傅里叶下确界半格中的形态表示理论：频域深度学习算子的通用分解
Morphological Representation Theory in the Fourier Inf-Semilattice: Universal Decomposition of Frequency-Domain Deep Learning Operators
- Mines Paris, PSL University（巴黎矿业大学，PSL大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究建立傅里叶下确界半格下的频域形态表示理论，证明正齐次等算子的形态分解定理，推导深度网络、散射、池化等相关结构推论，为频域深度学习提供形态学算子理论基础。
AI中文摘要：
我们建立了作用于$L^2(\mathbb{R}^n)$频域的算子的形态表示理论。为该空间赋予**傅里叶下确界半格**序$\leq_{\\!\mathcal{F}}$（相位相等时的谱模不等式）后，卷积成为形态腐蚀，其伴随膨胀即为维纳逆滤波器。一个关键发现是，空间平移不变性在谱序中无实质意义，其结构地位被算子在谱模上的**正齐次性**取代。在此假设下，我们证明了傅里叶模格$\hat{L}$的**形态表示定理**：任意递增、上半连续、正齐次的算子$\Psi:\hat{L}\to\hat{L}$可精确分解为最大-乘积腐蚀$\hat{f}/\psi$的上确界，由最小形态基索引。由此得到三个结构推论：谱模映射$|\cdot|:L^2(\mathbb{R}^n)\to\hat{L}$是幂等交叉格投影（形态学版本的ReLU），使深度具有非平凡性；小波散射系数通过Littlewood–Paley密度构成该基的标准字典；空间池化是傅里叶腐蚀，因此池化后再反池化是理想带通开运算，U-Net跳跃连接是傅里叶顶帽变换，且步长池化保持形态结构当且仅当奈奎斯特条件成立。该框架带有自然的$\mathbb{C}^*$-群形态结构，将数学形态学定位为谱偏置、散射和热带几何理论的构造性算子理论补充。
英文摘要：
We develop a morphological representation theory for operators acting in the frequency domain of $L^2(\mathbb{R}^n)$. Equipping the space with the \emph{Fourier inf-semilattice} order $\leq_{\!\mathcal{F}}$ (spectral modulus inequality with phase equality), convolution becomes a morphological erosion and the adjoint dilation is the Wiener inverse filter. A key observation is that spatial translation-invariance is vacuous in the spectral order, replaced structurally by \emph{positive homogeneity} of the operator on spectral moduli. Under this hypothesis we prove the \emph{Morphological Representation Theorem} for the Fourier modulus lattice $\hat{L}$: any increasing, USC, positively homogeneous $Ψ:\hat{L}\to\hat{L}$ decomposes \emph{exactly} as a supremum of max-times erosions $\hat{f}/ψ$, indexed by a minimal morphological basis. Three structural corollaries follow: the spectral modulus map $|\cdot|:L^2(\mathbb{R}^n)\to\hat{L}$ is an idempotent cross-lattice projection (the morphological analogue of ReLU) that makes depth non-trivial; wavelet scattering coefficients form the canonical dictionary for the basis via Littlewood--Paley density; and spatial pooling is a Fourier erosion, so that pool-then-unpool is the ideal band-pass opening, U-Net skip connections are Fourier top-hat transforms, and strided pooling preserves the morphological structure if and only if the Nyquist condition holds. The framework carries a natural $\mathbb{C}^*$-group morphology structure, positioning mathematical morphology as the constructive operator-theoretic complement of spectral bias, scattering, and tropical geometry theories.
