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：图像与复杂核的卷积是摄影、科学成像与动画特效中的基础运算，但直接进行稠密卷积在资源受限设备上计算代价过高。现有近似方法（如模拟退火或低秩分解）或效率不足，或难以捕捉非凸核结构。本文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的空间变体、稠密、复杂核。本方法具有以下特点：（i）支持稀疏核可微分优化的分解机制；（ii）针对非凸形状的专用初始化策略，以避免陷入不良局部极小值；（iii）核空间插值方案，可将单核滤波扩展至空间变化滤波，无需重新训练且无额外运行时开销。在高斯核与非凸核上的实验表明，本方法比模拟退火具有更高保真度，且计算成本显著低于低秩分解。本方法为移动成像与实时渲染提供了实用解决方案，同时保持完全可微分性，便于集成至更广泛的学习流程中。