2层图像。

使用Laplacian of Gaussian能够很好地找到找到图像中的兴趣点,但是需要大量的计算量,所以使用Difference of Gaussian图像的极大极小值近似寻找特征点.DOG算子计算简单,是尺度归一化的LoG算子的近似,有关DOG寻找特征点的介绍及方法详见http://blog.csdn.net/abcjennifer/article/details/7639488,极值点检测用的Non-Maximal Suppression。

3. 除去不好的特征点

这一步本质上要去掉DoG局部曲率非常不对称的像素。(不理解)

通过拟和三维二次函数以精确确定关键点的位置尺度(达到亚像素精度),同时去除低对比度的关键点和不稳定的边缘响应点(因为DoG算子会产生较强的边缘响应),以增强匹配稳定性、提高抗噪声能力,在这里使用近似Harris Corner检测器。 计算过程摘录如下:(还没有自行推导) ①空间尺度函数泰勒展开式如下:

对上式求导,并令其为0,得到精确的位置, 得

②在已经检测到的特征点中,要去掉低对比度的特征点和不稳定的边缘响应点。去除低对比度的点:把公式(2)代入公式(1),即在DoG Space的极值点处D(x)取值,只取前两项可得:

该特征点就保留下来,否则丢弃。 ③边缘响应的去除一个定义不好的高斯差分算子的极值在横跨边缘的地方有较大的主曲率,而在垂直边缘的方向有较小的主曲率。主曲率通过一个2×2 的Hessian矩阵H求出:

导数由采样点相邻差估计得到。 D的主曲率和H的特征值成正比,令α为较大特征值,β为较小的特征值,则

令α=γβ,则

的值在两个特征值相等的时候最小,随着r的增大而增大,因此,为了检测主曲率是否在某域值r下,只需检测

if (α+β)/ αβ> (r+1)2 /r, throw it out. 在Lowe的文章中,取r=10。

4. 给特征点赋值一个128维方向参数

上一步中确定了每幅图中的特征点,为每个特征点计算一个方向,依照这个方向做进一步的计算, 利用关键点邻域像素的梯度方向分布特性为每个关键点指定方向参数,使算子具备旋转不变性。

为(x,y)处梯度的模值和方向公式。 其中L所用的尺度为每个关键点各自所在的尺度。至此,图像的关键点检测完毕,每个关键点有三个信息:位置所处尺度方向,由此可以确定一个SIFT特征区域。

个人理解上面求特征点梯度方向和赋值的算式,本质上是求特征点水平和垂直方向像素值构成的两个向量的夹角和距离。

梯度直方图的范围是0~360度,其中每10度一个柱,总共36个柱。随着距 中心点越远的领域其对直方图的贡献也响应减小。Lowe论文中还提到要使用高斯函数对直方图进行平滑,减少突变的影响。这主要是因为SIFT算法只考虑了尺度和旋转不变形,没有考虑仿射不变性。通过高斯平滑,可以使关键点附近的梯度幅值有较大权重,从而部分弥补没考虑仿射不变形产生的特征点不稳定。 通常离散的梯度直方图要进行插值拟合处理,以求取更精确的方向角度值。

在实际计算时,我们在以关键点为中心的邻域窗口内采样,并用直方图统计邻域像素的梯度方向。梯度直方图的范围是0~360度,其中每45度一个柱,总共8个柱, 或者每10度一个柱,总共36个柱。Lowe论文中还提到要使用高斯函数对直方图进行平滑,减少突变的影响。直方图的峰值则代表了该关键点处邻域梯度的主方向,即作为该关键点的方向。

直方图中的峰值就是主方向,其他的达到最大值80%的方向可作为辅助方向

由梯度方向直方图确定主梯度方向 该步中将建立所有scale中特征点的描述子(128维)

Identify peak and assign orientation and sum of magnitude to key point. ** The user may choose a threshold to exclude key points based on their** assigned sum of magnitudes.

直方图峰值代表该关键点处邻域内图像梯度的主方向,也就是该关键点的主方向。在梯度方向直方图中,当存在另一个相当于主峰值 80%能量的峰值时,则将这个方向认为是该关键点的辅方向。所以一个关键点可能检测得到多个方向,这可以增强匹配的鲁棒性。Lowe的论文指出大概有15%关键点具有多方向,但这些点对匹配的稳定性至为关键。 获得图像关键点主方向后,每个关键点有三个信息(x,y,σ,θ):位置、尺度、方向。由此我们可以确定一个SIFT特征区域。通常使用一个带箭头的圆或直接使用箭头表示SIFT区域的三个值:中心表示特征点位置,半径表示关键点尺度(r=2.5σ),箭头表示主方向。具有多个方向的关键点可以复制成多份,然后将方向值分别赋给复制后的关键点。如下图:

关键点描述子的生成步骤

通过对关键点周围图像区域分块,计算块内梯度直方图,生成具有独特性的向量,这个向量是该区域图像信息的一种抽象,具有唯一性。

5.关键点描述子的生成

首先将坐标轴旋转为关键点的方向,以确保旋转不变性。以关键点为中心取8×8的窗口。

Figure.16*16的图中其中1/4的特征点梯度方向及scale,右图为其加权到8个主方向后的效果。 图左部分的中央为当前关键点的位置,每个小格代表关键点邻域所在尺度空间的一个像素,利用公式求得每个像素的梯度幅值与梯度方向,箭头方向代表该像素的梯度方向,箭头长度代表梯度模值,然后用高斯窗口对其进行加权运算。

图中蓝色的圈代表高斯加权的范围(越靠近关键点的像素梯度方向信息贡献越大)。然后在每4×4的小块上计算8个方向的梯度方向直方图,绘制每个梯度方向的累加值,即可形成一个种子点,如图右部分示。此图中一个关键点由2×2共4个种子点组成,每个种子点有8个方向向量信息。这种邻域方向性信息联合的思想增强了算法抗噪声的能力,同时对于含有定位误差的特征匹配也提供了较好的容错性。

计算keypoint周围的16*16的window中每一个像素的梯度,而且使用高斯下降函数降低远离中心的权重。

在每个44的1/16象限中,通过加权梯度值加到直方图8个方向区间中的一个,计算出一个梯度方向直方图。 这样就可以对每个feature形成一个448=128维的描述子,每一维都可以表示44个格子中一个的scale/orientation. 将这个向量归一化之后,就进一步去除了光照的影响。

6. 根据SIFT进行Match

生成了A、B两幅图的描述子,(分别是k1128维和k2128维),就将两图中各个scale(所有scale)的描述子进行匹配,匹配上128维即可表示两个特征点match上了。

实际计算过程中,为了增强匹配的稳健性,Lowe建议对每个关键点使用4×4共16个种子点来描述,这样对于一个关键点就可以产生128个数据,即最终形成128维的SIFT特征向量。此时SIFT特征向量已经去除了尺度变化、旋转等几何变形因素的影响,再继续将特征向量的长度归一化,则可以进一步去除光照变化的影响。 当两幅图像的SIFT特征向量生成后,下一步我们采用关键点特征向量的欧式距离来作为两幅图像中关键点的相似性判定度量。取图像1中的某个关键点,并找出其与图像2中欧式距离最近的前两个关键点,在这两个关键点中,如果最近的距离除以次近的距离少于某个比例阈值,则接受这一对匹配点。降低这个比例阈值,SIFT匹配点数目会减少,但更加稳定。为了排除因为图像遮挡和背景混乱而产生的无匹配关系的关键点,Lowe提出了比较最近邻距离与次近邻距离的方法,距离比率ratio小于某个阈值的认为是正确匹配。因为对于错误匹配,由于特征空间的高维性,相似的距离可能有大量其他的错误匹配,从而它的ratio值比较高。Lowe推荐ratio的阈值为0.8。但作者对大量任意存在尺度、旋转和亮度变化的两幅图片进行匹配,结果表明ratio取值在0. 4~0. 6之间最佳,小于0. 4的很少有匹配点,大于0. 6的则存在大量错误匹配点。(如果这个地方你要改进,最好给出一个匹配率和ration之间的关系图,这样才有说服力)作者建议ratio的取值原则如下: ratio=0. 4 对于准确度要求高的匹配;ratio=0. 6 对于匹配点数目要求比较多的匹配; ratio=0. 5 一般情况下。也可按如下原则:当最近邻距离<200时ratio=0. 6,反之ratio=0. 4。ratio的取值策略能排分错误匹配点。

当两幅图像的SIFT特征向量生成后,下一步我们采用关键点特征向量的欧式距离来作为两幅图像中关键点的相似性判定度量。取图像1中的某个关键点,并找出其与图像2中欧式距离最近的前两个关键点,在这两个关键点中,如果最近的距离除以次近的距离少于某个比例阈值,则接受这一对匹配点。降低这个比例阈值,SIFT匹配点数目会减少,但更加稳定。

7.OpenCV3.2.0版本中SIFT源码

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/**********************************************************************************************\
 Implementation of SIFT is based on the code from http://blogs.oregonstate.edu/hess/code/sift/
 Below is the original copyright.

//    Copyright (c) 2006-2010, Rob Hess <hess@eecs.oregonstate.edu>
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//    The following patent has been issued for methods embodied in this
//    software: "Method and apparatus for identifying scale invariant features
//    in an image and use of same for locating an object in an image," David
//    G. Lowe, US Patent 6,711,293 (March 23, 2004). Provisional application
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#include "precomp.hpp"
#include <iostream>
#include <stdarg.h>
#include <opencv2/core/hal/hal.hpp>

namespace cv
{
namespace xfeatures2d
{

/*!
 SIFT implementation.

 The class implements SIFT algorithm by D. Lowe.
 */
class SIFT_Impl : public SIFT
{
public:
    explicit SIFT_Impl( int nfeatures = 0, int nOctaveLayers = 3,
                          double contrastThreshold = 0.04, double edgeThreshold = 10,
                          double sigma = 1.6);

    //! returns the descriptor size in floats (128)
    int descriptorSize() const;

    //! returns the descriptor type
    int descriptorType() const;

    //! returns the default norm type
    int defaultNorm() const;

    //! finds the keypoints and computes descriptors for them using SIFT algorithm.
    //! Optionally it can compute descriptors for the user-provided keypoints
    void detectAndCompute(InputArray img, InputArray mask,
                    std::vector<KeyPoint>& keypoints,
                    OutputArray descriptors,
                    bool useProvidedKeypoints = false);

    void buildGaussianPyramid( const Mat& base, std::vector<Mat>& pyr, int nOctaves ) const;
    void buildDoGPyramid( const std::vector<Mat>& pyr, std::vector<Mat>& dogpyr ) const;
    void findScaleSpaceExtrema( const std::vector<Mat>& gauss_pyr, const std::vector<Mat>& dog_pyr,
                               std::vector<KeyPoint>& keypoints ) const;

protected:
    CV_PROP_RW int nfeatures;
    CV_PROP_RW int nOctaveLayers;
    CV_PROP_RW double contrastThreshold;
    CV_PROP_RW double edgeThreshold;
    CV_PROP_RW double sigma;
};

Ptr<SIFT> SIFT::create( int _nfeatures, int _nOctaveLayers,
                     double _contrastThreshold, double _edgeThreshold, double _sigma )
{
    return makePtr<SIFT_Impl>(_nfeatures, _nOctaveLayers, _contrastThreshold, _edgeThreshold, _sigma);
}

/******************************* Defs and macros *****************************/

// default width of descriptor histogram array
static const int SIFT_DESCR_WIDTH = 4;

// default number of bins per histogram in descriptor array
static const int SIFT_DESCR_HIST_BINS = 8;

// assumed gaussian blur for input image
static const float SIFT_INIT_SIGMA = 0.5f;

// width of border in which to ignore keypoints
static const int SIFT_IMG_BORDER = 5;

// maximum steps of keypoint interpolation before failure
static const int SIFT_MAX_INTERP_STEPS = 5;

// default number of bins in histogram for orientation assignment
static const int SIFT_ORI_HIST_BINS = 36;

// determines gaussian sigma for orientation assignment
static const float SIFT_ORI_SIG_FCTR = 1.5f;

// determines the radius of the region used in orientation assignment
static const float SIFT_ORI_RADIUS = 3 * SIFT_ORI_SIG_FCTR;

// orientation magnitude relative to max that results in new feature
static const float SIFT_ORI_PEAK_RATIO = 0.8f;

// determines the size of a single descriptor orientation histogram
static const float SIFT_DESCR_SCL_FCTR = 3.f;

// threshold on magnitude of elements of descriptor vector
static const float SIFT_DESCR_MAG_THR = 0.2f;

// factor used to convert floating-point descriptor to unsigned char
static const float SIFT_INT_DESCR_FCTR = 512.f;

#if 0
// intermediate type used for DoG pyramids
typedef short sift_wt;
static const int SIFT_FIXPT_SCALE = 48;
#else
// intermediate type used for DoG pyramids
typedef float sift_wt;
static const int SIFT_FIXPT_SCALE = 1;
#endif

static inline void
unpackOctave(const KeyPoint& kpt, int& octave, int& layer, float& scale)
{
    octave = kpt.octave & 255;
    layer = (kpt.octave >> 8) & 255;
    octave = octave < 128 ? octave : (-128 | octave);
    scale = octave >= 0 ? 1.f/(1 << octave) : (float)(1 << -octave);
}

static Mat createInitialImage( const Mat& img, bool doubleImageSize, float sigma )
{
    Mat gray, gray_fpt;
    if( img.channels() == 3 || img.channels() == 4 )
    {
        cvtColor(img, gray, COLOR_BGR2GRAY);
        gray.convertTo(gray_fpt, DataType<sift_wt>::type, SIFT_FIXPT_SCALE, 0);
    }
    else
        img.convertTo(gray_fpt, DataType<sift_wt>::type, SIFT_FIXPT_SCALE, 0);

    float sig_diff;

    if( doubleImageSize )
    {
        sig_diff = sqrtf( std::max(sigma * sigma - SIFT_INIT_SIGMA * SIFT_INIT_SIGMA * 4, 0.01f) );
        Mat dbl;
        resize(gray_fpt, dbl, Size(gray_fpt.cols*2, gray_fpt.rows*2), 0, 0, INTER_LINEAR);
        GaussianBlur(dbl, dbl, Size(), sig_diff, sig_diff);
        return dbl;
    }
    else
    {
        sig_diff = sqrtf( std::max(sigma * sigma - SIFT_INIT_SIGMA * SIFT_INIT_SIGMA, 0.01f) );
        GaussianBlur(gray_fpt, gray_fpt, Size(), sig_diff, sig_diff);
        return gray_fpt;
    }
}


void SIFT_Impl::buildGaussianPyramid( const Mat& base, std::vector<Mat>& pyr, int nOctaves ) const
{
    std::vector<double> sig(nOctaveLayers + 3);
    pyr.resize(nOctaves*(nOctaveLayers + 3));

    // precompute Gaussian sigmas using the following formula:
    //  \sigma_{total}^2 = \sigma_{i}^2 + \sigma_{i-1}^2
    sig[0] = sigma;
    double k = std::pow( 2., 1. / nOctaveLayers );
    for( int i = 1; i < nOctaveLayers + 3; i++ )
    {
        double sig_prev = std::pow(k, (double)(i-1))*sigma;
        double sig_total = sig_prev*k;
        sig[i] = std::sqrt(sig_total*sig_total - sig_prev*sig_prev);
    }

    for( int o = 0; o < nOctaves; o++ )
    {
        for( int i = 0; i < nOctaveLayers + 3; i++ )
        {
            Mat& dst = pyr[o*(nOctaveLayers + 3) + i];
            if( o == 0  &&  i == 0 )
                dst = base;
            // base of new octave is halved image from end of previous octave
            else if( i == 0 )
            {
                const Mat& src = pyr[(o-1)*(nOctaveLayers + 3) + nOctaveLayers];
                resize(src, dst, Size(src.cols/2, src.rows/2),
                       0, 0, INTER_NEAREST);
            }
            else
            {
                const Mat& src = pyr[o*(nOctaveLayers + 3) + i-1];
                GaussianBlur(src, dst, Size(), sig[i], sig[i]);
            }
        }
    }
}


void SIFT_Impl::buildDoGPyramid( const std::vector<Mat>& gpyr, std::vector<Mat>& dogpyr ) const
{
    int nOctaves = (int)gpyr.size()/(nOctaveLayers + 3);
    dogpyr.resize( nOctaves*(nOctaveLayers + 2) );

    for( int o = 0; o < nOctaves; o++ )
    {
        for( int i = 0; i < nOctaveLayers + 2; i++ )
        {
            const Mat& src1 = gpyr[o*(nOctaveLayers + 3) + i];
            const Mat& src2 = gpyr[o*(nOctaveLayers + 3) + i + 1];
            Mat& dst = dogpyr[o*(nOctaveLayers + 2) + i];
            subtract(src2, src1, dst, noArray(), DataType<sift_wt>::type);
        }
    }
}


// Computes a gradient orientation histogram at a specified pixel
static float calcOrientationHist( const Mat& img, Point pt, int radius,
                                  float sigma, float* hist, int n )
{
    int i, j, k, len = (radius*2+1)*(radius*2+1);

    float expf_scale = -1.f/(2.f * sigma * sigma);
    AutoBuffer<float> buf(len*4 + n+4);
    float *X = buf, *Y = X + len, *Mag = X, *Ori = Y + len, *W = Ori + len;
    float* temphist = W + len + 2;

    for( i = 0; i < n; i++ )
        temphist[i] = 0.f;

    for( i = -radius, k = 0; i <= radius; i++ )
    {
        int y = pt.y + i;
        if( y <= 0 || y >= img.rows - 1 )
            continue;
        for( j = -radius; j <= radius; j++ )
        {
            int x = pt.x + j;
            if( x <= 0 || x >= img.cols - 1 )
                continue;

            float dx = (float)(img.at<sift_wt>(y, x+1) - img.at<sift_wt>(y, x-1));
            float dy = (float)(img.at<sift_wt>(y-1, x) - img.at<sift_wt>(y+1, x));

            X[k] = dx; Y[k] = dy; W[k] = (i*i + j*j)*expf_scale;
            k++;
        }
    }

    len = k;

    // compute gradient values, orientations and the weights over the pixel neighborhood
    cv::hal::exp32f(W, W, len);
    cv::hal::fastAtan2(Y, X, Ori, len, true);
    cv::hal::magnitude32f(X, Y, Mag, len);

    for( k = 0; k < len; k++ )
    {
        int bin = cvRound((n/360.f)*Ori[k]);
        if( bin >= n )
            bin -= n;
        if( bin < 0 )
            bin += n;
        temphist[bin] += W[k]*Mag[k];
    }

    // smooth the histogram
    temphist[-1] = temphist[n-1];
    temphist[-2] = temphist[n-2];
    temphist[n] = temphist[0];
    temphist[n+1] = temphist[1];
    for( i = 0; i < n; i++ )
    {
        hist[i] = (temphist[i-2] + temphist[i+2])*(1.f/16.f) +
            (temphist[i-1] + temphist[i+1])*(4.f/16.f) +
            temphist[i]*(6.f/16.f);
    }

    float maxval = hist[0];
    for( i = 1; i < n; i++ )
        maxval = std::max(maxval, hist[i]);

    return maxval;
}


//
// 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.
static bool adjustLocalExtrema( const std::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=0;
    int i = 0;

    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<sift_wt>(r, c+1) - img.at<sift_wt>(r, c-1))*deriv_scale,
                 (img.at<sift_wt>(r+1, c) - img.at<sift_wt>(r-1, c))*deriv_scale,
                 (next.at<sift_wt>(r, c) - prev.at<sift_wt>(r, c))*deriv_scale);

        float v2 = (float)img.at<sift_wt>(r, c)*2;
        float dxx = (img.at<sift_wt>(r, c+1) + img.at<sift_wt>(r, c-1) - v2)*second_deriv_scale;
        float dyy = (img.at<sift_wt>(r+1, c) + img.at<sift_wt>(r-1, c) - v2)*second_deriv_scale;
        float dss = (next.at<sift_wt>(r, c) + prev.at<sift_wt>(r, c) - v2)*second_deriv_scale;
        float dxy = (img.at<sift_wt>(r+1, c+1) - img.at<sift_wt>(r+1, c-1) -
                     img.at<sift_wt>(r-1, c+1) + img.at<sift_wt>(r-1, c-1))*cross_deriv_scale;
        float dxs = (next.at<sift_wt>(r, c+1) - next.at<sift_wt>(r, c-1) -
                     prev.at<sift_wt>(r, c+1) + prev.at<sift_wt>(r, c-1))*cross_deriv_scale;
        float dys = (next.at<sift_wt>(r+1, c) - next.at<sift_wt>(r-1, c) -
                     prev.at<sift_wt>(r+1, c) + prev.at<sift_wt>(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;

        if( std::abs(xi) > (float)(INT_MAX/3) ||
            std::abs(xr) > (float)(INT_MAX/3) ||
            std::abs(xc) > (float)(INT_MAX/3) )
            return false;

        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
    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<sift_wt>(r, c+1) - img.at<sift_wt>(r, c-1))*deriv_scale,
                   (img.at<sift_wt>(r+1, c) - img.at<sift_wt>(r-1, c))*deriv_scale,
                   (next.at<sift_wt>(r, c) - prev.at<sift_wt>(r, c))*deriv_scale);
        float t = dD.dot(Matx31f(xc, xr, xi));

        contr = img.at<sift_wt>(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
        float v2 = img.at<sift_wt>(r, c)*2.f;
        float dxx = (img.at<sift_wt>(r, c+1) + img.at<sift_wt>(r, c-1) - v2)*second_deriv_scale;
        float dyy = (img.at<sift_wt>(r+1, c) + img.at<sift_wt>(r-1, c) - v2)*second_deriv_scale;
        float dxy = (img.at<sift_wt>(r+1, c+1) - img.at<sift_wt>(r+1, c-1) -
                     img.at<sift_wt>(r-1, c+1) + img.at<sift_wt>(r-1, c-1)) * cross_deriv_scale;
        float tr = dxx + dyy;
        float det = dxx * dyy - dxy * dxy;

        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;
    kpt.response = std::abs(contr);

    return true;
}


//
// Detects features at extrema in DoG scale space.  Bad features are discarded
// based on contrast and ratio of principal curvatures.
void SIFT_Impl::findScaleSpaceExtrema( const std::vector<Mat>& gauss_pyr, const std::vector<Mat>& dog_pyr,
                                  std::vector<KeyPoint>& keypoints ) const
{
    int nOctaves = (int)gauss_pyr.size()/(nOctaveLayers + 3);
    int threshold = cvFloor(0.5 * contrastThreshold / nOctaveLayers * 255 * SIFT_FIXPT_SCALE);
    const int n = SIFT_ORI_HIST_BINS;
    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 sift_wt* currptr = img.ptr<sift_wt>(r);
                const sift_wt* prevptr = prev.ptr<sift_wt>(r);
                const sift_wt* nextptr = next.ptr<sift_wt>(r);

                for( int c = SIFT_IMG_BORDER; c < cols-SIFT_IMG_BORDER; c++)
                {
                    sift_wt val = currptr[c];

                    // find local extrema with pixel accuracy
                    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;
                        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);
                        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 = 360.f - (float)((360.f/n) * bin);
                                if(std::abs(kpt.angle - 360.f) < FLT_EPSILON)
                                    kpt.angle = 0.f;
                                keypoints.push_back(kpt);
                            }
                        }
                    }
                }
            }
        }
}


static void calcSIFTDescriptor( const Mat& img, Point2f ptf, float ori, float scl,
                               int d, int n, float* dst )
{
    Point pt(cvRound(ptf.x), cvRound(ptf.y));
    float cos_t = cosf(ori*(float)(CV_PI/180));
    float sin_t = sinf(ori*(float)(CV_PI/180));
    float bins_per_rad = n / 360.f;
    float exp_scale = -1.f/(d * d * 0.5f);
    float hist_width = SIFT_DESCR_SCL_FCTR * scl;
    int radius = cvRound(hist_width * 1.4142135623730951f * (d + 1) * 0.5f);
    // Clip the radius to the diagonal of the image to avoid autobuffer too large exception
    radius = std::min(radius, (int) sqrt(((double) img.cols)*img.cols + ((double) img.rows)*img.rows));
    cos_t /= hist_width;
    sin_t /= hist_width;

    int i, j, k, len = (radius*2+1)*(radius*2+1), histlen = (d+2)*(d+2)*(n+2);
    int rows = img.rows, cols = img.cols;

    AutoBuffer<float> buf(len*6 + histlen);
    float *X = buf, *Y = X + len, *Mag = Y, *Ori = Mag + len, *W = Ori + len;
    float *RBin = W + len, *CBin = RBin + len, *hist = CBin + len;

    for( i = 0; i < d+2; i++ )
    {
        for( j = 0; j < d+2; j++ )
            for( k = 0; k < n+2; k++ )
                hist[(i*(d+2) + j)*(n+2) + k] = 0.;
    }

    for( i = -radius, k = 0; i <= radius; i++ )
        for( j = -radius; j <= radius; j++ )
        {
            // Calculate sample's histogram array coords rotated relative to ori.
            // Subtract 0.5 so samples that fall e.g. in the center of row 1 (i.e.
            // r_rot = 1.5) have full weight placed in row 1 after interpolation.
            float c_rot = j * cos_t - i * sin_t;
            float r_rot = j * sin_t + i * cos_t;
            float rbin = r_rot + d/2 - 0.5f;
            float cbin = c_rot + d/2 - 0.5f;
            int r = pt.y + i, c = pt.x + j;

            if( rbin > -1 && rbin < d && cbin > -1 && cbin < d &&
                r > 0 && r < rows - 1 && c > 0 && c < cols - 1 )
            {
                float dx = (float)(img.at<sift_wt>(r, c+1) - img.at<sift_wt>(r, c-1));
                float dy = (float)(img.at<sift_wt>(r-1, c) - img.at<sift_wt>(r+1, c));
                X[k] = dx; Y[k] = dy; RBin[k] = rbin; CBin[k] = cbin;
                W[k] = (c_rot * c_rot + r_rot * r_rot)*exp_scale;
                k++;
            }
        }

    len = k;
    cv::hal::fastAtan2(Y, X, Ori, len, true);
    cv::hal::magnitude32f(X, Y, Mag, len);
    cv::hal::exp32f(W, W, len);

    for( k = 0; k < len; k++ )
    {
        float rbin = RBin[k], cbin = CBin[k];
        float obin = (Ori[k] - ori)*bins_per_rad;
        float mag = Mag[k]*W[k];

        int r0 = cvFloor( rbin );
        int c0 = cvFloor( cbin );
        int o0 = cvFloor( obin );
        rbin -= r0;
        cbin -= c0;
        obin -= o0;

        if( o0 < 0 )
            o0 += n;
        if( o0 >= n )
            o0 -= n;

        // histogram update using tri-linear interpolation
        float v_r1 = mag*rbin, v_r0 = mag - v_r1;
        float v_rc11 = v_r1*cbin, v_rc10 = v_r1 - v_rc11;
        float v_rc01 = v_r0*cbin, v_rc00 = v_r0 - v_rc01;
        float v_rco111 = v_rc11*obin, v_rco110 = v_rc11 - v_rco111;
        float v_rco101 = v_rc10*obin, v_rco100 = v_rc10 - v_rco101;
        float v_rco011 = v_rc01*obin, v_rco010 = v_rc01 - v_rco011;
        float v_rco001 = v_rc00*obin, v_rco000 = v_rc00 - v_rco001;

        int idx = ((r0+1)*(d+2) + c0+1)*(n+2) + o0;
        hist[idx] += v_rco000;
        hist[idx+1] += v_rco001;
        hist[idx+(n+2)] += v_rco010;
        hist[idx+(n+3)] += v_rco011;
        hist[idx+(d+2)*(n+2)] += v_rco100;
        hist[idx+(d+2)*(n+2)+1] += v_rco101;
        hist[idx+(d+3)*(n+2)] += v_rco110;
        hist[idx+(d+3)*(n+2)+1] += v_rco111;
    }

    // finalize histogram, since the orientation histograms are circular
    for( i = 0; i < d; i++ )
        for( j = 0; j < d; j++ )
        {
            int idx = ((i+1)*(d+2) + (j+1))*(n+2);
            hist[idx] += hist[idx+n];
            hist[idx+1] += hist[idx+n+1];
            for( k = 0; k < n; k++ )
                dst[(i*d + j)*n + k] = hist[idx+k];
        }
    // copy histogram to the descriptor,
    // apply hysteresis thresholding
    // and scale the result, so that it can be easily converted
    // to byte array
    float nrm2 = 0;
    len = d*d*n;
    for( k = 0; k < len; k++ )
        nrm2 += dst[k]*dst[k];
    float thr = std::sqrt(nrm2)*SIFT_DESCR_MAG_THR;
    for( i = 0, nrm2 = 0; i < k; i++ )
    {
        float val = std::min(dst[i], thr);
        dst[i] = val;
        nrm2 += val*val;
    }
    nrm2 = SIFT_INT_DESCR_FCTR/std::max(std::sqrt(nrm2), FLT_EPSILON);

#if 1
    for( k = 0; k < len; k++ )
    {
        dst[k] = saturate_cast<uchar>(dst[k]*nrm2);
    }
#else
    float nrm1 = 0;
    for( k = 0; k < len; k++ )
    {
        dst[k] *= nrm2;
        nrm1 += dst[k];
    }
    nrm1 = 1.f/std::max(nrm1, FLT_EPSILON);
    for( k = 0; k < len; k++ )
    {
        dst[k] = std::sqrt(dst[k] * nrm1);//saturate_cast<uchar>(std::sqrt(dst[k] * nrm1)*SIFT_INT_DESCR_FCTR);
    }
#endif
}

static void calcDescriptors(const std::vector<Mat>& gpyr, const std::vector<KeyPoint>& keypoints,
                            Mat& descriptors, int nOctaveLayers, int firstOctave )
{
    int d = SIFT_DESCR_WIDTH, n = SIFT_DESCR_HIST_BINS;

    for( size_t i = 0; i < keypoints.size(); i++ )
    {
        KeyPoint kpt = keypoints[i];
        int octave, layer;
        float scale;
        unpackOctave(kpt, octave, layer, scale);
        CV_Assert(octave >= firstOctave && layer <= nOctaveLayers+2);
        float size=kpt.size*scale;
        Point2f ptf(kpt.pt.x*scale, kpt.pt.y*scale);
        const Mat& img = gpyr[(octave - firstOctave)*(nOctaveLayers + 3) + layer];

        float angle = 360.f - kpt.angle;
        if(std::abs(angle - 360.f) < FLT_EPSILON)
            angle = 0.f;
        calcSIFTDescriptor(img, ptf, angle, size*0.5f, d, n, descriptors.ptr<float>((int)i));
    }
}

//////////////////////////////////////////////////////////////////////////////////////////

SIFT_Impl::SIFT_Impl( int _nfeatures, int _nOctaveLayers,
           double _contrastThreshold, double _edgeThreshold, double _sigma )
    : nfeatures(_nfeatures), nOctaveLayers(_nOctaveLayers),
    contrastThreshold(_contrastThreshold), edgeThreshold(_edgeThreshold), sigma(_sigma)
{
}

int SIFT_Impl::descriptorSize() const
{
    return SIFT_DESCR_WIDTH*SIFT_DESCR_WIDTH*SIFT_DESCR_HIST_BINS;
}

int SIFT_Impl::descriptorType() const
{
    return CV_32F;
}

int SIFT_Impl::defaultNorm() const
{
    return NORM_L2;
}


void SIFT_Impl::detectAndCompute(InputArray _image, InputArray _mask,
                      std::vector<KeyPoint>& keypoints,
                      OutputArray _descriptors,
                      bool useProvidedKeypoints)
{
    int firstOctave = -1, actualNOctaves = 0, actualNLayers = 0;
    Mat image = _image.getMat(), mask = _mask.getMat();

    if( image.empty() || image.depth() != CV_8U )
        CV_Error( Error::StsBadArg, "image is empty or has incorrect depth (!=CV_8U)" );

    if( !mask.empty() && mask.type() != CV_8UC1 )
        CV_Error( Error::StsBadArg, "mask has incorrect type (!=CV_8UC1)" );

    if( useProvidedKeypoints )
    {
        firstOctave = 0;
        int maxOctave = INT_MIN;
        for( size_t i = 0; i < keypoints.size(); i++ )
        {
            int octave, layer;
            float scale;
            unpackOctave(keypoints[i], octave, layer, scale);
            firstOctave = std::min(firstOctave, octave);
            maxOctave = std::max(maxOctave, octave);
            actualNLayers = std::max(actualNLayers, layer-2);
        }

        firstOctave = std::min(firstOctave, 0);
        CV_Assert( firstOctave >= -1 && actualNLayers <= nOctaveLayers );
        actualNOctaves = maxOctave - firstOctave + 1;
    }

    Mat base = createInitialImage(image, firstOctave < 0, (float)sigma);
    std::vector<Mat> gpyr, dogpyr;
    int nOctaves = actualNOctaves > 0 ? actualNOctaves : cvRound(std::log( (double)std::min( base.cols, base.rows ) ) / std::log(2.) - 2) - firstOctave;

    //double t, tf = getTickFrequency();
    //t = (double)getTickCount();
    buildGaussianPyramid(base, gpyr, nOctaves);
    buildDoGPyramid(gpyr, dogpyr);

    //t = (double)getTickCount() - t;
    //printf("pyramid construction time: %g\n", t*1000./tf);

    if( !useProvidedKeypoints )
    {
        //t = (double)getTickCount();
        findScaleSpaceExtrema(gpyr, dogpyr, keypoints);
        KeyPointsFilter::removeDuplicated( keypoints );

        if( nfeatures > 0 )
            KeyPointsFilter::retainBest(keypoints, nfeatures);
        //t = (double)getTickCount() - t;
        //printf("keypoint detection time: %g\n", t*1000./tf);

        if( firstOctave < 0 )
            for( size_t i = 0; i < keypoints.size(); i++ )
            {
                KeyPoint& kpt = keypoints[i];
                float scale = 1.f/(float)(1 << -firstOctave);
                kpt.octave = (kpt.octave & ~255) | ((kpt.octave + firstOctave) & 255);
                kpt.pt *= scale;
                kpt.size *= scale;
            }

        if( !mask.empty() )
            KeyPointsFilter::runByPixelsMask( keypoints, mask );
    }
    else
    {
        // filter keypoints by mask
        //KeyPointsFilter::runByPixelsMask( keypoints, mask );
    }

    if( _descriptors.needed() )
    {
        //t = (double)getTickCount();
        int dsize = descriptorSize();
        _descriptors.create((int)keypoints.size(), dsize, CV_32F);
        Mat descriptors = _descriptors.getMat();

        calcDescriptors(gpyr, keypoints, descriptors, nOctaveLayers, firstOctave);
        //t = (double)getTickCount() - t;
        //printf("descriptor extraction time: %g\n", t*1000./tf);
    }
}

}
}

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