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、科学成像和动画特效中的基础操作，然而在资源受限设备上直接进行密集卷积的计算代价过高。现有近似方法（如模拟退火或低秩分解）要么效率不足，要么无法捕捉非凸核。我们提出一种可微核分解框架，通过一组稀疏核样本来表示目标空间变化、密集、复杂的核。该方法包含：（i）一种支持稀疏核可微优化的分解方案，（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，以及（iii）一种核空间插值方案，可将单核滤波扩展至空间变化滤波，且无需重新训练或增加运行时开销。在高斯核与非凸核上的实验表明，我们的方法相比模拟退火实现了更高保真度，同时比低秩分解显著降低了计算成本。该方法为移动成像和实时渲染提供了实用解决方案，且完全可微，便于集成到更广泛的机器学习流程中。