The Laplacian operator transforms the image into its Laplacian field, which usually is sparse and satisfies a stable distribution. On the other hand, an image can be uniquely reconstructed from its Laplacian field via solving a Poisson equation with a proper boundary condition. Such uniqueness is mathematically guaranteed. Thanks to these properties, we propose to use the sparse Laplacian field to present the image. We first show that the Laplacian field is sparse and satisfies a stable distribution on hundreds images. Then, we show that the image can be accurately reconstruct from its Laplacian field. For the reconstruction task, we propose a shared-kernel wavelet neural network, which solves the Poisson equation and has three advantages. First, it has less than {\bf 0.0002M} parameters, which is compact enough for most of devices. Second, it has linear computation complexity, leading to a real-time reconstruction. Third, it achieves higher accuracy than previous methods. Several numerical experiments are conducted to show the effectiveness and efficiency of the sparse Laplacian field and the proposed Poisson solver. The proposed method can be applied in a large range of applications such as image compression, low light enhancement, object tracking, etc.

翻译：拉普拉斯算子将图像变换为其拉普拉斯场，该场通常稀疏且满足稳定分布。另一方面，通过求解带适当边界条件的泊松方程，图像可唯一地从其拉普拉斯场重建，这种唯一性在数学上得到保证。基于这些特性，我们提出使用稀疏拉普拉斯场表示图像。首先，我们证明在数百张图像中拉普拉斯场具有稀疏性且满足稳定分布。然后，我们证明图像可以从其拉普拉斯场精确重建。针对重建任务，我们提出一种共享核小波神经网络来求解泊松方程，该网络具有三个优势：第一，其参数不足{\bf 0.0002M}，可紧凑适配大多数设备；第二，具有线性计算复杂度，可实现实时重建；第三，相比现有方法具有更高精度。通过多个数值实验验证了稀疏拉普拉斯场及所提泊松求解器的有效性与高效性。该方法可广泛应用于图像压缩、低光增强、目标跟踪等领域。