Image convolution with complex kernels is a fundamental operation in photography, scientific imaging, and animation effects, yet direct dense convolution is computationally prohibitive on resource-limited devices. Existing approximations, such as simulated annealing or low-rank decompositions, either lack efficiency or fail to capture non-convex kernels. We introduce a differentiable kernel decomposition framework that represents a target spatially-variant, dense, complex kernel using a set of sparse kernel samples. Our approach features (i) a decomposition that enables differentiable optimization of sparse kernels, (ii) a dedicated initialization strategy for non-convex shapes to avoid poor local minima, and (iii) a kernel-space interpolation scheme that extends single-kernel filtering to spatially varying filtering without retraining and additional runtime overhead. Experiments on Gaussian and non-convex kernels show that our method achieves higher fidelity than simulated annealing and significantly lower cost than low-rank decompositions. Our approach provides a practical solution for mobile imaging and real-time rendering, while remaining fully differentiable for integration into broader learning pipelines.

翻译：复数核图像卷积是摄影、科学成像和动画效果中的基本操作，然而直接进行稠密卷积在资源受限设备上计算代价极高。现有近似方法如模拟退火或低秩分解，要么效率不足，要么无法捕获非凸核函数。我们提出一种可微分核分解框架，通过一组稀疏核样本来表示目标空间变异稠密复数核。该方法包含：(i) 实现稀疏核可微分优化的分解策略，(ii) 针对非凸形状设计的专用初始化方法以避免局部极小值问题，以及(iii) 无需重新训练且无额外运行时开销即可将单核滤波扩展至空间变异滤波的核空间插值方案。在高斯核与非凸核上的实验表明，本方法相比模拟退火具有更高保真度，且成本显著低于低秩分解。该方案为移动成像和实时渲染提供了实用解决方案，同时保持完全可微分特性以集成至更广泛的训练流程中。