An accurate detection of the centerlines of linear objects is a challenging
topic in many sensitive real-world applications such X-ray imaging, remote
sensing and lane marking detection in road traffic. Model-based approaches
using Hough and Radon transforms are often used but, are not recommended for
thick line detection, whereas approaches based on image derivatives need
further step-by-step processing, making their efficiency dependent on each step
outcomes. In this paper, we aim to detect linear structures found in images by
considering the 3D representation of the image gray levels as a finite mixture
model of statistical distribution. The latter, which we named linear anchored
Gaussian distribution could be parametrized by a scale value {\sigma}
describing the linear structure thickness and a line equation, parametrized, in
turn, by a radius \r{ho} and an orientation angle {\theta}, describing the
linear structure centerline location. Expectation-Maximization (EM) algorithm
is used for the mixture model parameter estimation, where a new paradigm, using
the background subtraction for the likelihood function computation, is
proposed. For the EM algorithm, two {\theta} parameter initialization schemes
are used: the first one is based on a random choice of the first component of
{\theta} vector, whereas the second is based on the image Hessian with a
simultaneous computation of the mixture model components number. Experiments on
real world images and synthetic images corrupted by blur and additive noise
show the good performance of the proposed methods, where the algorithm using
background subtraction and Hessian-based {\theta} initialization provides an
outstanding accuracy of the linear structure detection despite irregular image
background and presence of blur and noise.

通过考虑图像灰度的三维表示作为统计分布的有限混合模型，本文旨在检测图像中的线性结构。采用背景减法和基于 Hessian 的参数初始化方案的期望最大化算法，提供了对线性结构的准确性检测，尽管图像背景不规则且存在模糊和噪声。

线性锚定高斯混合模型用于粗线形状物体的位置和宽度计算

Linear Anchored Gaussian Mixture Model for Location and Width  Computation of Objects in Thick Line Shape

In this paper, we propose a novel model for high-dimensional data, called the
Hybrid Orthogonal Projection and Estimation (HOPE) model, which combines a
linear orthogonal projection and a finite mixture model under a unified
generative modeling framework. The HOPE model itself can be learned
unsupervised from unlabelled data based on the maximum likelihood estimation as
well as discriminatively from labelled data. More interestingly, we have shown
the proposed HOPE models are closely related to neural networks (NNs) in a
sense that each hidden layer can be reformulated as a HOPE model. As a result,
the HOPE framework can be used as a novel tool to probe why and how NNs work,
more importantly, to learn NNs in either supervised or unsupervised ways. In
this work, we have investigated the HOPE framework to learn NNs for several
standard tasks, including image recognition on MNIST and speech recognition on
TIMIT. Experimental results have shown that the HOPE framework yields
significant performance gains over the current state-of-the-art methods in
various types of NN learning problems, including unsupervised feature learning,
supervised or semi-supervised learning.

该论文提出了一种新的高维数据模型 —— 混合正交投影估计（HOPE）模型，它将线性正交投影和有限混合模型结合到一个统一的生成建模框架中，并展示了该模型与神经网络密切相关，可用作理解和学习神经网络的新工具，在标准任务上取得了显著的性能优势。