张量投票：计算机视觉与机器学习中的感知组织方法

计算机视觉

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"Tensor Voting: A Perceptual Organization Approach to Computer Vision and Machine Learning" Tensor投票（Tensor Voting）是一种基于计算机视觉和机器学习的感知组织方法，由Morgan & Claypool出版社出版，作者是Philippos Mordohai和Gérard Medioni。该技术深受心理学中的Gestalt原理启发，Gestalt原理指出人类视觉系统倾向于依据特定规则将图像特征组织成更高级别的结构。这些规则包括邻近性、相似性、平行性、封闭性、对称性、连接性、垂直性和连续性等。 Tensor Voting的核心概念是利用张量数学来处理和分析图像数据，以实现对图像特征的自动组织和理解。张量在这里是一种多维数组，能够表示和存储多种类型的关系和属性。在Tensor Voting中，张量被用来编码空间、方向和其他属性的信息，从而帮助识别图像中的模式和结构。算法的流程大致如下： 1. **特征提取**：首先，从图像中提取诸如边缘、角点等低级特征。 2. **张量构造**：接着，为每个特征构建一个张量，张量的元素包含特征之间的空间、方向和强度关系。 3. **投票过程**：特征之间进行相互投票，投票的结果会更新张量，这个过程反映了Gestalt原则中的邻近性、相似性等规则。 4. **整合信息**：通过整合这些投票结果，可以识别出更高层次的结构，如线、面、物体边界等。 5. **决策与分割**：最终，根据张量的统计特性进行分割或分类决策，完成图像的理解和解析。 Tensor Voting的优势在于其自适应性和鲁棒性，它能够处理噪声、不完整数据和部分遮挡的情况。此外，由于其基于感知原则，它在处理复杂场景时能提供更自然的解释。然而，该方法也存在计算复杂度高和参数调整困难等问题。 Tensor Voting不仅仅局限于计算机视觉领域，还可以应用于机器学习任务，如目标检测、图像分割和场景理解。通过与其他机器学习技术结合，例如支持向量机（SVM）或深度学习模型，可以进一步提升性能。 Tensor Voting是一种强大的工具，它利用心理学理论来增强计算机对视觉信息的处理能力，对于理解和解析复杂图像场景具有重要意义。尽管存在挑战，但随着计算能力的增强和技术的发展，Tensor Voting在计算机视觉和机器学习领域的应用前景依然广阔。

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6 TENSOR VOTING: A PERCEPTUAL ORGANIZATION APPROACH

(a) Input data (b) Inferred curve

FIGURE 1.5: Tensor voting is able to infer the correct intrinsic dimensionality of the data, which is

1D, despite the fact that it appears as 2D if observed globally. The correct perceptual structure, a curve,

is inferred without having to examine potential surface hypotheses.

Additional advantages brought about by the representation and voting schemes are noise

robustness and the ability to employ a least-commitment approach. As shown in numerous

experiments and publications [25, 60, 104], tensor voting is robust to very large percentages of

outliers in the data. This is due to the fact that random outliers cast inconsistent votes, which do

not affect the solution signiﬁcantly. This does not hold when there is a systematic source of errors,

as is thecase in manycomputer vision problems. Examplesof such problems are shown in theap-

propriate chapters. Regarding the avoidance ofprematuredecisions, the capability of the second-

order tensor toencompass saliencies for all structure types allowsus not having to decidewhether

a token is an inlier or an outlier before all the necessary information has been accumulated.

Finally, we believe that a model-free, data-driven design is more appropriate for a general

approach since it is easier to generalize to new domains and more ﬂexible in terms of the types

of solutions it can infer. The main assumption in the algorithms described here is smoothness,

which is a very weak and general model. Moreover, the absence of global computations increases

the amount of data that can be processed since computational and storage complexity scale

linearly with the number of tokens, if their density does not increase.

1.3 OUTLINE

The book is organized in three parts:

•

The original tensor voting framework in 2D and 3D and its application to computer

vision problems.

•

Tensor voting in high dimensions with applications to machine learning problems.

•

First-order voting for boundary inference and its application to ﬁgure completion.

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INTRODUCTION 7

We beginby illustrating thetensor voting framework in 2D and 3D.We show how tokens,

which are simple primitives that potentially belong to perceptual structures, can be represented

by second-order, symmetric, nonnegative deﬁnite tensors. Then, we present the mechanism

for information propagation we have designed. The results of this propagation are structural

saliency values for each potential structure type along with the preferred normal and tangent

orientations at each token. Experimental results demonstrate the effectiveness of our approach,

as well as its robustness to noise.

A large part of our research efforts is devoted to the development of a stereo reconstruction

approach, which is presented in Chapter 3. Stereovision can be cast as a perceptual organization

problem under the premise that solutions must comprise coherent structures. These structures

become salient due to the alignment of potential pixel correspondences reconstructed in a 3D

space. Tensor voting is performed to infer the correct matches that are generated by the true

scene surfaces as inliers of smooth perceptual structures. The retained matches are grouped

into smooth surfaces and inconsistent matches are rejected. Disparity hypotheses for pixels

that remain unmatched are generated based on the color information of nearby surfaces and

validated by ensuring the good continuation of the surfaces via tensor voting. Thus, information

is propagated frommore to less reliablepixels considering both geometric and color information.

Arecent,major enhancement of the frameworkisanefﬁcient N D implementation, which

is described in Chapter 4. We present a new implementation of tensor voting that signiﬁcantly

reduces computational time and storage requirements, especially in high-dimensional spaces,

and thus can be applied to machine learning problems, as well as a variety of new domains.

This work is based on the observation that the Gestalt principles still apply in spaces of higher

dimensionality. The computational and storage requirements of the original implementation

prevented its wide application to problems in high dimensions. This is no longer the case with

the new implementation which opens up an array of potential applications mostly in the ﬁeld

of instance-based learning.

Chapter 5 presents our approach to machine learning problems. We address unsuper-

vised manifold learning from observations in high-dimensional spaces using the new efﬁcient

implementation of tensor voting. We are able to estimate local dimensionality and structure,

measure geodesic distances, and perform nonlinear interpolation. We ﬁrst show that we can

obtain reliable dimensionality estimates at each point. Then, we present a quantitative eval-

uation of our results in the estimation of a local manifold structure using synthetic datasets

with known ground truth. We also present results on datasets with varying dimensionality and

intersections under severe noise corruption, which would have been impossible to process with

current state-of-the-art methods. We also address function approximation from samples, which

is an important problem with many applications in machine learning. We propose a noniter-

ative, local, nonparametric approach that can successfully approximate nonlinear functions in

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CHAPTER 2

TENSOR VOTING

The tensor voting framework is an approach for perceptual organization that is able to infer

salient structures based on thesupport thetokens which comprise them receive from their neigh-

bors. It is based on the Gestalt principles of proximity and good continuation and can tolerate

very large numbers of outliers. Data tokens are represented by tensors and their saliency is com-

puted based on informationpropagatedamong neighboring tokensvia tensorvoting.The tokens

can be any type of local primitive, such as points or surfels, that is localized in space and poten-

tially has orientation preferences associated with it, but no dimensions. The framework has been

developed over the past several years beginning in the work of Guy [24] which was followed by

the work of Lee [49] and Tang [103]. Parts of the work presented here can also be found in [64].

A shortcoming of the original framework was its inability to detect terminations of

the inferred perceptual structures. This has been addressed with the addition of ﬁrst order

information to the framework [112]. To avoid confusion we make the distinction between ﬁrst

and second order information throughout, even though the description of the ﬁrst additions

comes a later in Chapter 6. We begin by brieﬂy going over other perceptual organization

approaches and proceed to describe the original, second order formulation of tensor voting in

2-D and 3-D.

2.1 RELATED WORK

Perceptual organization has been an active research area since the beginning of the previous

century based on the work of the Gestalt psychologists [42, 43, 118]. Important issues include

noise robustness, initialization requirements, handling of discontinuities, ﬂexibility in the types

that can be represented, and computational complexity. This section reviews related work which

can be classiﬁed in the following categories. More detailed descriptions can be found in [60, 64]

where work on perceptual organization based on regularization, relaxation labeling, level set

methods, clustering and robust estimation is also presented.

•

symbolic methods

•

methods based on local interactions

•

methods inspired by psychophysiology and neuroscience.

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10 TENSOR VOTING: A PERCEPTUAL ORGANIZATION APPROACH

Symbolic Methods. Following the paradigm set by Marr [58], many researchers developed

methods for hierarchical grouping of symbolic data. Lowe [56] developed a system for 3-D

object recognition based on perceptual organization of image edgels. Groupings are selected

among the numerous possibilities according to the Gestalt principles, viewpoint invariance and

low likelihood of being accidental formations. Later, Mohan and Nevatia [63] and Dolan and

Riseman [17] also proposed perceptual organization approaches based on the Gestalt principles.

Both are symbolic and operate in a hierarchical bottom-up fashion starting from edgels and

increasing the level of abstraction at each iteration. The latter approach aims at inferring curvi-

linear structures, while the former aims at segmentation and extraction of3-D scene descriptions

from collations of features that have high likelihood of being projections of scene objects. Along

the same lines is Jacobs’ [34] technique for inferring salient convex groups among clutter since

they most likely correspond to world objects. The criteria to determine the non-accidentalness

of the potential structures are convexity, proximity and the contrast of the edgels.

Methods Based on Local Interactions. Shashua and Ullman [99] ﬁrst addressed the issue of struc-

tural saliency and how prominent curves are formed from tokens that are not salient in isolation.

They deﬁne a locally connected network that assigns a saliency value to every image location

according to the length and smoothness of curvature of curves going through that location. In

[79], Parent and Zucker infer trace points and their curvature based on spatial integration of

local information. An important aspect of this method is its robustness to noise. This work was

extended to surface inference in three dimensions by Sander and Zucker [86]. Sarkar and Boyer

[89] employ a voting scheme to detect a hierarchy of tokens. Voting in parameter space has to

be performed separately for each type of structure, thus making the computational complexity

prohibitive for generalization to 3-D. The inability of previous techniques to simultaneously

handle surfaces, curves and junctions was addressed in the precursor of our research, the work

of Guy and Medioni [25, 26]. A uniﬁed framework where all types of perceptual structures

can be represented is proposed along with a preliminary version of the voting scheme presented

here. The major advantages of [25, 26] are noise robustness and computational efﬁciency, since

it is not iterative. How this methodology evolved is presented in the remaining sections of this

chapter.

Methods Inspired by Psychophysiology and Neuroscience. Finally, there is an important class of

perceptual organization methods that are inspired by human perception and research in psy-

chophysiology and neuroscience. Grossberg and Mingolla [22] and Grossberg and Todor-

ovic [23] developed the Boundary Contour System and the Feature Contour System that can

group fragmented and even illusory edges to form closed boundaries and regions by feature

cooperation in a neural network. Heitger and von der Heydt [29], in a classic paper on neural

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张量投票：计算机视觉与机器学习中的感知组织方法

Tensor Voting Framework：2D medioni张量框架的实现-matlab开发

tensor voting理论详解（含有伪代码实现）

tensor voting的 matlab实现

tensor 和longtensor

sparsetensor转tensor

将tensor变成tensor元组

tensor数组转tensor

tensor变量和longtensor变量

如何判断tensor是空tensor

tensor_input =torch.cat([tensor1, tensor2, tensor3], dim=0) RuntimeError: zero-dimensional tensor (at position 0) cannot be concatenated

两个tensor合成一个tensor

torch.tensor 转 tensorflow tensor

torch中的tensor与tensor

tensorflow float tensor 转double tensor

Tensor input, Tensor weight

将list转换为LongTensor类型的tensor

can't convert CUDA tensor to numpy. Use Tensor.cpu() to copy the tensor to host memory first.

tensor数据类型转换

torch::Tensor 转为 caffe2::Tensor

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