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盲源分离 外文翻译

盲源分离 外文翻译
盲源分离 外文翻译

英文原文

An information-maximisation approach to

blind separation and blind deconvolution

Anthony J. Bell and Terrence J. Sejnowski

Computational Neurobiology Laboratory

The Salk Institute

100010 N.Torrey Pines Road

La Jolla, California 92037

Abstract

We derive a new self-organising learning algorithm which maximises the infor- mation transferred in a network of non-linear units. The algorithm does not assume any knowledge of the input distributions, and is defined here for the zero-noise limit. Under these conditions, information maximisation has extra properties not found in the linear case(Linsker 1989). The non-linearities in the transfer function are able to pick up higher-order moments of the input distributions and perform something akin to true redundancy reduction between units in the output representation. This enable- s the network to separate statistically independent components in the inputs: a higher -order generalisation of Principal Components Analysis.

We apply the network to the source separation (or cocktail party)problem, succe- ssfully separating unknown mixtures of up to ten speakers. We also show that a vari- ant on the network architecture is able to perform blind deconvolution (cancellation of unknown echoes and reverberation in a speech signal). Finally, we derive depend- encies of information transfer on time delays. We suggest that information maximi- sation provides a unifying framework for problems in …blind? signal processing.

1 Introduction

This paper presents a convergence of two lines of research. The first, the developme- nt of information theoretic unsupervised learning rules for neural networks has been pioneered by Linsker 1992, Becker & Hinton 1992, Atick & Redlich 1993, Plumble- y & Fallside 1988 and others. The second is the use,in signal processing, of higher-order statistics, for separating out mixtures of independent sources (blind sep- aration) or reversing the effect of an unknown filter (blind deconvolution). Methods exist for solving these problems, but it is fair to say that many of them are ad hoc. The literature displays a diversity of approaches and justifications—for historical re- views see (Comon 1994) and (Haykin 1994a).

In this paper, we supply a common theoretical framework for these problems through the use of information-theoretic objective functions applied to neural netwo- rks with non-linear units. The resulting learning rules have enabled a principled app- roach to the signal processing problems, and opened a new application area for info- rmation theoretic unsupervised learning.

Blind separation techniques can be used in any domain where an array of N r- eceivers picks up linear mixtures of N source signals. Examples include speech sep- aration (the …cocktail party problem?), processing of arrays of r adar or sonar signals, and processing of multi-sensor biomedical recordings. A previous approach has been implemented in analog VLSI circuitry for real-time source separation (Vittoz et al 1989, Cohen et al 1992). The application areas of blind deconvolution techniques in- clude the cancellation of acoustic reverberations (for example the …barrel effect? obs- erved when using speaker phones),the processing of geophysical data (seismic deco- nvolution) and the restorationof images.

The approach we take to th ese problems is a generalisation of Linsker?s infomax principle to non-linear units with arbitrarily distributed inputs uncorrupted by any k- nown noise sources. The principle is that described by

Laughlin (1981)(see Fig.1a): when inputs are to be passed through a sigmoid function, maximum information tra- nsmission can be achieved when the sloping part of the sigmoid is optimally lined up with the high density parts of the inputs. As we show, this can be achieved in an adaptive manner, using a stochastic gradient ascent rule. The generalisation of this r- ule to multiple units leads to a system which, in maximising information transfer, al- so reduces the redundancy between the units in the output layer. It is this latter proc- ess, also called

Independent Component Analysis (ICA), which enables the network to solve the blind separation task.

The paper is organised as follows. Section 2 describes the new information ma- ximisation learning algorithm, applied, respectively to a single input, an N N → mapping, a causal filter, a system with time delays and a …flexible? non-linearity. Section 3 describes the blind separation and blind deconvolution problems. Section 4 dis- cusses the conditions under which the information

maximisation process can find factorial codes (perform ICA),and therefore solve the separation and deconvo- lution problems. Section 5 presents results on the separation and deconvolution of speech signals. Section 6 attempts to place the theory and results in the context of previous work and mentions the limitations of the approach.

A brief report of this research appears in Bell & Sejnowski (1995)

2 Information maximisation

The basic problem tackled here is how to maximise the mutual information that the output Y of a neural network processor contains about its input X..

This is defined as:

(,)()()I Y X H Y H Y X =-∣ (1)

where ()H Y is the entropy of the output, while ()H Y X ∣ is whatever entropy the output has w hich didn?t come from the input. In the case that we have no noise (or r- ather, we don?t know what is noise and what is signal in the input),the mapping bet- ween X and Y is deterministic and ()H Y X ∣ has its

lowest possible value: it diverges to -∞. This divergence is one of the

consequences of the generalisation of infor- mation theory to continuous variables. What we call ()H Y is really the …differenti - al? entropy of Y with respect to some reference,such as the noise-level or the accura- cy of our discretisation of the variables in X and 1Y - .To avoid su- ch complexities, we consider here only the gradient of information theoretic quantities with respect to some parameter, w, in our network. Such gradients are as well-behaved as

discrete-variable entropies,because the reference terms involved in the definition of differential entropies disappear. The above equation can be differe- ntiated as follows, with respect to a parameter, w, involved in the mapping from X to Y :

(,)()I Y X H Y w w

??=?? (2) because ()H Y X ∣ does not depend on w . This can be seen by considering a system which avoids infinities: ()Y G X N =+, where G is some invertible transformatio- n and N is additive noise on the outputs. In this

case,()()H Y X H N ∣=( Nadal & Parga 1995). Whatever the level of this additive noise, maximisation of the mutual i- nformation, (,)I Y X , is equivalent to the maximisation of the output entropy, ()H Y , because ()()0w H N ??=.There is nothing mysterious about the determinis- tic case, despite the fact that ()H Y X ∣ tends to minus infinity as the noise variance goes to zero.

Thus for invertible continuous deterministic mappings, the mutual

information between inputs and outputs can be maximised by maximising the entropy of the out- puts alone.

2.1 For one input and one output

When we pass a single input x through a transforming function ()g x to give an output variable y , both (,)I y x and ()H Y are maximised when we align high density parts of the probability density function (pdf) of x with highly sloping parts of the function ()g x . This is the idea of “matching a

neuron?s input-output function to the expected distribution of signals” that we find in (Laughlin 1981) .

When ()g x is monotonically increasing or decreasing (ie: has a unique inver- se). the pdf of the output, ()y f y , can be written as a function of the pdf of the input, ()x f x .

()

()x y f x f y y x =∣??∣ (3)

where the bars denote absolute value. The entropy of the output, ()H Y ,is given by:

()[ln ()]()ln ()y y y H y E f y f y f y dy ∞

-∞=-=-? (4) where []E denotes expected value. Substituting (3) into (4) gives

[]()ln ln ()x y H y E E f x x ???=-?????

(5) The second term on the right (the entropy of x) may be considered to be unaffected by alterations in a parameter w determining g(x).Therefore in order to maximise the entropy of y by changing w , we need only concentrate on

maximising the first te- rm, which is the average log of how the input affects the output. This can be done by considering the …training set?of 'x s to approximate the density ()x f x , and derivin- g an …online? stochastic gradient ascent learning rule:

1

ln H y y y w w w x x w x -?????????????∝== ? ? ????????????? (6) In the case of the logistic transfer function:

11u y e

-=+, 0u wx w =+ (7) in which the input is multiplied by a weight w and added to a bias-weight

0w ,the terms above evaluate as:

(1)y wy y x

?=-? (8) (1)(1(12))y y y wx y w x ????=-+- ?????

(9) Dividing (9) by (8) gives the learning rule for the logistic function, as calculated fr- om the general rule of (6):

1(12)w x y w

?∝+- (10) Similar reasoning leads to the rule for the bias-weight:

012w y ?∝- (11)

The effect of these two rules can be seen in https://www.doczj.com/doc/bc4805258.html,. For example, if the input pdf ()x f x were gaussian, then the 0w ?-rule would centre the steepest part of the sigm- oid curve on the peak of ()x f x , matching input density to output slope, in a manner suggested intuitively by (3).The w ?-rule would then scale the slope of the sigmoid curve to match the variance of ()x f x . For example, narrow pdf?s would lead to sharply-sloping sigmoids. The w ?-rule is anti-Hebbian 2, with an anti-decay term. The anti-Hebbian term keeps y away from one uninformative situation: that of y being saturated at 0 or 1. But an anti-Hebbian rule alone makes weights go to zero, so the anti-decay term (1w ) keeps y away from the other uninformative situation: when w is so small that y stays around 0.5.

The effect of these two balanced forces is to produce an output pdf,

()y f y ,that is cl- ose to the flat unit distribution, which is the maximum entropy distribution for a var- iable bounded between 0 and 1.Fig.1b shows a family of these distributions, with the most informative one occurring at opt w .

A rule which maximises information for one input and one output may be suggestive for structures such as synapses and photoreceptors which must

position the gain of their non-linearity at a level appropriate to the average value and size of the input fl- uctuations (Laughlin 1981).However, to see the

advantages of this approach in artif- icial neural networks, we now analyse the case of multi-dimensional inputs and out- puts.

2.2 For an N N → network

Consider a network with an input vector x , a weight matrix W , a bias vector 0w and a monotonically transformed output vector 0()y g Wx w =+. Analogously to (3) ,the multivariate probability density function of y can be written:

()()x y f x f y J

= (12) where J is the absolute value of the Jacobian of the transformation. The Jacobian is the determinant of the matrix of partial derivatives:

1111det n n n n y y x x J y y x x ???? ??????= ???????? ??????

(13) The derivation proceeds as in the previous section except instead of maximizing ln y x ??,now we maximise ln J 。This latter quantity represents

the log of the volume of space in y into which points in x are mapped 。

中文翻译

一种解决盲分离和盲信号去卷积的信息最大化方

安东尼 J. 贝尔和 Terrence J. Sejnowski

计算的神经生物学实验室

索尔客学院

我们获得了新的自我学习算法-尽量增大在非线的单位网络中信息转化。该算法不使用任何输入分配信号,同时定义是在零噪音条件限制下。在这些情况之下,数据最大值化有在线性的情形下没有发现的其他的性质(Linsker 1989)。在转变过程中的非线性,能够在输出表现上获得高次序瞬间的输入分配和执行一些对真实的多余减少的同类型方法。这使网络能够在输入中以统计方法来分开独立的成份: 主要成份分析的高低次序一般化。

我们应用网络来处理源分离(或酒会)问题,成功地分开了多达十个人的话语。我们也说明在网络体系中的变量是能够做盲解卷的(对一个声音信号不知道卷积方式的反过程)。最后,我们得到信息转化在时延上的属性。我们表明数据最大值化提供了一个‘盲'信号处理问题的统一的架构。

1 介绍

这篇论文介绍了研究的二条路线的集中。第一个,神经网络在信息理论上的无人监督学习规则的发展已经被Linsker 1992、Becker & Hinton 1992 、Atick & Redlich 1993、Plumbley & Fallside 1988以及另外一些人所提出。其次是在信号的处理方面,使用高次序统计学,为了分开独立信源的混合体(盲分离)或者为了反向转化未知过滤后的结果(盲解卷)。存在解决这些问题的方法,但是公平的说他们中的大多数都是特例。文献显示了对历史的评论调查(Comon 1994)和(Haykin 1994a)的方法和辩论的不同。

在这篇论文里,通过使用被应用于带有非线性单位的神经网络的信息理论目标函数,我们为这些问题提供了一个公有理论框架。学习规则已经产生出一个逼近算法来解决信号处理问题,而且为信息理论上的无人监督方法打开一个新的应用领域。

盲分离技术可以应用于用一组N个接收器获得N个信号源线性混合的任何领域。例子包括讲话分离('鸡尾酒会问题'),一组雷达信号或声纳信号的处理, 和多感应生物医学的录音处理。早期的方法已经可以用类似VLSI电路来实现实时信源分离.(Vittoz等1989,科恩等1992)盲解卷技术的应用区域包括声音反响的消除,(例如,当使用话筒时观察到的听筒影响),地球物理学数据的处理(地震波的解卷)和图象的修复。

我们解决这些问题的方法是Linsker的infomax原则通过任何已知的噪音来源对未分离的任意分配输入的非线性单位的一般化。Laughlin(1981)是这样描述这个原则的。当输入数据通过一个S字形函数时,在S字形的倾斜部份和输入数据高密度部分最佳地排成一行时信息最大化传输可以完成。正如我们所述,通过使用随机梯度上升规则,这可以用适应方式实现。这一规则对多个单位的一般化产生一个在数据转化最大化方面起到减少输出层单位之间的多余部分的系统。这个最新的算法,即独立的成份分析(ICA),他能使网络能够解决盲分离工作。

论文是由以下各项组成。第2部分描述新的信息最大值化学习算法、应用、分别到单个输入、N N

映射、一个因果滤波器、一个有时间延迟的系统和一'灵活的'非线性。第3部分描述盲分离和盲解卷问题。第4部分讨论在什么情况下数据最大值化过程能找到阶乘编码(运行ICA),因此来解决分离和解卷问题。第5部分介绍在分离和解卷语音信号方面的结果。第6部分尝试把理论和结果用于早期相关工作并且提到了方法的局限性。Bell 和Sejnowski在1995年发表了这个研究的简短报告。

2 信息最大值化

这里要处理的基本问题是该如何使互信息最大化,互信息即是类神经网路处理器的输出Y中包含输入X。

这被定义为:

(,)()()I Y X H Y H Y X =-∣ (1)

()H Y 是输出结果的熵,而()H Y X ∣是输出结果中没有来自输入信号部分的熵。在没有噪声(或者我们不知道在输入中什么是噪音什么是信息)的情况下,X 和Y 之间的映射是确定的以及()H Y X ∣到最低的可能数值:它到无穷小。这种分歧是对连续变量的信息论的一般化结果之一。我们所称的()H Y 只是Y 的‘微分'熵函数,Y 涉及到一些叁考值,像是噪声级别或我们在X 和Y 中的变量的离散化的准确性。为了避免这样的麻烦,在我们的网络中,我们只考虑有关于一些叁数的信息理论上的量的梯度—W 。这类梯度象离散变量熵函数一样正式,因为有关于微分熵的定义的叁考术语消失。上述等式可以依照下列各项来区别,使用有关变量—W ,W 与X 到Y 的映射有关。

(,)()I Y X H Y w w

??=?? (2) 因为()H Y X ∣并不依赖w 。这可以从考虑避免无限大的系统见出:()Y G X N =+,这里的G 是反向转化的函数和N 是输出信号的附加噪声。在这种情况下,()()H Y X H N ∣=( Nadal 和Parga 1995)。这种附加噪声的任何级别,互信息—(,)I Y X 的最大值与输出熵—()H Y 的最大值相等,因为()()0w H N ??=。尽管当噪声趋于零()H Y X ∣趋于负无穷大的事实,关于上面确定的事情也没有任何神秘的东西。如此对于连续确定性映射的逆转,在输入和输出之间的互信息值可以通过单个输出值的最大熵来最大化。

2.1 对于输入和输出

当我们把一个单独输入信号X 通过转化函数()g x ,得到输出变量y .当我们把x 的概率密度函数的高密度部分和函数()g x 非常倾斜部分结合起来时,(,)I y x 和()H Y 有最大值。这是把神经原的输入到输出的转化函数与信号的预期分配相结合的思想,我们可以在(Laughlin 1981)中找到。

当()g x 单调地增加或减少时,输出的信号的概率密度函数()y f y 可以当作输入信号概率密度函数()x f x 的函数;

()

()x y f x f y y x =∣??∣ (3)

这里竖条表示绝对值。输出值的熵函数()H Y 由下式表示:

()[ln ()]()ln ()y y y H y E f y f y f y dy ∞

-∞=-=-? (4) 这里[]E 指示数学期望。用公式(3) 代入公式(4)中得到

[]()ln ln ()x y H y E E f x x ???=-?????

(5) 等式右边的第二个式子(x 的熵函数)可以认为不会因为决定()g x 的叁数w 的改变而受到影响。因此为了通过改变w 来使y 的熵达到最大值,因此我们只需要使第一个式子值最大,这个式子反映输入影响输出的平均值。这可以通过认为x 的训练规则与密度()x f x 近似来完成,而且源自一随机倾度上升学问规则:

1

ln H y y y w w w x x w x -?????????????∝== ? ? ????????????? (6) 在实现逻辑转换功能时:

11u y e

-=+,0u wx w =+ (7) 在式子中输入x 与分量w 相乘,然后与次分量0w 相加上面的式子可等

价于:

(1)y wy y x

?=-? (8) (1)(1(12))y y y wx y w x ????=-+- ?????

(9) 当计算公式6的一般规则时,通过公式8(把学习规则和逻辑函数互换)来拆开公式9:

1(12)w x y w ?∝

+- (10) 类似方法得到次分量的式子: 012w y ?∝- (11)

例如,如果输入信号的概率密度函数()x f x 是高斯函数,然后0w ?式子会在()x f x 的最高处集中S 字形曲线的最陡峭部份,使输入信号概率密度与输出信号斜率相等,用由公式3直接推导出的方式。然后0w ?式子会度量S 字形曲线的倾斜度配合()x f x 的变化。例如,狭窄的概率密度函数会产生尖

锐倾斜的S 字形。0w ?规则是一个带无衰减anti-Hebbian 算法。Anti-Hebbian 算法使y 脱离一个不知道任何输入的环境:y 一定是0或1。但是当anti-Hebbian 算法将单位变成零,因此无衰减算法1w 使y 远离另一个不知道任何输入的环境:当w 足够小时,y 大约为0.5。

这两个均衡规则的作用是导出输出值的概率密度函数()x f x ,这是接近

最佳的单元分配,()x f x 是分布于0和1之间的变量的最大熵。图.1b 说明了

这些在opt w 处含最多信息的分量的整体。用于单输入单输出系统的信息最大化法则可能提示了一些象神经键和光感应器之类的结构,这类结构能够确定他们近似于输入波形平均大小和尺寸的曲线增益位置(Laughlin 1981)。然而正是看到这种算法在人工神经网络的优点,我们现在开始分析多变量的输入和输出。

2.2对于一个N N →网络

考虑一个输入矢量为x 、分量矩阵为W 、偏量矢量为0w 和单调转化输

出矢量0()y g Wx w =+的网络,与公式3相似,y 的多变量概率密度函数可以写为:

()()x y f x f y J

= (12) 这里J 是转换矩阵的雅克比行列式的绝对值,雅克比行列式是由分量的导数组成的矩阵的行列式。

1111det n n n n y y x x J y y x x ???? ??????= ???????? ??????

(13) 由上式引出,我们现在使ln J 最大化,而不是使ln y x ??最大化。ln J 的大小代表着指向被映射x 的y 的空间量。通过使lnJ 最大化,我们试图扩展我们的x 均匀指向y 的训练规则。对于g 是逻辑函数的E 皱型单位,()y g u =,0u Wx w =+。

图像处理中值滤波器中英文对照外文翻译文献

中英文资料对照外文翻译 一、英文原文 A NEW CONTENT BASED MEDIAN FILTER ABSTRACT In this paper the hardware implementation of a contentbased median filter suitabl e for real-time impulse noise suppression is presented. The function of the proposed ci rcuitry is adaptive; it detects the existence of impulse noise in an image neighborhood and applies the median filter operator only when necessary. In this way, the blurring o f the imagein process is avoided and the integrity of edge and detail information is pre served. The proposed digital hardware structure is capable of processing gray-scale im ages of 8-bit resolution and is fully pipelined, whereas parallel processing is used to m inimize computational time. The architecturepresented was implemented in FPGA an d it can be used in industrial imaging applications, where fast processing is of the utm ost importance. The typical system clock frequency is 55 MHz. 1. INTRODUCTION Two applications of great importance in the area of image processing are noise filtering and image enhancement [1].These tasks are an essential part of any image pro cessor,whether the final image is utilized for visual interpretation or for automatic an alysis. The aim of noise filtering is to eliminate noise and its effects on the original im age, while corrupting the image as little as possible. To this end, nonlinear techniques (like the median and, in general, order statistics filters) have been found to provide mo re satisfactory results in comparison to linear methods. Impulse noise exists in many p ractical applications and can be generated by various sources, including a number of man made phenomena, such as unprotected switches, industrial machines and car ign ition systems. Images are often corrupted by impulse noise due to a noisy sensor or ch annel transmission errors. The most common method used for impulse noise suppressi on n forgray-scale and color images is the median filter (MF) [2].The basic drawback o f the application of the MF is the blurringof the image in process. In the general case,t he filter is applied uniformly across an image, modifying pixels that arenot contamina ted by noise. In this way, the effective elimination of impulse noise is often at the exp ense of an overalldegradation of the image and blurred or distorted features[3].In this paper an intelligent hardware structure of a content based median filter (CBMF) suita ble for impulse noise suppression is presented. The function of the proposed circuit is to detect the existence of noise in the image window and apply the corresponding MF

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forced concrete structure reinforced with an overviewRein Since the reform and opening up, with the national economy's rapid and sustained development of a reinforced concrete structure built, reinforced with the development of technology has been great. Therefore, to promote the use of advanced technology reinforced connecting to improve project quality and speed up the pace of construction, improve labor productivity, reduce costs, and is of great significance. Reinforced steel bars connecting technologies can be divided into two broad categories linking welding machinery and steel. There are six types of welding steel welding methods, and some apply to the prefabricated plant, and some apply to the construction site, some of both apply. There are three types of machinery commonly used reinforcement linking method primarily applicable to the construction site. Ways has its own characteristics and different application, and in the continuous development and improvement. In actual production, should be based on specific conditions of work, working environment and technical requirements, the choice of suitable methods to achieve the best overall efficiency. 1、steel mechanical link 1.1 radial squeeze link Will be a steel sleeve in two sets to the highly-reinforced Department with superhigh pressure hydraulic equipment (squeeze tongs) along steel sleeve radial squeeze steel casing, in squeezing out tongs squeeze pressure role of a steel sleeve plasticity deformation closely integrated with reinforced through reinforced steel sleeve and Wang Liang's Position will be two solid steel bars linked Characteristic: Connect intensity to be high, performance reliable, can bear high stress draw and pigeonhole the load and tired load repeatedly.

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