Loading... ## 1、前言 在基于PCA的人脸识别算法中，二维人脸图像要先转换为一维向量。这就导致人脸图像向量将达到高维空间，由于其协方差矩阵维度很高而且训练样本很少，因此很难评估协方差矩阵的准确性。虽然我们可以通过SVD的方法来简化计算，但是将二维图像转换为一维向量，丢失了图像行列的相关信息。而2DPCA是基于二维图像矩阵的，这种处理方法不需要事先把图像转成一维向量，图像的协方差矩阵可以通过原始图像矩阵直接构造出来。 ## 3、2DPCA推导 #### 2.1 首先定义一个线性变换方程 $$ Y = AX $$ - 上式中A为原始图像矩阵，X是一组正交基（特征子空间），Y是图像投影到特征子空间后的数据。 #### 2.2 构造投影特征的协方差矩阵 $$ S_X = E(Y-EY)(Y-EY)^T \\ S_x = E[AX-E(AX)][AX-E(AX)]^T \\ S_x = E[(A-EA)X][(A-EA)X]^T $$ #### 2.3 构造图像协方差矩阵 $$ G_t = E[(A-EA)^T)(A-EA)] $$ - 矩阵$G_t$叫做图像协方差矩阵。容易验证，$G_t$是一个`nxn`正定矩阵。我们可以使用训练样本来评估$G_t$，假设共有M个训练图像样本，第j个训练样本由$A_j(j=1,2,...,M)$由`mxn`矩阵构成，并且所有样本的平均图像为$\bar{A} $。那么，$G_t$可简化为 $$ G_t = \frac{1}{M}\sum_{j=1}^{M}(A_j-\bar{A})^T(A_j-\bar{A}a) $$ #### 2.4 计算图像协方差矩阵的特征值和特征向量 - 得到图像协方差矩阵$G_t$后，计算$G_t$的特征值$\lambda_i$和特征向量$v_i$。 - 对特征值进行排序，取前d个特征值对应的特征向量作为特征子空间。 $$ w = \{v_1,v_2,...,v_d\} $$ #### 2.5 特征提取（将图像映射到特征子空间） $$ Y_k = w^TA_k $$ - $w$为特征向量构成的矩阵（其维度为`mxt`），$A_k$为第k个图像样本。 $$ V = [Y_1,Y_2,...,Y_d] $$ #### 2.6 分类识别 $$ d = \sum_{j=1}^{d}||Y_k^{(i)}-Y_k^{(j)}||_2 $$ #### 2.7 图像重建 $$ \hat{A} = wV $$ ## 3、2DPCA图像重建结果 - `t=26`是前t个特征值的和占所有特征值和的前99%取值。   - 代码实现 ```cpp void PcaAchieve::PCA2D_achieve(){ //数据集的划分按自己情况... std::string train_image_path = "../images/ORL_faces_train_3/s"; int per_class_num = 8; //每一个类别的样本数 int class_nums = 40; //类别数 //图像样本矩阵 (40 x 112 x 92) std::vector<cv::Mat> all_train_image; //平均脸 （112 x 92） cv::Mat average_face(112, 92, CV_32FC1, cv::Scalar(0)); for (int i = 1; i <= 40; i++) { std::string train_image_sub_path = train_image_path + std::to_string(i); cv::Mat per_image(112, 92, CV_32FC1, cv::Scalar(0)); for (int j = 1; j <= per_class_num; j++) { std::string image_path = train_image_sub_path + "/" + std::to_string(j) + ".png"; std::cout << image_path << std::endl; cv::Mat image = cv::imread(image_path, 0); image.convertTo(image, 5, 1.0/255); per_image += image; } cv::Mat average_image = per_image / 8; average_face += average_image; all_train_image.push_back(average_image); } average_face /= class_nums; cv::imshow("average_face", average_face); cv::waitKey(0); //计算协方差矩阵 (112 x 92) x (92 x 112) cv::Mat conv(112, 112, CV_32FC1, cv::Scalar(0)); for (int i = 0; i < class_nums; i++) { //去均值脸 cv::Mat decenter_image = all_train_image[i] - average_face; conv += decenter_image * cv::Mat(decenter_image.t()); } cv::Mat eigen_values; cv::Mat eigen_vectors; cv::eigen(conv, eigen_values, eigen_vectors); float eigen_value_sum; for (int j = 0; j < eigen_values.rows; j++) { eigen_value_sum += eigen_values.at<float>(j, 0); std::cout << eigen_values.at<float>(j, 0) << std::endl; } float lamb = 0; int t = 0; for (; t < eigen_values.rows; t++) { lamb += eigen_values.at<float>(t, 0); std::cout << lamb / eigen_value_sum << std::endl; if (lamb / eigen_value_sum >= 0.99) { break; } } std::cout << "get_t:" << t << std::endl; t = 5; //(112 x t) cv::Mat L_eigen_vec(112, t, 5, cv::Scalar(0)); for (int i = 0; i < t; i++) { L_eigen_vec(cv::Rect(i, 0, 1, 112)) += eigen_vectors(cv::Rect(i, 0, 1, 112)); } cv::Mat temp(t, 112, 5, cv::Scalar(0)); for (int i = 0; i < t; i++) { temp(cv::Rect(0, i, 112, 1)) += eigen_vectors(cv::Rect(0, i, 112, 1)); } std::cout << "-------------"; cv::Mat test_image = cv::imread("../images/ORL_faces_test_3/s1/10.png", 0); test_image.convertTo(test_image, CV_32FC1, 1.0/255); test_image -= average_face; //将样本映射到特征子空间 (t x 112) x (112 x 92) = (t x 92) cv::Mat Y_test = cv::Mat(L_eigen_vec.t()) * test_image; //样本重建 (112 x t) x (t x 92) cv::Mat rebuild_image = L_eigen_vec * Y_test; rebuild_image += average_face; cv::imshow("rebuild_image", rebuild_image); cv::waitKey(0); } ``` ## 4、2DPCA人脸识别结果 - 2DPCA识别人脸准确率 | 特征子空间维度 | (112 x 1) | (112 x 2) | (112 x 3) | (112 x 5) | (112 x 10) | (112 x 26) | (112 x 40) | | -------------- | --------- | --------- | --------- | --------- | ---------- | ---------- | ---------- | | 识别准确率(%) | 41.25 | 58.75 | 70 | 73.75 | 83.75 | 90 | 91.25 | - 代码 ```cpp void recg_face_with_2dpca() { /***人脸识别部分***/ //将训练集投影到特征子空间 std::vector<cv::Mat> train_sub; for (auto iter = all_train_image.begin(); iter != all_train_image.end(); iter++) { train_sub.push_back(cv::Mat(L_eigen_vec.t()) * (*iter-average_face)); } //读取测试集图像 std::string test_image_path = "../images/ORL_faces_test_3/s"; int right = 0; for (int i = 1; i <= 40; i++) { std::string test_image_sub_path = test_image_path + std::to_string(i); cv::Mat per_image(112, 92, CV_32FC1, cv::Scalar(0)); for (int j = 0; j < 2 ; j++) { std::string image_path = test_image_sub_path + "/" + std::to_string(j+9) + ".png"; cv::Mat test_image = cv::imread(image_path, 0); test_image.convertTo(test_image, 5, 1.0 / 255); test_image -= average_face; cv::Mat test_image_sub = cv::Mat(L_eigen_vec.t()) * test_image; int result = compute_face_dist(train_sub, test_image_sub); std::cout << "result:" << result << std::endl; if (result == i) right++; } } std::cout << t << " recg right :" << right << "recg accracy:" << (float)(right / 80.0) << std::endl; } int PcaAchieve::compute_face_dist(std::vector<cv::Mat>& train_sub, cv::Mat test_image) { std::vector<float> dist_vector; for (auto iter = train_sub.begin(); iter != train_sub.end(); iter++) { float temp = cv::norm(*iter, test_image); dist_vector.push_back(temp); } auto position = std::min_element(dist_vector.begin(), dist_vector.end()); int result = position - dist_vector.begin(); return result + 1; } ``` ## 5、参考资料 [1] Jian Yang, D. Zhang, A. F. Frangi and Jing-yu Yang, "Two-dimensional PCA: a new approach to appearance-based face representation and recognition," in IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 26, no. 1, pp. 131-137, Jan. 2004, doi: 10.1109/TPAMI.2004.1261097. [2] [图像的 2DPCA 与 2D2DPCA 特征提取](https://blog.csdn.net/lifeng_math/article/details/50474740?utm_medium=distribute.pc_relevant.none-task-blog-BlogCommendFromBaidu-2.control&depth_1-utm_source=distribute.pc_relevant.none-task-blog-BlogCommendFromBaidu-2.control) [3] [【python】双向二维PCA（2D-2D PCA）算法实现](https://blog.csdn.net/w450468524/article/details/54895477) Last modification：January 22nd, 2021 at 08:17 pm © 允许规范转载