stitching_detail算法介绍（二）

在上一篇中，我们配置好了stitching_detail样例，在看到效果之后我们需要深入去看代码，理解原理。

参考博文:http://blog.csdn.net/skeeee/article/details/19480693

一、stitching_detail程序运行流程

1.命令行调用程序，输入源图像以及程序的参数

2.特征点检测，判断是使用surf还是orb，默认是surf。

3.对图像的特征点进行匹配，使用最近邻和次近邻方法，将两个最优的匹配的置信度保存下来。

4.对图像进行排序以及将置信度高的图像保存到同一个集合中，删除置信度比较低的图像间的匹配，得到能正确匹配的图像序列。这样将置信度高于门限的所有匹配合并到一个集合中。

5.对所有图像进行相机参数粗略估计，然后求出旋转矩阵

6.使用光束平均法进一步精准的估计出旋转矩阵。

7.波形校正，水平或者垂直

8.拼接

9.融合，多频段融合，光照补偿，

二、stitching_detail程序接口介绍

img1 img2 img3 输入图像

--preview 以预览模式运行程序，比正常模式要快，但输出图像分辨率低，拼接的分辨率compose_megapix 设置为0.6

--try_gpu (yes|no) 是否使用GPU(图形处理器)，默认为no

/* 运动估计参数 */

--work_megapix <--work_megapix <float>> 图像匹配的分辨率大小，图像的面积尺寸变为work_megapix*100000，默认为0.6

--features (surf|orb) 选择surf或者orb算法进行特征点计算，默认为surf

--match_conf <float> 特征点检测置信等级，最近邻匹配距离与次近邻匹配距离的比值，surf默认为0.65，orb默认为0.3

--conf_thresh <float> 两幅图来自同一全景图的置信度，默认为1.0

--ba (reproj|ray) 光束平均法的误差函数选择，默认是ray方法

--ba_refine_mask (mask) ---------------

--wave_correct (no|horiz|vert) 波形校验 水平，垂直或者没有 默认是horiz

--save_graph <file_name> 将匹配的图形以点的形式保存到文件中， Nm代表匹配的数量，NI代表正确匹配的数量，C表示置信度

/*图像融合参数:*/

--warp (plane|cylindrical|spherical|fisheye|stereographic|compressedPlaneA2B1|compressedPlaneA1.5B1|compressedPlanePortraitA2B1

|compressedPlanePortraitA1.5B1|paniniA2B1|paniniA1.5B1|paniniPortraitA2B1|paniniPortraitA1.5B1|mercator|transverseMercator)

选择融合的平面，默认是球形

--seam_megapix <float> 拼接缝像素的大小 默认是0.1 ------------

--seam (no|voronoi|gc_color|gc_colorgrad) 拼接缝隙估计方法 默认是gc_color

--compose_megapix <float> 拼接分辨率，默认为-1

--expos_comp (no|gain|gain_blocks) 光照补偿方法，默认是gain_blocks

--blend (no|feather|multiband) 融合方法，默认是多频段融合

--blend_strength <float> 融合强度，0-100.默认是5.

--output <result_img> 输出图像的文件名，默认是result,jpg

命令使用实例，以及程序运行时的提示：

三、程序代码分析

1.参数读入

程序参数读入分析，将程序运行是输入的参数以及需要拼接的图像读入内存，检查图像是否多于2张。

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int retval = parseCmdArgs(argc, argv);
if (retval)
return retval;

// Check if have enough images
int num_images = static_cast<int>(img_names.size());
if (num_images < 2)
{
LOGLN("Need more images");
return -1;
}

2.特征点检测

判断选择是surf还是orb特征点检测（默认是surf）以及对图像进行预处理（尺寸缩放），然后计算每幅图形的特征点，以及特征点描述子

2.1 计算work_scale，将图像resize到面积在work_megapix*10^6以下，（work_megapix 默认是0.6）

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work_scale = min(1.0, sqrt(work_megapix * 1e6 / full_img.size().area()));

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resize(full_img, img, Size(), work_scale, work_scale);

图像大小是740*830，面积大于6*10^5，所以计算出work_scale = 0.98,然后对图像resize。

2.2 计算seam_scale，也是根据图像的面积小于seam_megapix*10^6，（seam_megapix 默认是0.1），seam_work_aspect目前还没用到

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seam_scale = min(1.0, sqrt(seam_megapix * 1e6 / full_img.size().area()));
seam_work_aspect = seam_scale / work_scale; //seam_megapix = 0.1 seam_work_aspect = 0.69

2.3 计算图像特征点，以及计算特征点描述子，并将img_idx设置为i。

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(*finder)(img, features[i]);//matcher.cpp 348
features[i].img_idx = i;

特征点描述的结构体定义如下：

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struct detail::ImageFeatures
Structure containing image keypoints and descriptors.
struct CV_EXPORTS ImageFeatures
{
int img_idx;//
Size img_size;
std::vector<KeyPoint> keypoints;
Mat descriptors;
};

2.4 将源图像resize到seam_megapix*10^6，并存入image[]中

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resize(full_img, img, Size(), seam_scale, seam_scale);
images[i] = img.clone();

3.图像匹配

对任意两副图形进行特征点匹配，然后使用查并集法，将图片的匹配关系找出，并删除那些不属于同一全景图的图片。

3.1 使用最近邻和次近邻匹配，对任意两幅图进行特征点匹配。

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vector<MatchesInfo> pairwise_matches;//Structure containing information about matches between two images.
BestOf2NearestMatcher matcher(try_gpu, match_conf);//最近邻和次近邻法
matcher(features, pairwise_matches);//对每两个图片进行matcher，20-》400 matchers.cpp 502

介绍一下BestOf2NearestMatcher 函数：

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//Features matcher which finds two best matches for each feature and leaves the best one only if the ratio between descriptor distances is greater than the threshold match_conf.
detail::BestOf2NearestMatcher::BestOf2NearestMatcher(bool try_use_gpu=false,float match_conf=0.3f,
intnum_matches_thresh1=6, int num_matches_thresh2=6)
Parameters: try_use_gpu – Should try to use GPU or not
match_conf – Match distances ration threshold
num_matches_thresh1 – Minimum number of matches required for the 2D projective
transform estimation used in the inliers classification step
num_matches_thresh2 – Minimum number of matches required for the 2D projective
transform re-estimation on inliers

函数的定义（只是设置一下参数，属于构造函数）：

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BestOf2NearestMatcher::BestOf2NearestMatcher(bool try_use_gpu, float match_conf, int num_matches_thresh1, int num_matches_thresh2)
{
#ifdef HAVE_OPENCV_GPU
if (try_use_gpu && getCudaEnabledDeviceCount() > 0)
impl_ = new GpuMatcher(match_conf);
else
#else
(void)try_use_gpu;
#endif
impl_ = new CpuMatcher(match_conf);

is_thread_safe_ = impl_->isThreadSafe();
num_matches_thresh1_ = num_matches_thresh1;
num_matches_thresh2_ = num_matches_thresh2;
}

以及MatchesInfo的结构体定义：

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Structure containing information about matches between two images. It’s assumed that there is a homography between those images.
struct CV_EXPORTS MatchesInfo
{
MatchesInfo();
MatchesInfo(const MatchesInfo &other);
const MatchesInfo& operator =(const MatchesInfo &other);
int src_img_idx, dst_img_idx; // Images indices (optional)
std::vector<DMatch> matches;
std::vector<uchar> inliers_mask; // Geometrically consistent matches mask
int num_inliers; // Number of geometrically consistent matches
Mat H; // Estimated homography
double confidence; // Confidence two images are from the same panorama
};

求出图像匹配的结果如下（具体匹配参见sift特征点匹配），任意两幅图都进行匹配（3*3=9），其中1-》2和2-》1只计算了一次，以1-》2为准，：

3.2 根据任意两幅图匹配的置信度，将所有置信度高于conf_thresh（默认是1.0）的图像合并到一个全集中。

我们通过函数的参数 save_graph打印匹配结果如下（我稍微修改了一下输出）：

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"outimage101.jpg" -- "outimage102.jpg"[label="Nm=866, Ni=637, C=2.37864"];
"outimage101.jpg" -- "outimage103.jpg"[label="Nm=165, Ni=59, C=1.02609"];
"outimage102.jpg" -- "outimage103.jpg"[label="Nm=460, Ni=260, C=1.78082"];

Nm代表匹配的数量，NI代表正确匹配的数量，C表示置信度
通过查并集方法，查并集介绍请参见博文：
http://blog.csdn.net/skeeee/article/details/20471949
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vector<int> indices = leaveBiggestComponent(features, pairwise_matches, conf_thresh);//将置信度高于门限的所有匹配合并到一个集合中
vector<Mat> img_subset;
vector<string> img_names_subset;
vector<Size> full_img_sizes_subset;
for (size_t i = 0; i < indices.size(); ++i)
{
img_names_subset.push_back(img_names[indices[i]]);
img_subset.push_back(images[indices[i]]);
full_img_sizes_subset.push_back(full_img_sizes[indices[i]]);
}

images = img_subset;
img_names = img_names_subset;
full_img_sizes = full_img_sizes_subset;

4.根据单应性矩阵粗略估计出相机参数（焦距）

4.1 焦距参数的估计

根据前面求出的任意两幅图的匹配，我们根据两幅图的单应性矩阵H，求出符合条件的f，（4副图，16个匹配，求出8个符合条件的f），然后求出f的均值或者中值当成所有图形的粗略估计的f。

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estimateFocal(features, pairwise_matches, focals);

函数的主要源码如下：

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for (int i = 0; i < num_images; ++i)
{
for (int j = 0; j < num_images; ++j)
{
const MatchesInfo &m = pairwise_matches[i*num_images + j];
if (m.H.empty())
continue;
double f0, f1;
bool f0ok, f1ok;
focalsFromHomography(m.H, f0, f1, f0ok, f1ok);//Tries to estimate focal lengths from the given homography 79
//under the assumption that the camera undergoes rotations around its centre only.
if (f0ok && f1ok)
all_focals.push_back(sqrt(f0 * f1));
}
}

其中函数focalsFromHomography的定义如下：

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Tries to estimate focal lengths from the given homography
under the assumption that the camera undergoes rotations around its centre only.
Parameters
H – Homography.
f0 – Estimated focal length along X axis.
f1 – Estimated focal length along Y axis.
f0_ok – True, if f0 was estimated successfully, false otherwise.
f1_ok – True, if f1 was estimated successfully, false otherwise.

函数的源码：

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void focalsFromHomography(const Mat& H, double &f0, double &f1, bool &f0_ok, bool &f1_ok)
{
CV_Assert(H.type() == CV_64F && H.size() == Size(3, 3));//Checks a condition at runtime and throws exception if it fails

const double* h = reinterpret_cast<const double*>(H.data);//强制类型转换

double d1, d2; // Denominators
double v1, v2; // Focal squares value candidates

//具体的计算过程有点看不懂啊
f1_ok = true;
d1 = h[6] * h[7];
d2 = (h[7] - h[6]) * (h[7] + h[6]);
v1 = -(h[0] * h[1] + h[3] * h[4]) / d1;
v2 = (h[0] * h[0] + h[3] * h[3] - h[1] * h[1] - h[4] * h[4]) / d2;
if (v1 < v2) std::swap(v1, v2);
if (v1 > 0 && v2 > 0) f1 = sqrt(std::abs(d1) > std::abs(d2) ? v1 : v2);
else if (v1 > 0) f1 = sqrt(v1);
else f1_ok = false;

f0_ok = true;
d1 = h[0] * h[3] + h[1] * h[4];
d2 = h[0] * h[0] + h[1] * h[1] - h[3] * h[3] - h[4] * h[4];
v1 = -h[2] * h[5] / d1;
v2 = (h[5] * h[5] - h[2] * h[2]) / d2;
if (v1 < v2) std::swap(v1, v2);
if (v1 > 0 && v2 > 0) f0 = sqrt(std::abs(d1) > std::abs(d2) ? v1 : v2);
else if (v1 > 0) f0 = sqrt(v1);
else f0_ok = false;
}

求出的焦距有8个



求出的焦距取中值或者平均值，然后就是所有图片的焦距。

并构建camera参数，将矩阵写入camera：

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cameras.assign(num_images, CameraParams());
for (int i = 0; i < num_images; ++i)
cameras[i].focal = focals[i];

4.2 根据匹配的内点构建最大生成树，然后广度优先搜索求出根节点，并求出camera的R矩阵，K矩阵以及光轴中心

camera其他参数：

aspect = 1.0，ppx，ppy目前等于0，最后会赋值成图像中心点的。

K矩阵的值：



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Mat CameraParams::K() const
{
Mat_<double> k = Mat::eye(3, 3, CV_64F);
k(0,0) = focal; k(0,2) = ppx;
k(1,1) = focal * aspect; k(1,2) = ppy;
return k;
}

R矩阵的值：



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void operator ()(const GraphEdge &edge)
{
int pair_idx = edge.from * num_images + edge.to;

Mat_<double> K_from = Mat::eye(3, 3, CV_64F);
K_from(0,0) = cameras[edge.from].focal;
K_from(1,1) = cameras[edge.from].focal * cameras[edge.from].aspect;
K_from(0,2) = cameras[edge.from].ppx;
K_from(0,2) = cameras[edge.from].ppy;

Mat_<double> K_to = Mat::eye(3, 3, CV_64F);
K_to(0,0) = cameras[edge.to].focal;
K_to(1,1) = cameras[edge.to].focal * cameras[edge.to].aspect;
K_to(0,2) = cameras[edge.to].ppx;
K_to(0,2) = cameras[edge.to].ppy;

Mat R = K_from.inv() * pairwise_matches[pair_idx].H.inv() * K_to;
cameras[edge.to].R = cameras[edge.from].R * R;
}

光轴中心的值

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for (int i = 0; i < num_images; ++i)
{
cameras[i].ppx += 0.5 * features[i].img_size.width;
cameras[i].ppy += 0.5 * features[i].img_size.height;
}

5.使用Bundle Adjustment方法对所有图片进行相机参数校正

Bundle Adjustment（光束法平差）算法主要是解决所有相机参数的联合。这是全景拼接必须的一步，因为多个成对的单应性矩阵合成全景图时，会忽略全局的限制，造成累积误差。因此每一个图像都要加上光束法平差值，使图像被初始化成相同的旋转和焦距长度。

光束法平差的目标函数是一个具有鲁棒性的映射误差的平方和函数。即每一个特征点都要映射到其他的图像中，计算出使误差的平方和最小的相机参数。具体的推导过程可以参见Automatic Panoramic Image Stitching using Invariant Features.pdf的第五章，本文只介绍一下opencv实现的过程（完整的理论和公式 暂时还没看懂，希望有人能一起交流）

opencv中误差描述函数有两种如下：（opencv默认是BundleAdjusterRay ）：

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BundleAdjusterReproj // BundleAdjusterBase(7, 2)//最小投影误差平方和
Implementation of the camera parameters refinement algorithm which minimizes sum of the reprojection error squares

BundleAdjusterRay // BundleAdjusterBase(4, 3)//最小特征点与相机中心点的距离和
Implementation of the camera parameters refinement algorithm which minimizes sum of the distances between the
rays passing through the camera center and a feature.

5.1 首先计算cam_params_的值：

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setUpInitialCameraParams(cameras);

函数主要源码：

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cam_params_.create(num_images_ * 4, 1, CV_64F);
SVD svd;//奇异值分解
for (int i = 0; i < num_images_; ++i)
{
cam_params_.at<double>(i * 4, 0) = cameras[i].focal;

svd(cameras[i].R, SVD::FULL_UV);
Mat R = svd.u * svd.vt;
if (determinant(R) < 0)
R *= -1;

Mat rvec;
Rodrigues(R, rvec);
CV_Assert(rvec.type() == CV_32F);
cam_params_.at<double>(i * 4 + 1, 0) = rvec.at<float>(0, 0);
cam_params_.at<double>(i * 4 + 2, 0) = rvec.at<float>(1, 0);
cam_params_.at<double>(i * 4 + 3, 0) = rvec.at<float>(2, 0);
}

计算cam_params_的值，先初始化cam_params(num_images_*4,1,CV_64F);
cam_params_[i*4+0] = cameras[i].focal;

cam_params_后面3个值，是cameras[i].R先经过奇异值分解，然后对u*vt进行Rodrigues运算，得到的rvec第一行3个值赋给cam_params_。

奇异值分解的定义：

在矩阵M的奇异值分解中 M = UΣV*

·U的列(columns)组成一套对M的正交"输入"或"分析"的基向量。这些向量是MM*的特征向量。

·V的列(columns)组成一套对M的正交"输出"的基向量。这些向量是M*M的特征向量。

·Σ对角线上的元素是奇异值，可视为是在输入与输出间进行的标量的"膨胀控制"。这些是M*M及MM*的奇异值，并与U和V的行向量相对应。

5.2 删除置信度小于门限的匹配对

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// Leave only consistent image pairs
edges_.clear();
for (int i = 0; i < num_images_ - 1; ++i)
{
for (int j = i + 1; j < num_images_; ++j)
{
const MatchesInfo& matches_info = pairwise_matches_[i * num_images_ + j];
if (matches_info.confidence > conf_thresh_)
edges_.push_back(make_pair(i, j));
}
}

5.3 使用LM算法计算camera参数。

首先初始化LM的参数（具体理论还没有看懂）

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//计算所有内点之和
for (size_t i = 0; i < edges_.size(); ++i)
total_num_matches_ += static_cast<int>(pairwise_matches[edges_[i].first * num_images_ +
edges_[i].second].num_inliers);

CvLevMarq solver(num_images_ * num_params_per_cam_,
total_num_matches_ * num_errs_per_measurement_,
term_criteria_);

Mat err, jac;
CvMat matParams = cam_params_;
cvCopy(&matParams, solver.param);

int iter = 0;
for(;;)//类似于while（1）,但是比while（1）效率高
{
const CvMat* _param = 0;
CvMat* _jac = 0;
CvMat* _err = 0;

bool proceed = solver.update(_param, _jac, _err);

cvCopy(_param, &matParams);

if (!proceed || !_err)
break;

if (_jac)
{
calcJacobian(jac); //构造雅阁比行列式
CvMat tmp = jac;
cvCopy(&tmp, _jac);
}

if (_err)
{
calcError(err);//计算err
LOG_CHAT(".");
iter++;
CvMat tmp = err;
cvCopy(&tmp, _err);
}
}

计算camera

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obtainRefinedCameraParams(cameras);//Gets the refined camera parameters.

函数源代码：

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void BundleAdjusterRay::obtainRefinedCameraParams(vector<CameraParams> &cameras) const
{
for (int i = 0; i < num_images_; ++i)
{
cameras[i].focal = cam_params_.at<double>(i * 4, 0);

Mat rvec(3, 1, CV_64F);
rvec.at<double>(0, 0) = cam_params_.at<double>(i * 4 + 1, 0);
rvec.at<double>(1, 0) = cam_params_.at<double>(i * 4 + 2, 0);
rvec.at<double>(2, 0) = cam_params_.at<double>(i * 4 + 3, 0);
Rodrigues(rvec, cameras[i].R);

Mat tmp;
cameras[i].R.convertTo(tmp, CV_32F);
cameras[i].R = tmp;
}
}

求出根节点，然后归一化旋转矩阵R

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// Normalize motion to center image
Graph span_tree;
vector<int> span_tree_centers;
findMaxSpanningTree(num_images_, pairwise_matches, span_tree, span_tree_centers);
Mat R_inv = cameras[span_tree_centers[0]].R.inv();
for (int i = 0; i < num_images_; ++i)
cameras[i].R = R_inv * cameras[i].R;

6 波形校正

前面几节把相机旋转矩阵计算出来，但是还有一个因素需要考虑，就是由于拍摄者拍摄图片的时候不一定是水平的，轻微的倾斜会导致全景图像出现飞机曲线，因此我们要对图像进行波形校正，主要是寻找每幅图形的“上升向量”（up_vector），使他校正成

。



波形校正的效果图
opencv实现的源码如下（也是暂时没看懂，囧）

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<span style="white-space:pre"> </span>vector<Mat> rmats;
for (size_t i = 0; i < cameras.size(); ++i)
rmats.push_back(cameras[i].R);
waveCorrect(rmats, wave_correct);//Tries to make panorama more horizontal (or vertical).
for (size_t i = 0; i < cameras.size(); ++i)
cameras[i].R = rmats[i];

其中waveCorrect(rmats, wave_correct)源码如下：

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void waveCorrect(vector<Mat> &rmats, WaveCorrectKind kind)
{
LOGLN("Wave correcting...");
#if ENABLE_LOG
int64 t = getTickCount();
#endif

Mat moment = Mat::zeros(3, 3, CV_32F);
for (size_t i = 0; i < rmats.size(); ++i)
{
Mat col = rmats[i].col(0);
moment += col * col.t();//相机R矩阵第一列转置相乘然后相加
}
Mat eigen_vals, eigen_vecs;
eigen(moment, eigen_vals, eigen_vecs);//Calculates eigenvalues and eigenvectors of a symmetric matrix.

Mat rg1;
if (kind == WAVE_CORRECT_HORIZ)
rg1 = eigen_vecs.row(2).t();//如果是水平校正，去特征向量的第三行
else if (kind == WAVE_CORRECT_VERT)
rg1 = eigen_vecs.row(0).t();//如果是垂直校正，特征向量第一行
else
CV_Error(CV_StsBadArg, "unsupported kind of wave correction");

Mat img_k = Mat::zeros(3, 1, CV_32F);
for (size_t i = 0; i < rmats.size(); ++i)
img_k += rmats[i].col(2);//R函数第3列相加
Mat rg0 = rg1.cross(img_k);//rg1与img_k向量积
rg0 /= norm(rg0);//归一化？

Mat rg2 = rg0.cross(rg1);

double conf = 0;
if (kind == WAVE_CORRECT_HORIZ)
{
for (size_t i = 0; i < rmats.size(); ++i)
conf += rg0.dot(rmats[i].col(0));//Computes a dot-product of two vectors.数量积
if (conf < 0)
{
rg0 *= -1;
rg1 *= -1;
}
}
else if (kind == WAVE_CORRECT_VERT)
{
for (size_t i = 0; i < rmats.size(); ++i)
conf -= rg1.dot(rmats[i].col(0));
if (conf < 0)
{
rg0 *= -1;
rg1 *= -1;
}
}

Mat R = Mat::zeros(3, 3, CV_32F);
Mat tmp = R.row(0);
Mat(rg0.t()).copyTo(tmp);
tmp = R.row(1);
Mat(rg1.t()).copyTo(tmp);
tmp = R.row(2);
Mat(rg2.t()).copyTo(tmp);

for (size_t i = 0; i < rmats.size(); ++i)
rmats[i] = R * rmats[i];

LOGLN("Wave correcting, time: " << ((getTickCount() - t) / getTickFrequency()) << " sec");
}

7.单应性矩阵变换

由图像匹配，Bundle Adjustment算法以及波形校验，求出了图像的相机参数以及旋转矩阵，接下来就对图形进行单应性矩阵变换，亮度的增量补偿以及多波段融合（图像金字塔）。首先介绍的就是单应性矩阵变换：

源图像的点（x，y，z=1），图像的旋转矩阵R，图像的相机参数矩阵K，经过变换后的同一坐标（x_,y_,z_），然后映射到球形坐标（u，v，w），他们之间的关系如下：

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void SphericalProjector::mapForward(float x, float y, float &u, float &v)
{
float x_ = r_kinv[0] * x + r_kinv[1] * y + r_kinv[2];
float y_ = r_kinv[3] * x + r_kinv[4] * y + r_kinv[5];
float z_ = r_kinv[6] * x + r_kinv[7] * y + r_kinv[8];

u = scale * atan2f(x_, z_);
float w = y_ / sqrtf(x_ * x_ + y_ * y_ + z_ * z_);
v = scale * (static_cast<float>(CV_PI) - acosf(w == w ? w : 0));
}



根据映射公式，对图像的上下左右四个边界求映射后的坐标，然后确定变换后图像的左上角和右上角的坐标，

如果是球形拼接，则需要再加上判断（暂时还没研究透）：

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float tl_uf = static_cast<float>(dst_tl.x);
float tl_vf = static_cast<float>(dst_tl.y);
float br_uf = static_cast<float>(dst_br.x);
float br_vf = static_cast<float>(dst_br.y);

float x = projector_.rinv[1];
float y = projector_.rinv[4];
float z = projector_.rinv[7];
if (y > 0.f)
{
float x_ = (projector_.k[0] * x + projector_.k[1] * y) / z + projector_.k[2];
float y_ = projector_.k[4] * y / z + projector_.k[5];
if (x_ > 0.f && x_ < src_size.width && y_ > 0.f && y_ < src_size.height)
{
tl_uf = min(tl_uf, 0.f); tl_vf = min(tl_vf, static_cast<float>(CV_PI * projector_.scale));
br_uf = max(br_uf, 0.f); br_vf = max(br_vf, static_cast<float>(CV_PI * projector_.scale));
}
}

x = projector_.rinv[1];
y = -projector_.rinv[4];
z = projector_.rinv[7];
if (y > 0.f)
{
float x_ = (projector_.k[0] * x + projector_.k[1] * y) / z + projector_.k[2];
float y_ = projector_.k[4] * y / z + projector_.k[5];
if (x_ > 0.f && x_ < src_size.width && y_ > 0.f && y_ < src_size.height)
{
tl_uf = min(tl_uf, 0.f); tl_vf = min(tl_vf, static_cast<float>(0));
br_uf = max(br_uf, 0.f); br_vf = max(br_vf, static_cast<float>(0));
}
}

然后利用反投影将图形反投影到变换的图像上，像素计算默认是二维线性插值。
反投影的公式：

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void SphericalProjector::mapBackward(float u, float v, float &x, float &y)
{
u /= scale;
v /= scale;

float sinv = sinf(static_cast<float>(CV_PI) - v);
float x_ = sinv * sinf(u);
float y_ = cosf(static_cast<float>(CV_PI) - v);
float z_ = sinv * cosf(u);

float z;
x = k_rinv[0] * x_ + k_rinv[1] * y_ + k_rinv[2] * z_;
y = k_rinv[3] * x_ + k_rinv[4] * y_ + k_rinv[5] * z_;
z = k_rinv[6] * x_ + k_rinv[7] * y_ + k_rinv[8] * z_;

if (z > 0) { x /= z; y /= z; }
else x = y = -1;
}

然后将反投影求出的x，y值写入Mat矩阵xmap和ymap中

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xmap.create(dst_br.y - dst_tl.y + 1, dst_br.x - dst_tl.x + 1, CV_32F);
ymap.create(dst_br.y - dst_tl.y + 1, dst_br.x - dst_tl.x + 1, CV_32F);

float x, y;
for (int v = dst_tl.y; v <= dst_br.y; ++v)
{
for (int u = dst_tl.x; u <= dst_br.x; ++u)
{
projector_.mapBackward(static_cast<float>(u), static_cast<float>(v), x, y);
xmap.at<float>(v - dst_tl.y, u - dst_tl.x) = x;
ymap.at<float>(v - dst_tl.y, u - dst_tl.x) = y;
}
}

最后使用opencv自带的remap函数将图像重新绘制:

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remap(src, dst, xmap, ymap, interp_mode, border_mode);//重映射，xmap，yamp分别是u，v反投影对应的x，y值，默认是双线性插值

8.光照补偿

图像拼接中，由于拍摄的图片有可能因为光圈或者光线的问题，导致相邻图片重叠区域出现亮度差，所以在拼接时就需要对图像进行亮度补偿，（opencv只对重叠区域进行了亮度补偿，这样会导致图像融合处虽然光照渐变，但是图像整体的光强没有柔和的过渡）。

首先，将所有图像，mask矩阵进行分块（大小在32*32附近）。

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for (int img_idx = 0; img_idx < num_images; ++img_idx)
{
Size bl_per_img((images[img_idx].cols + bl_width_ - 1) / bl_width_,
(images[img_idx].rows + bl_height_ - 1) / bl_height_);
int bl_width = (images[img_idx].cols + bl_per_img.width - 1) / bl_per_img.width;
int bl_height = (images[img_idx].rows + bl_per_img.height - 1) / bl_per_img.height;
bl_per_imgs[img_idx] = bl_per_img;
for (int by = 0; by < bl_per_img.height; ++by)
{
for (int bx = 0; bx < bl_per_img.width; ++bx)
{
Point bl_tl(bx * bl_width, by * bl_height);
Point bl_br(min(bl_tl.x + bl_width, images[img_idx].cols),
min(bl_tl.y + bl_height, images[img_idx].rows));

block_corners.push_back(corners[img_idx] + bl_tl);
block_images.push_back(images[img_idx](Rect(bl_tl, bl_br)));
block_masks.push_back(make_pair(masks[img_idx].first(Rect(bl_tl, bl_br)),
masks[img_idx].second));
}
}
}

然后，求出任意两块图像的重叠区域的平均光强，

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//计算每一块区域的光照均值sqrt(sqrt（R）+sqrt(G)+sqrt(B))
//光照均值是对称矩阵，所以一次循环计算两个光照值，（i，j），与（j，i）
for (int i = 0; i < num_images; ++i)
{
for (int j = i; j < num_images; ++j)
{
Rect roi;
//判断image[i]与image[j]是否有重叠部分
if (overlapRoi(corners[i], corners[j], images[i].size(), images[j].size(), roi))
{
subimg1 = images[i](Rect(roi.tl() - corners[i], roi.br() - corners[i]));
subimg2 = images[j](Rect(roi.tl() - corners[j], roi.br() - corners[j]));

submask1 = masks[i].first(Rect(roi.tl() - corners[i], roi.br() - corners[i]));
submask2 = masks[j].first(Rect(roi.tl() - corners[j], roi.br() - corners[j]));
intersect = (submask1 == masks[i].second) & (submask2 == masks[j].second);

N(i, j) = N(j, i) = max(1, countNonZero(intersect));

double Isum1 = 0, Isum2 = 0;
for (int y = 0; y < roi.height; ++y)
{
const Point3_<uchar>* r1 = subimg1.ptr<Point3_<uchar> >(y);
const Point3_<uchar>* r2 = subimg2.ptr<Point3_<uchar> >(y);
for (int x = 0; x < roi.width; ++x)
{
if (intersect(y, x))
{
Isum1 += sqrt(static_cast<double>(sqr(r1[x].x) + sqr(r1[x].y) + sqr(r1[x].z)));
Isum2 += sqrt(static_cast<double>(sqr(r2[x].x) + sqr(r2[x].y) + sqr(r2[x].z)));
}
}
}
I(i, j) = Isum1 / N(i, j);
I(j, i) = Isum2 / N(i, j);
}
}
}

建立方程，求出每个光强的调整系数

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Mat_<double> A(num_images, num_images); A.setTo(0);
Mat_<double> b(num_images, 1); b.setTo(0);//beta*N(i,j)
for (int i = 0; i < num_images; ++i)
{
for (int j = 0; j < num_images; ++j)
{
b(i, 0) += beta * N(i, j);
A(i, i) += beta * N(i, j);
if (j == i) continue;
A(i, i) += 2 * alpha * I(i, j) * I(i, j) * N(i, j);
A(i, j) -= 2 * alpha * I(i, j) * I(j, i) * N(i, j);
}
}

solve(A, b, gains_);//求方程的解A*gains = B

gains_原理分析:

num_images ：表示图像分块的个数，零num_images = n

N矩阵，大小n*n，N(i,j)表示第i幅图像与第j幅图像重合的像素点数，N(i,j)=N(j,i)

I矩阵，大小n*n，I(i,j)与I(j,i)表示第i，j块区域重合部分的像素平均值，I(i，j)是重合区域中i快的平均亮度值，



参数alpha和beta，默认值是alpha=0.01，beta=100.

A矩阵，大小n*n，公式图片不全



b矩阵，大小n*1，



然后根据求解矩阵

gains_矩阵，大小1*n，A*gains = B

然后将gains_进行线性滤波

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Mat_<float> ker(1, 3);
ker(0,0) = 0.25; ker(0,1) = 0.5; ker(0,2) = 0.25;

int bl_idx = 0;
for (int img_idx = 0; img_idx < num_images; ++img_idx)
{
Size bl_per_img = bl_per_imgs[img_idx];
gain_maps_[img_idx].create(bl_per_img);

for (int by = 0; by < bl_per_img.height; ++by)
for (int bx = 0; bx < bl_per_img.width; ++bx, ++bl_idx)
gain_maps_[img_idx](by, bx) = static_cast<float>(gains[bl_idx]);
//用分解的核函数对图像做卷积。首先，图像的每一行与一维的核kernelX做卷积；然后，运算结果的每一列与一维的核kernelY做卷积
sepFilter2D(gain_maps_[img_idx], gain_maps_[img_idx], CV_32F, ker, ker);
sepFilter2D(gain_maps_[img_idx], gain_maps_[img_idx], CV_32F, ker, ker);
}

然后构建一个gain_maps的三维矩阵，gain_main[图像的个数][图像分块的行数][图像分块的列数]，然后对没一副图像的gain进行滤波。

9.Seam Estimation

缝隙估计有6种方法，默认就是第三种方法，seam_find_type == "gc_color"，该方法是利用最大流方法检测。

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if (seam_find_type == "no")
seam_finder = new detail::NoSeamFinder();//Stub seam estimator which does nothing.
else if (seam_find_type == "voronoi")
seam_finder = new detail::VoronoiSeamFinder();//Voronoi diagram-based seam estimator. 泰森多边形缝隙估计
else if (seam_find_type == "gc_color")
{
#ifdef HAVE_OPENCV_GPU
if (try_gpu && gpu::getCudaEnabledDeviceCount() > 0)
seam_finder = new detail::GraphCutSeamFinderGpu(GraphCutSeamFinderBase::COST_COLOR);
else
#endif
seam_finder = new detail::GraphCutSeamFinder(GraphCutSeamFinderBase::COST_COLOR);//Minimum graph cut-based seam estimator
}
else if (seam_find_type == "gc_colorgrad")
{
#ifdef HAVE_OPENCV_GPU
if (try_gpu && gpu::getCudaEnabledDeviceCount() > 0)
seam_finder = new detail::GraphCutSeamFinderGpu(GraphCutSeamFinderBase::COST_COLOR_GRAD);
else
#endif
seam_finder = new detail::GraphCutSeamFinder(GraphCutSeamFinderBase::COST_COLOR_GRAD);
}
else if (seam_find_type == "dp_color")
seam_finder = new detail::DpSeamFinder(DpSeamFinder::COLOR);
else if (seam_find_type == "dp_colorgrad")
seam_finder = new detail::DpSeamFinder(DpSeamFinder::COLOR_GRAD);
if (seam_finder.empty())
{
cout << "Can't create the following seam finder '" << seam_find_type << "'\n";
return 1;
}

程序解读可参见： http://blog.csdn.net/chlele0105/article/details/12624541 http://blog.csdn.net/zouxy09/article/details/8534954 http://blog.csdn.net/yangtrees/article/details/7930640

10.多波段融合

由于以前进行处理的图片都是以work_scale（3.1节有介绍）进行缩放的，所以图像的内参，corner（同一坐标后的顶点），mask（融合的掩码）都需要重新计算。以及将之前计算的光照增强的gain也要计算进去。

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// Read image and resize it if necessary
full_img = imread(img_names[img_idx]);
if (!is_compose_scale_set)
{
if (compose_megapix > 0)
compose_scale = min(1.0, sqrt(compose_megapix * 1e6 / full_img.size().area()));
is_compose_scale_set = true;

// Compute relative scales
//compose_seam_aspect = compose_scale / seam_scale;
compose_work_aspect = compose_scale / work_scale;

// Update warped image scale
warped_image_scale *= static_cast<float>(compose_work_aspect);
warper = warper_creator->create(warped_image_scale);

// Update corners and sizes
for (int i = 0; i < num_images; ++i)
{
// Update intrinsics
cameras[i].focal *= compose_work_aspect;
cameras[i].ppx *= compose_work_aspect;
cameras[i].ppy *= compose_work_aspect;

// Update corner and size
Size sz = full_img_sizes[i];
if (std::abs(compose_scale - 1) > 1e-1)
{
sz.width = cvRound(full_img_sizes[i].width * compose_scale);//取整
sz.height = cvRound(full_img_sizes[i].height * compose_scale);
}

Mat K;
cameras[i].K().convertTo(K, CV_32F);
Rect roi = warper->warpRoi(sz, K, cameras[i].R);//Returns Projected image minimum bounding box
corners[i] = roi.tl();//! the top-left corner
sizes[i] = roi.size();//! size of the real buffer
}
}
if (abs(compose_scale - 1) > 1e-1)
resize(full_img, img, Size(), compose_scale, compose_scale);
else
img = full_img;
full_img.release();
Size img_size = img.size();

Mat K;
cameras[img_idx].K().convertTo(K, CV_32F);

// Warp the current image
warper->warp(img, K, cameras[img_idx].R, INTER_LINEAR, BORDER_REFLECT, img_warped);
// Warp the current image mask
mask.create(img_size, CV_8U);
mask.setTo(Scalar::all(255));
warper->warp(mask, K, cameras[img_idx].R, INTER_NEAREST, BORDER_CONSTANT, mask_warped);
// Compensate exposure
compensator->apply(img_idx, corners[img_idx], img_warped, mask_warped);//光照补偿
img_warped.convertTo(img_warped_s, CV_16S);
img_warped.release();
img.release();
mask.release();

dilate(masks_warped[img_idx], dilated_mask, Mat());
resize(dilated_mask, seam_mask, mask_warped.size());
mask_warped = seam_mask & mask_warped;

对图像进行光照补偿，就是将对应区域乘以gain：

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//块光照补偿
void BlocksGainCompensator::apply(int index, Point /*corner*/, Mat &image, const Mat &/*mask*/)
{
CV_Assert(image.type() == CV_8UC3);

Mat_<float> gain_map;
if (gain_maps_[index].size() == image.size())
gain_map = gain_maps_[index];
else
resize(gain_maps_[index], gain_map, image.size(), 0, 0, INTER_LINEAR);

for (int y = 0; y < image.rows; ++y)
{
const float* gain_row = gain_map.ptr<float>(y);
Point3_<uchar>* row = image.ptr<Point3_<uchar> >(y);
for (int x = 0; x < image.cols; ++x)
{
row[x].x = saturate_cast<uchar>(row[x].x * gain_row[x]);
row[x].y = saturate_cast<uchar>(row[x].y * gain_row[x]);
row[x].z = saturate_cast<uchar>(row[x].z * gain_row[x]);
}
}
}

进行多波段融合，首先初始化blend，确定blender的融合的方式，默认是多波段融合MULTI_BAND，以及根据corners顶点和图像的大小确定最终全景图的尺寸。

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<span> </span>//初始化blender
if (blender.empty())
{
blender = Blender::createDefault(blend_type, try_gpu);
Size dst_sz = resultRoi(corners, sizes).size();//计算最后图像的大小
float blend_width = sqrt(static_cast<float>(dst_sz.area())) * blend_strength / 100.f;
if (blend_width < 1.f)
blender = Blender::createDefault(Blender::NO, try_gpu);
else if (blend_type == Blender::MULTI_BAND)
{
MultiBandBlender* mb = dynamic_cast<MultiBandBlender*>(static_cast<Blender*>(blender));
mb->setNumBands(static_cast<int>(ceil(log(blend_width)/log(2.)) - 1.));
LOGLN("Multi-band blender, number of bands: " << mb->numBands());
}
else if (blend_type == Blender::FEATHER)
{
FeatherBlender* fb = dynamic_cast<FeatherBlender*>(static_cast<Blender*>(blender));
fb->setSharpness(1.f/blend_width);
LOGLN("Feather blender, sharpness: " << fb->sharpness());
}
blender->prepare(corners, sizes);//根据corners顶点和图像的大小确定最终全景图的尺寸
}

然后对每幅图图形构建金字塔，层数可以由输入的系数确定，默认是5层。
先对顶点以及图像的宽和高做处理，使其能被2^num_bands除尽，这样才能将进行向下采样num_bands次，首先从源图像pyrDown向下采样，在由最底部的图像pyrUp向上采样，把对应层得上采样和下采样的相减，就得到了图像的高频分量，存储到每一个金字塔中。然后根据mask，将每幅图像的各层金字塔分别写入最终的金字塔层src_pyr_laplace中。

最后将各层的金字塔叠加，就得到了最终完整的全景图。

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blender->feed(img_warped_s, mask_warped, corners[img_idx]);//将图像写入金字塔中

源码：

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void MultiBandBlender::feed(const Mat &img, const Mat &mask, Point tl)
{
CV_Assert(img.type() == CV_16SC3 || img.type() == CV_8UC3);
CV_Assert(mask.type() == CV_8U);

// Keep source image in memory with small border
int gap = 3 * (1 << num_bands_);
Point tl_new(max(dst_roi_.x, tl.x - gap),
max(dst_roi_.y, tl.y - gap));
Point br_new(min(dst_roi_.br().x, tl.x + img.cols + gap),
min(dst_roi_.br().y, tl.y + img.rows + gap));

// Ensure coordinates of top-left, bottom-right corners are divided by (1 << num_bands_).
// After that scale between layers is exactly 2.
//
// We do it to avoid interpolation problems when keeping sub-images only. There is no such problem when
// image is bordered to have size equal to the final image size, but this is too memory hungry approach.
//将顶点调整成能被2^num_bank次方除尽的值
tl_new.x = dst_roi_.x + (((tl_new.x - dst_roi_.x) >> num_bands_) << num_bands_);
tl_new.y = dst_roi_.y + (((tl_new.y - dst_roi_.y) >> num_bands_) << num_bands_);
int width = br_new.x - tl_new.x;
int height = br_new.y - tl_new.y;
width += ((1 << num_bands_) - width % (1 << num_bands_)) % (1 << num_bands_);
height += ((1 << num_bands_) - height % (1 << num_bands_)) % (1 << num_bands_);
br_new.x = tl_new.x + width;
br_new.y = tl_new.y + height;
int dy = max(br_new.y - dst_roi_.br().y, 0);
int dx = max(br_new.x - dst_roi_.br().x, 0);
tl_new.x -= dx; br_new.x -= dx;
tl_new.y -= dy; br_new.y -= dy;

int top = tl.y - tl_new.y;
int left = tl.x - tl_new.x;
int bottom = br_new.y - tl.y - img.rows;
int right = br_new.x - tl.x - img.cols;

// Create the source image Laplacian pyramid
Mat img_with_border;
copyMakeBorder(img, img_with_border, top, bottom, left, right,
BORDER_REFLECT);//给图像设置一个边界，BORDER_REFLECT边界颜色任意
vector<Mat> src_pyr_laplace;
if (can_use_gpu_ && img_with_border.depth() == CV_16S)
createLaplacePyrGpu(img_with_border, num_bands_, src_pyr_laplace);
else
createLaplacePyr(img_with_border, num_bands_, src_pyr_laplace);//创建高斯金字塔，每一层保存的全是高频信息

// Create the weight map Gaussian pyramid
Mat weight_map;
vector<Mat> weight_pyr_gauss(num_bands_ + 1);

if(weight_type_ == CV_32F)
{
mask.convertTo(weight_map, CV_32F, 1./255.);//将mask的0，255归一化成0,1
}
else// weight_type_ == CV_16S
{
mask.convertTo(weight_map, CV_16S);
add(weight_map, 1, weight_map, mask != 0);
}

copyMakeBorder(weight_map, weight_pyr_gauss[0], top, bottom, left, right, BORDER_CONSTANT);

for (int i = 0; i < num_bands_; ++i)
pyrDown(weight_pyr_gauss[i], weight_pyr_gauss[i + 1]);

int y_tl = tl_new.y - dst_roi_.y;
int y_br = br_new.y - dst_roi_.y;
int x_tl = tl_new.x - dst_roi_.x;
int x_br = br_new.x - dst_roi_.x;

// Add weighted layer of the source image to the final Laplacian pyramid layer
if(weight_type_ == CV_32F)
{
for (int i = 0; i <= num_bands_; ++i)
{
for (int y = y_tl; y < y_br; ++y)
{
int y_ = y - y_tl;
const Point3_<short>* src_row = src_pyr_laplace[i].ptr<Point3_<short> >(y_);
Point3_<short>* dst_row = dst_pyr_laplace_[i].ptr<Point3_<short> >(y);
const float* weight_row = weight_pyr_gauss[i].ptr<float>(y_);
float* dst_weight_row = dst_band_weights_[i].ptr<float>(y);

for (int x = x_tl; x < x_br; ++x)
{
int x_ = x - x_tl;
dst_row[x].x += static_cast<short>(src_row[x_].x * weight_row[x_]);
dst_row[x].y += static_cast<short>(src_row[x_].y * weight_row[x_]);
dst_row[x].z += static_cast<short>(src_row[x_].z * weight_row[x_]);
dst_weight_row[x] += weight_row[x_];
}
}
x_tl /= 2; y_tl /= 2;
x_br /= 2; y_br /= 2;
}
}
else// weight_type_ == CV_16S
{
for (int i = 0; i <= num_bands_; ++i)
{
for (int y = y_tl; y < y_br; ++y)
{
int y_ = y - y_tl;
const Point3_<short>* src_row = src_pyr_laplace[i].ptr<Point3_<short> >(y_);
Point3_<short>* dst_row = dst_pyr_laplace_[i].ptr<Point3_<short> >(y);
const short* weight_row = weight_pyr_gauss[i].ptr<short>(y_);
short* dst_weight_row = dst_band_weights_[i].ptr<short>(y);

for (int x = x_tl; x < x_br; ++x)
{
int x_ = x - x_tl;
dst_row[x].x += short((src_row[x_].x * weight_row[x_]) >> 8);
dst_row[x].y += short((src_row[x_].y * weight_row[x_]) >> 8);
dst_row[x].z += short((src_row[x_].z * weight_row[x_]) >> 8);
dst_weight_row[x] += weight_row[x_];
}
}
x_tl /= 2; y_tl /= 2;
x_br /= 2; y_br /= 2;
}
}
}

其中，金字塔构建的源码：

[cpp]
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void createLaplacePyr(const Mat &img, int num_levels, vector<Mat> &pyr)
{
#ifdef HAVE_TEGRA_OPTIMIZATION
if(tegra::createLaplacePyr(img, num_levels, pyr))
return;
#endif

pyr.resize(num_levels + 1);

if(img.depth() == CV_8U)
{
if(num_levels == 0)
{
img.convertTo(pyr[0], CV_16S);
return;
}

Mat downNext;
Mat current = img;
pyrDown(img, downNext);

for(int i = 1; i < num_levels; ++i)
{
Mat lvl_up;
Mat lvl_down;

pyrDown(downNext, lvl_down);
pyrUp(downNext, lvl_up, current.size());
subtract(current, lvl_up, pyr[i-1], noArray(), CV_16S);

current = downNext;
downNext = lvl_down;
}

{
Mat lvl_up;
pyrUp(downNext, lvl_up, current.size());
subtract(current, lvl_up, pyr[num_levels-1], noArray(), CV_16S);

downNext.convertTo(pyr[num_levels], CV_16S);
}
}
else
{
pyr[0] = img;
//构建高斯金字塔
for (int i = 0; i < num_levels; ++i)
pyrDown(pyr[i], pyr[i + 1]);//先高斯滤波，在亚采样，得到比pyr【i】缩小一半的图像
Mat tmp;
for (int i = 0; i < num_levels; ++i)
{
pyrUp(pyr[i + 1], tmp, pyr[i].size());//插值（偶数行，偶数列赋值为0），然后高斯滤波，核是5*5。
subtract(pyr[i], tmp, pyr[i]);//pyr[i] = pyr[i]-tmp，得到的全是高频信息
}
}
}

最终把所有层得金字塔叠加的程序：

[cpp]
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Mat result, result_mask;
blender->blend(result, result_mask);//将多层金字塔图形叠加

源码：

[cpp]
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void MultiBandBlender::blend(Mat &dst, Mat &dst_mask)
{
for (int i = 0; i <= num_bands_; ++i)
normalizeUsingWeightMap(dst_band_weights_[i], dst_pyr_laplace_[i]);

if (can_use_gpu_)
restoreImageFromLaplacePyrGpu(dst_pyr_laplace_);
else
restoreImageFromLaplacePyr(dst_pyr_laplace_);

dst_ = dst_pyr_laplace_[0];
dst_ = dst_(Range(0, dst_roi_final_.height), Range(0, dst_roi_final_.width));
dst_mask_ = dst_band_weights_[0] > WEIGHT_EPS;
dst_mask_ = dst_mask_(Range(0, dst_roi_final_.height), Range(0, dst_roi_final_.width));
dst_pyr_laplace_.clear();
dst_band_weights_.clear();

Blender::blend(dst, dst_mask);
}