Abstract
The paper introduces the weighted convolution, a novel approach to the convolution for signals defined on regular grids (e.g., 2D images) through the application of an optimal density function to scale the contribution of neighbouring pixels based on their distance from the central pixel. This choice differs from the traditional uniform convolution, which treats all neighbouring pixels equally. Our weighted convolution can be applied to convolutional neural network problems to improve the approximation accuracy. Given a convolutional network, we define a framework to compute the optimal density function through a minimisation model. The framework separates the optimisation of the convolutional kernel weights (using stochastic gradient descent) from the optimisation of the density function (using DIRECT-L). Experimental results on a learning model for an image-to-image task (e.g., image denoising) show that the weighted convolution significantly reduces the loss (up to 53% improvement) and increases the test accuracy compared to standard convolution. While this method increases execution time by 11%, it is robust across several hyperparameters of the learning model. Future work will apply the weighted convolution to real-case 2D and 3D image convolutional learning problems.
Abstract (translated)
该论文介绍了加权卷积，这是一种针对在规则网格（例如二维图像）上定义的信号的新颖卷积方法。通过应用最优密度函数来调整邻近像素对中心像素贡献的比例，其依据是这些像素与中心像素的距离。这一选择不同于传统的均匀卷积法，后者对所有相邻像素一视同仁。我们的加权卷积可以应用于卷积神经网络问题中以提高逼近精度。 对于给定的卷积网络，我们通过最小化模型定义了一个框架来计算最优密度函数。该框架将卷积核权重的优化（使用随机梯度下降法）与密度函数的优化（使用DIRECT-L方法）分开进行。在学习模型处理图像到图像任务（例如图像去噪）时所取得的实验结果表明，加权卷积显著降低了损失（最高可达53%的改进），并提高了测试精度，相比标准卷积而言。 虽然此方法增加了11%的执行时间，但它对于学习模型中的多个超参数都是稳健的。未来的研究将把加权卷积应用于实际案例中的二维和三维图像卷积学习问题。
URL
https://arxiv.org/abs/2505.24527
