Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A foundation for exact binarized morphological neural networks

Theodore Aouad, Hugues Talbot

TL;DR

This work offers a mathematically grounded foundation for binarized neural networks by marrying Binary Morphological Neural Networks (BiMoNN) with Mathematical Morphology (MM). It introduces the Binary Structuring Element (BiSE) neuron and the BiSEL/DenseLUI layers to realize morphologically meaningful, binarizable architectures that extend to grayscale and RGB inputs, along with exact and approximate binarization schemes. A key contribution is a morphological regularization framework that guides networks toward activable, morphology-consistent representations, improving binarized performance on tasks like denoising and MNIST classification. The approach challenges traditional STE-based BWNNs, showing that learning and binarization can be exact or approximately accurate when guided by MM principles, with practical benefits for efficiency and interpretability. Overall, BiMoNN establishes a robust pathway for morphology-aware binarization with demonstrated empirical potential and publicly available code.

Abstract

Training and running deep neural networks (NNs) often demands a lot of computation and energy-intensive specialized hardware (e.g. GPU, TPU...). One way to reduce the computation and power cost is to use binary weight NNs, but these are hard to train because the sign function has a non-smooth gradient. We present a model based on Mathematical Morphology (MM), which can binarize ConvNets without losing performance under certain conditions, but these conditions may not be easy to satisfy in real-world scenarios. To solve this, we propose two new approximation methods and develop a robust theoretical framework for ConvNets binarization using MM. We propose as well regularization losses to improve the optimization. We empirically show that our model can learn a complex morphological network, and explore its performance on a classification task.

A foundation for exact binarized morphological neural networks

TL;DR

This work offers a mathematically grounded foundation for binarized neural networks by marrying Binary Morphological Neural Networks (BiMoNN) with Mathematical Morphology (MM). It introduces the Binary Structuring Element (BiSE) neuron and the BiSEL/DenseLUI layers to realize morphologically meaningful, binarizable architectures that extend to grayscale and RGB inputs, along with exact and approximate binarization schemes. A key contribution is a morphological regularization framework that guides networks toward activable, morphology-consistent representations, improving binarized performance on tasks like denoising and MNIST classification. The approach challenges traditional STE-based BWNNs, showing that learning and binarization can be exact or approximately accurate when guided by MM principles, with practical benefits for efficiency and interpretability. Overall, BiMoNN establishes a robust pathway for morphology-aware binarization with demonstrated empirical potential and publicly available code.

Abstract

Training and running deep neural networks (NNs) often demands a lot of computation and energy-intensive specialized hardware (e.g. GPU, TPU...). One way to reduce the computation and power cost is to use binary weight NNs, but these are hard to train because the sign function has a non-smooth gradient. We present a model based on Mathematical Morphology (MM), which can binarize ConvNets without losing performance under certain conditions, but these conditions may not be easy to satisfy in real-world scenarios. To solve this, we propose two new approximation methods and develop a robust theoretical framework for ConvNets binarization using MM. We propose as well regularization losses to improve the optimization. We empirically show that our model can learn a complex morphological network, and explore its performance on a classification task.
Paper Structure (53 sections, 20 theorems, 113 equations, 3 figures, 2 tables)

This paper contains 53 sections, 20 theorems, 113 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. General Binary Morphological Neural Network
  3. Binary Structuring Element neuron
  4. Morphological Equivalence
  5. Binary Morphological Neural Network
  6. DenseLUI
  7. Gray-Scale / RGB Inputs
  8. Training considerations
  9. Positivity
  10. Dual reparametrization
  11. Projected
  12. Projected Reparam
  13. Initialization
  14. Binarization
  15. Exact BiSE Binarization
  16. ...and 38 more sections

Key Result

Theorem 2.3

For a given structuring element (SE) $S \subset \Omega$, and an almost binary parameter $\delta \in ]0, \frac{1}{2}]$, a set of reparametrized weights $\mathbf W \in \mathbb{R}^{\Omega}$ and bias $b \in \mathbb{R}$, we define: Let $\psi \in \{\oplus, \ominus\}$ be a dilation or erosion. Then: In this case, $\forall s \in S, \mathbf W_s \geq 0$ and $b \geq 0$ and we say that a BiSE $\chi$ with we

Figures (3)

  • Figure 1: BiSEL vs Conv Layer. Input $\mathbf x$ with 3 channels. Output $\phi(\mathbf x)$ with 2 channels.
  • Figure 2: Gray to level-set for 5 different values, generating 5 input channels.
  • Figure :

Theorems & Definitions (36)

  • Definition 2.1: BiSE neuron
  • Definition 2.2: Almost Binary Image
  • Theorem 2.3: Dilation - Erosion Equivalence
  • Definition 2.4: BiSEL
  • Proposition 3.1: Linear Check
  • Proposition 3.2
  • Lemma A.1
  • proof
  • Proposition A.2
  • proof
  • ...and 26 more