3D高斯损失函数（2）添加BA优化和结构损失

在PyTorch中添加重投影误差（Reprojection Error）作为损失函数通常用于计算机视觉任务，特别是涉及多视图几何（如立体视觉或多视角重建）的问题。重投影误差衡量了3D点在投影到不同视角后的图像平面上的偏差。以下是如何在PyTorch中实现和使用重投影误差作为损失函数的步骤：

定义重投影误差函数：

重投影误差通常是2D图像点的实际位置与通过投影矩阵计算得到的预测位置之间的距离。
实现损失函数：

使用PyTorch的操作来计算重投影误差，并将其封装在一个损失函数中。
假设你有以下变量：

points_3d: 原始的3D点，形状为 (N, 3)
camera_matrix: 相机的内参矩阵，形状为 (3, 3)
extrinsics: 相机的外参矩阵，形状为 (3, 4)
points_2d: 实际2D图像点，形状为 (N, 2)
以下是一个简单的示例代码：

import torch
import torch.nn as nn
import torch.nn.functional as F

class ReprojectionLoss(nn.Module):
    def __init__(self):
        super(ReprojectionLoss, self).__init__()

    def forward(self, points_3d, points_2d, camera_matrix, extrinsics):
        # 添加一个列 [1] 到 3D 点的末尾 (N, 3) -> (N, 4)
        ones = torch.ones(points_3d.shape[0], 1, device=points_3d.device)
        points_3d_h = torch.cat([points_3d, ones], dim=1)  # (N, 4)

        # 计算投影点
        projected_points_2d_h = camera_matrix @ extrinsics @ points_3d_h.T  # (3, N)
        projected_points_2d = projected_points_2d_h[:2] / projected_points_2d_h[2]  # 归一化 (2, N)
        projected_points_2d = projected_points_2d.T  # (N, 2)

        # 计算重投影误差 (实际2D点和预测2D点之间的距离)
        reprojection_error = torch.norm(points_2d - projected_points_2d, dim=1)  # (N,)
        
        # 返回平均误差作为损失
        return reprojection_error.mean()

# 示例数据
points_3d = torch.rand((10, 3), requires_grad=True)
points_2d = torch.rand((10, 2))
camera_matrix = torch.eye(3)
extrinsics = torch.cat([torch.eye(3), torch.zeros(3, 1)], dim=1)

# 实例化并计算损失
loss_fn = ReprojectionLoss()
loss = loss_fn(points_3d, points_2d, camera_matrix, extrinsics)
print("Loss:", loss.item())

# 使用优化器最小化损失
optimizer = torch.optim.Adam([points_3d], lr=0.01)
optimizer.zero_grad()
loss.backward()
optimizer.step()

　　在这个示例中，我们定义了一个自定义的损失函数 ReprojectionLoss，计算了实际2D点和预测2D点之间的欧几里得距离作为重投影误差，并返回平均误差作为损失。在训练过程中，你可以使用这个损失函数来更新你的模型参数。、

 

 

可以将结构相似性指数（SSIM）损失与重投影误差结合起来作为总损失函数。在此示例中，我们将两者的权重设置为50%，即两者各占50%的权重。你可以使用PyTorch实现SSIM损失。

首先，我们需要实现或引入SSIM损失函数。然后，我们可以将两个损失结合起来进行训练。

实现SSIM损失函数
下面是一个简单的SSIM损失实现：

import torch
import torch.nn.functional as F
import numpy as np

class SSIMLoss(nn.Module):
    def __init__(self, window_size=11, size_average=True):
        super(SSIMLoss, self).__init__()
        self.window_size = window_size
        self.size_average = size_average
        self.channel = 1
        self.window = self.create_window(window_size, self.channel)

    def create_window(self, window_size, channel):
        def gaussian(window_size, sigma):
            gauss = torch.Tensor(
                [np.exp(-(x - window_size // 2)**2 / float(2 * sigma**2)) for x in range(window_size)])
            return gauss / gauss.sum()

        _1D_window = gaussian(window_size, 1.5).unsqueeze(1)
        _2D_window = _1D_window.mm(_1D_window.t()).float().unsqueeze(0).unsqueeze(0)
        window = _2D_window.expand(channel, 1, window_size, window_size).contiguous()
        return window

    def _ssim(self, img1, img2, window, window_size, channel, size_average=True):
        mu1 = F.conv2d(img1, window, padding=window_size // 2, groups=channel)
        mu2 = F.conv2d(img2, window, padding=window_size // 2, groups=channel)

        mu1_sq = mu1.pow(2)
        mu2_sq = mu2.pow(2)
        mu1_mu2 = mu1 * mu2

        sigma1_sq = F.conv2d(img1 * img1, window, padding=window_size // 2, groups=channel) - mu1_sq
        sigma2_sq = F.conv2d(img2 * img2, window, padding=window_size // 2, groups=channel) - mu2_sq
        sigma12 = F.conv2d(img1 * img2, window, padding=window_size // 2, groups=channel) - mu1_mu2

        C1 = 0.01 ** 2
        C2 = 0.03 ** 2

        ssim_map = ((2 * mu1_mu2 + C1) * (2 * sigma12 + C2)) / ((mu1_sq + mu2_sq + C1) * (sigma1_sq + sigma2_sq + C2))

        if size_average:
            return ssim_map.mean()
        else:
            return ssim_map.mean(1).mean(1).mean(1)

    def forward(self, img1, img2):
        (_, channel, _, _) = img1.size()
        window = self.window.type_as(img1)

        return 1 - self._ssim(img1, img2, window, self.window_size, channel, self.size_average)

　　

结合两种损失函数
假设你有以下变量：

images_pred: 预测的图像
images_gt: 真实图像
我们可以将SSIM损失和重投影误差结合起来：

import torch
import torch.nn as nn

class CombinedLoss(nn.Module):
    def __init__(self, reprojection_loss_weight=0.5, ssim_loss_weight=0.5):
        super(CombinedLoss, self).__init__()
        self.reprojection_loss_weight = reprojection_loss_weight
        self.ssim_loss_weight = ssim_loss_weight
        self.reprojection_loss_fn = ReprojectionLoss()
        self.ssim_loss_fn = SSIMLoss()

    def forward(self, points_3d, points_2d, camera_matrix, extrinsics, images_pred, images_gt):
        reprojection_loss = self.reprojection_loss_fn(points_3d, points_2d, camera_matrix, extrinsics)
        ssim_loss = self.ssim_loss_fn(images_pred, images_gt)

        combined_loss = (self.reprojection_loss_weight * reprojection_loss +
                         self.ssim_loss_weight * ssim_loss)
        return combined_loss

# 示例数据
points_3d = torch.rand((10, 3), requires_grad=True)
points_2d = torch.rand((10, 2))
camera_matrix = torch.eye(3)
extrinsics = torch.cat([torch.eye(3), torch.zeros(3, 1)], dim=1)
images_pred = torch.rand((1, 1, 256, 256), requires_grad=True)
images_gt = torch.rand((1, 1, 256, 256))

# 实例化并计算损失
combined_loss_fn = CombinedLoss()
loss = combined_loss_fn(points_3d, points_2d, camera_matrix, extrinsics, images_pred, images_gt)
print("Combined Loss:", loss.item())

# 使用优化器最小化损失
optimizer = torch.optim.Adam([points_3d, images_pred], lr=0.01)
optimizer.zero_grad()
loss.backward()
optimizer.step()

　　在这个示例中，我们定义了一个 CombinedLoss 类，它结合了重投影误差和SSIM损失。然后，我们使用这个组合损失函数来计算总损失，并用优化器最小化损失。你可以根据需要调整重投影误差和SSIM损失的权重。