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mnn模型量化剪枝蒸馏英文回答：Quantization, pruning, and distillation are three commonly used techniques in model compression to reduce the size and improve the efficiency of deep neural networks. In this answer, I will explain the process of quantization, pruning, and distillation in the context of MNN model compression.Quantization is the process of reducing the precision of weights and activations in a neural network. By quantizing the model, we can represent the weights and activations using fewer bits, which leads to a smaller model size and faster inference time. For example, instead of using 32-bit floating-point numbers, we can use 8-bit integers to represent the weights and activations. This reduces the memory footprint and allows for more efficient computation on hardware with limited resources.Pruning, on the other hand, involves removing unnecessary connections or neurons from a neural network. The idea behind pruning is that not all connections or neurons contribute equally to the network's performance. By removing the less important connections or neurons, we can reduce the model size and improve the inference speed without sacrificing much accuracy. Pruning can be done based on various criteria, such as weight magnitude or activation importance. For example, we can prune connections with small weights or neurons that have low activation values.Distillation is a technique that involves training a smaller "student" network to mimic the behavior of a larger "teacher" network. The teacher network is usually a larger and more accurate model, while the student network is a smaller and less accurate model. The student network is trained to match the output probabilities of the teacher network, using a combination of the teacher's soft targets and the ground truth labels. The idea behind distillation is that the student network can learn from the teacher network's knowledge and generalize better than if it wastrained from scratch. This allows us to compress the knowledge of the larger model into a smaller model without sacrificing much accuracy.To illustrate the process of quantization, pruning, and distillation, let's consider the example of compressing a large image classification model.First, we can start by quantizing the weights and activations of the model. For instance, we can convert the 32-bit floating-point weights to 8-bit integers. This reduces the model size and allows for faster inference on hardware with limited resources.Next, we can apply pruning to remove unnecessary connections or neurons from the model. For example, we can prune connections with small weights or neurons that have low activation values. This further reduces the model size and improves the inference speed.Finally, we can use distillation to train a smaller student network to mimic the behavior of the larger teachernetwork. The student network is trained to match the output probabilities of the teacher network, using a combination of the teacher's soft targets and the ground truth labels. This allows us to compress the knowledge of the larger model into a smaller model without sacrificing much accuracy.中文回答：量化、剪枝和蒸馏是模型压缩中常用的三种技术，用于减小深度神经网络的大小并提高效率。
面向小样本学习的轻量化知识蒸馏面向小样本学习的轻量化知识蒸馏随着深度学习的迅猛发展，人工智能技术在多个领域展现出了巨大的潜力。
然而，深度神经网络需要大量的标记样本进行训练，以获得良好的性能。
这在许多实际应用中，如医疗诊断和工业控制等领域，往往是一个挑战。
因此，如何在小样本学习中取得良好的效果成为了一个研究热点。
在小样本学习中，一个常见的方法是使用迁移学习来利用已有的知识。
知识蒸馏（knowledge distillation）作为一种迁移学习的方法，可以帮助将复杂模型的知识转移到简化的模型上，以便在小样本学习任务中发挥作用。
知识蒸馏的基本思想是将复杂模型（教师模型）的知识转化为简化模型（学生模型）可以理解的形式。
这样，学生模型在学习过程中可以依靠教师模型的知识来辅助训练，从而在小样本学习任务中获得更好的性能。
知识蒸馏方法通常通过两个步骤来实现：首先，使用教师模型对大规模数据进行训练，得到教师模型的预测结果和中间层特征；然后，使用这些预测结果和特征来训练学生模型。
通过这种方式，学生模型可以从教师模型的丰富知识中受益，提高在小样本学习任务中的性能。
然而，传统的知识蒸馏方法存在一个问题，就是学生模型往往比教师模型更大，带来了额外的计算和存储开销。
为了解决这个问题，研究者们提出了一种轻量化知识蒸馏的方法，即将教师模型的复杂信息进行简化和压缩，以满足学生模型的轻量化需求。
轻量化知识蒸馏的主要思想是通过模型的压缩和简化，减少参数和计算量，从而在小样本学习任务中保持高性能。
具体来说，有以下几个步骤：首先，对教师和学生模型进行结构压缩。
通常情况下，教师模型拥有较多的参数和层数，而学生模型需要更小更轻量化。
因此，可以通过剪枝、裁剪或者网络结构优化等方法，减少教师模型的参数和层数，使其适应学生模型的轻量化需求。
其次，对教师模型的知识进行压缩。
教师模型中的知识既包括预测结果，也包括中间层特征。
对于预测结果，可以使用软标签来替代硬标签，软标签是一种概率分布形式，可以提供更加丰富的信息。
迁移学习中的无监督迁移和半监督迁移方法研究迁移学习是一种通过将已学习的知识应用于新任务中的机器学习方法。
在实际应用中，由于数据的不完整性和不平衡性，以及标签的稀缺性等问题，传统的监督学习方法往往难以取得理想的效果。
为了解决这些问题，研究者们提出了无监督迁移和半监督迁移方法。
本文将对这两种方法进行深入研究。
无监督迁移是指在源领域和目标领域之间没有标签信息的情况下进行知识迁移。
无监督迁移通过挖掘源领域和目标领域之间的相似性来实现知识传递。
最常用的无监督迁移方法之一是领域自适应。
领域自适应通过对源领域和目标领域进行特征空间上的映射，使得两个领域能够在特征空间上具有相似性。
常见的映射方法有最大均值差异（MMD）和核最大均值差异（KMMD）等。
另一种常见的无监督迁移方法是主成分分析（PCA）和独立成分分析（ICA）。
PCA通过对源领域和目标领域的数据进行降维，找到它们之间的共享特征，从而实现知识迁移。
ICA则通过对源领域和目标领域进行独立分析，找到它们之间的独立特征，并进行知识迁移。
除了无监督迁移方法外，半监督迁移方法也是一种常用的迁移学习方法。
半监督迁移是指在源领域和目标领域之间只有部分标签信息的情况下进行知识迁移。
半监督迁移通过利用已有的标签信息和未标签信息来实现知识传递。
最常用的半监督迁移方法之一是自训练（self-training）。
自训练通过使用已有的标签信息来训练模型，并利用模型对未标签数据进行预测，从而获得更多的标签信息。
另一种常见的半监督迁移方法是共享分布式表示学习（SDRL）。
SDRL通过对源领域和目标领域进行表示学习，将它们映射到一个共享特征空间中，并利用已有的标签信息来训练模型。
在这个共享特征空间中，源领域和目标领域之间的相似性得到了保留，从而实现了知识迁移。
除了以上介绍的无监督迁移和半监督迁移方法，还有许多其他的方法被用于解决迁移学习问题。
例如，基于图的迁移学习方法通过构建源领域和目标领域之间的图结构来实现知识传递。
embedding的数学推理在数学领域中，embedding是一种常见的概念。
它指的是将一个数学结构嵌入到另一个更大或更复杂的数学结构中的过程。
通过embedding，我们可以将一个抽象或较简单的数学对象表示为更具体或更复杂的数学对象的一部分。
在本文中，我们将探讨embedding在数学推理中的应用和意义。
首先，我们来了解一下embedding的基本概念。
在数学中，embedding通常是通过将一个结构映射到另一个结构的方式实现的。
比如，我们可以将一个整数嵌入到一个实数集合中，或者将一个图形嵌入到一个三维空间中。
通过这种映射，我们可以保留原始结构的一些特性，并在更大或更复杂的结构中进行推理和分析。
其次，让我们来看一下embedding在数学推理中的应用。
首先，embedding可以帮助我们理解和解决复杂的数学问题。
通过将问题中的抽象对象嵌入到一个更具体的数学结构中，我们可以更好地理解问题的性质和规律，并从中寻找解决方案。
例如，在代数学中，通过将一个抽象的向量空间嵌入到一个更具体的欧几里德空间中，我们可以更容易地推导出向量的性质和操作规则。
此外，embedding还可以帮助我们建立数学对象之间的关联和联系。
通过将一个数学对象嵌入到另一个对象中，我们可以将它们之间的关系转化为更容易理解和处理的问题。
例如，在图论中，通过将一个图嵌入到一个更大的图中，我们可以将图的结构性质转化为更大图的性质，从而更好地研究和解决问题。
最后，embedding在数学推理中的使用要注意一些技巧和限制。
首先，我们要确保嵌入的过程是保持结构特性的，即在嵌入后，原始结构的一些重要性质仍然得以保留。
其次，我们要注意嵌入的精确性，以确保嵌入后的结构能够准确地代表原始结构。
同时，我们还需要注意嵌入的操作是否可逆，即能否从嵌入后的结构中恢复出原始结构。
总而言之，embedding是一种在数学推理中常用的技术，它可以帮助我们理解和解决复杂的数学问题，建立数学对象之间的关联和联系。
embedding 原理和实例-概述说明以及解释1.引言1.1 概述Embedding 是一种常见的机器学习技术，用于将高维数据映射到低维空间，从而更好地表示数据的特征。
通过Embedding 技术，我们可以将离散的数据转换为连续的向量表示，从而更好地进行数据分析、模型训练和预测。
Embedding 技术在自然语言处理、推荐系统、图像处理等领域都有广泛的应用。
在自然语言处理中，Embedding 可以将单词或者短语表示为向量，从而实现词向量的语义相似性和文本分类等任务。
在推荐系统中，Embedding 可以将用户和物品表示为向量，从而实现个性化推荐和推荐结果的相关性计算。
在图像处理中，Embedding 可以将图像表示为向量，从而实现图像的检索和相似性匹配。
本文将深入探讨Embedding 技术的概念、原理和实现方式，以及在自然语言处理中的具体应用。
通过对Embedding 技术的详细介绍，可以帮助读者更好地理解和应用这一重要的机器学习技术。
1.2 文章结构文章结构本文将分为引言、正文和结论三个部分展开论述embedding的原理和实例。
在引言部分中，将对embedding进行概述，介绍本文的结构和目的。
在正文部分中，将详细讨论embedding的概念和定义，探讨embedding的原理和实现方式，并深入探讨embedding在自然语言处理中的应用。
在结论部分中，将总结embedding的重要性，展望embedding在未来的发展，并进行结语。
整个文章将围绕embedding 展开，希望能为读者带来关于embedding的全面了解。
1.3 目的Embedding是机器学习和自然语言处理中的一个重要概念，通过将高维稀疏的数据转换成低维稠密的表示，有利于提高模型的性能和效率。
本文旨在通过对Embedding的概念、原理和实现方式进行深入探讨，以及分析Embedding在自然语言处理中的具体应用，希望读者能够更加全面地了解Embedding的重要性和应用价值。
零样本语义分割项目-概述说明以及解释1.引言1.1 概述概述:零样本语义分割项目是一项新兴的研究领域，致力于解决传统语义分割模型依赖于大量标注数据的问题。
在传统的语义分割任务中，模型需要通过有标签的数据进行训练和学习，但是对于新颖的类别或场景，缺乏对应的标注数据会导致模型性能下降。
零样本学习的概念则是在没有任何有关新类别的标注数据的情况下，使模型能够识别并分割出这些新类别的对象。
这项技术对于实际应用中的场景和类别的扩展至关重要。
零样本语义分割项目的目标是让模型具备一定的泛化能力，能够在没有见过的类别上进行准确的语义分割。
在本项目中，我们将探讨零样本学习的基本概念、语义分割任务中的挑战以及当前相关技术和方法的研究进展。
最终，我们希望通过这篇文章与读者分享零样本语义分割项目的意义和前景，为未来的研究和应用提供启示。
1.2 文章结构本文主要分为三个部分，即引言、正文和结论。
在引言部分，将首先介绍零样本语义分割项目的背景和意义，然后对文章的结构和目的进行概述，引导读者进入主题。
在正文部分，将分为三个小节进行阐述。
首先，将介绍什么是零样本学习，解释其中的基本概念和原理。
然后，将讨论零样本语义分割所面临的挑战，包括数据稀缺性、模型泛化能力等方面的问题。
最后，将介绍相关的技术和方法，包括迁移学习、元学习等在零样本语义分割中的应用。
在结论部分，将对前文进行总结，概括讨论的重点和收获。
同时，展望未来零样本语义分割项目的发展方向和潜力，为读者提供未来研究的参考。
最后，对全文进行结论性的总结，强调本文的重要性和价值。
1.3 目的本文的目的是介绍零样本语义分割项目的概念、挑战以及相关技术和方法。
通过深入讨论零样本学习的特点和应用场景，帮助读者了解这一新兴领域的重要性和研究现状。
同时，通过对相关技术和方法的介绍，希望能够激发更多学术界和工业界的关注和研究，推动零样本学习在语义分割领域的发展。
最终，本文旨在为读者提供关于零样本语义分割项目的全面理解，促进该领域的创新和进步。
邻域嵌入的张量学习摘要：本文旨在探讨如何使用张量学习方法来实现邻域嵌入的目的。
本研究提出的方法可以将大量的特征数据转换为嵌入空间中的低维度表征，以及介绍一种用于确定邻域嵌入的新的张量学习方法。
本文的研究结果表明，通过使用张量学习的邻域嵌入方法，能够使得最优邻域被找到，进而提高相关任务的性能。
关键词：张量学习，邻域嵌入，低维表征，最优邻域正文：本文旨在探讨如何使用张量学习方法来实现邻域嵌入的目的。
邻域嵌入是一种被广泛用于机器学习领域中的向量表示方法，它可以将大量的特征数据转换为低维度嵌入空间中的表征。
本文提出一种新的张量学习方法，用于确定邻域嵌入。
我们使用张量学习的邻域嵌入方法，根据输入的特征向量生成一个低维的表征空间，其中可以通过比较表征的相似度来确定最优的邻域。
本文的研究结果表明，通过使用张量学习的邻域嵌入方法，可以获得更好的结果，并有助于提高相关任务的性能。
最后，我们还提出了未来工作的建议，以进一步探究邻域嵌入的应用。
1. 因此，本文将详细探讨邻域嵌入的性能优势以及如何使用张量学习方法来实现这一目的。
首先，我们介绍了一种新的张量学习方法，可将特征数据从不同的维度嵌入到低维表征空间中，其中有助于提高基于相似度的相关任务的性能。
接下来，我们介绍了如何使用张量学习来实现邻域嵌入，以及如何定位最优邻域。
最后，我们还展示了如何使用张量学习的邻域嵌入方法来改进相关任务的性能。
2. 为验证我们的方法，我们使用了几个公开可用的数据集，例如MNIST、CIFAR10等，以在每项任务上进行实验，以评估张量学习的邻域嵌入方法。
本文的实验结果表明，使用张量学习的邻域嵌入方法能够有效改进相关任务的性能，并将大量数据映射到一个低维度表征空间。
3. 最后，本文还提出了未来工作的建议。
特别是，我们可以进一步利用张量学习方法来实现复杂的邻域嵌入，可以更好地优化空间表示。
此外，我们还可以利用深度神经网络来改进邻域嵌入性能，以更有效地提取高维特征和模式。
On Zerotree Quantization for Embedded Wavelet Packet ImageCodingN.M.Rajpoot1,2,F.G.Meyer3,R.G.Wilson1,and R.R.Coifman21Department of Computer Science,University of Warwick,Coventry,United Kingdom Department of2Mathematics and3Computer Science,Yale University,New Haven,CTe-mail:rajpoot-nasir@AbstractWavelet packets are an effective representation tool for adaptive waveform analysis of a given signal.We first combine the wavelet packet representation with zerotree quantization for image coding.A general ze-rotree structure is defined which can adapt itself to any arbitrary wavelet packet basis.We then describe an efficient coding algorithm based on this structure.Fi-nally,the hypothesis for prediction of coefficients from coarser scale tofiner scale is tested and its effective-ness is compared with that of zerotree hypothesis for wavelet coefficients.1IntroductionThere has been a surge of interest in wavelet trans-forms for image and video coding applications,in re-cent years.This is mainly due to the nice localiza-tion properties of wavelets in both time(or space) and frequency.The zerotree quantization,proposed by Shapiro[8],is an effective way of exploiting the self-similarities among high-frequency subbands at vari-ous resolutions.The main thrust of this quantization strategy is in the prediction of corresponding wavelet coefficients in higher frequency subbands at thefiner scales,by exploiting the parent-offspring dependen-cies.This prediction works well,in terms of efficiently coding the wavelet coefficients,due to the statistical characteristics of subbands at various resolutions,and due to the scale-invariance of edges in high frequency subbands of similar orientation.Various extensions of zerotree quantization,such as[3,9],have been pro-posed ever since its introduction.Wavelet packets[1]were developed in order to adapt the underlying wavelet bases to the contents of a signal.The basic idea is to allow non-octave subband decomposition to adaptively select the best basis for a particular signal.Results from various image cod-ing schemes based on wavelet packets show that they are particularly good in coding images with oscilla-tory patterns,a special form of texture.While wavelet packet bases are well adapted to the corresponding signal(image),one loses the parent-offspring depen-dencies,or the spatial orientation trees,as defined in [8,7].In this work,we address the following question: can the zerotree quantization be applied to wavelet packet transformed images?The question is valid since one does not generally observe as much self-similarities among the wavelet packet subbands as among the wavelet subbands.Moreover,even if there does exist self-similarity among wavelet packet sub-bands,the question arises:how would the spatial ori-entation trees,or zerotrees,be defined in order to pre-dict the insignificance of corresponding wavelet pack-ets at afiner scale,given a wavelet packet at a coarser scale?The issue was addressed partially by Xiong et al.in[10].In their work,however,the tree decompo-sition was restricted not to have a basis causing the parenting conflict,a problem described in the next sec-tion.We present a solution to this problem and define a set of rules to construct the zerotree structure for a given wavelet packet geometry,thus offering a general structure for an arbitrary wavelet packet decomposi-tion.This generalized zerotree structure is termed as the compatible zerotree for reasons mentioned in the next section.The new tree structure provides an effi-cient way of encoding the wavelet packet coefficients. We combine the idea of zerotree quantization with wavelet packet transforms for image coding purposes. 2Compatible Zerotree Quantization The wavelet packet basis[1]is adaptively selected in order to tailor the representation to the contents of a signal(an image,in our case).The nodes in a full subband tree(or short-term Fourier transform, namely STFT,tree)are pruned following a series of split/merge decisions using certain criterion(see[2] for entropy-based best basis selection and[6]for op-timizing the best basis from a rate-distortion view-point).Suppose the best basis has been selected us-ing one of these methods.The issue is how to organize the spatial orientation trees so as to exploit the self-similarities,if any,among the subbands.The basis selected by any of the above methods does not,in gen-eral,yield the parent-offspring relationships like those in wavelet subbands.Moreover,there can be an in-stance in the wavelet packet tree where one or more of the child nodes are at a coarser scale than the parent node.This results in the association of each coefficient of such a child node to multiple parent coefficients,in the parent node,giving rise to a parenting conflict. To make the point clear,let us consider the segmen-tation shown in Figure1,for a3-level wavelet packet transform.The nodes C1,C2,C3,and C4at a coarser scale are children of the node P at afiner scale.The problem with such a situation is that there are four candidates in P claiming the parenthood of the set of four coefficients belonging to the four children nodes. In[10],the best basis was selected in such a way that whenever a conflict like this arises,the four children are merged so as to resolve the conflict.This subopti-mal approach constrains selection of the best basis at the loss of the freedom of adapting the wavelet packet basis to the contents of a particularsignal.Figure1:Sample segmentation in a3-level wavelet packet decomposition and the parent-offspring dependencies for compatible zerotree originating from T1.We organize the zerotree structure,called compat-ible zerotrees,dynamically so that there is no restric-tion on selection of the wavelet packet basis(we call these the compatible zerotrees,since they are gener-ated taking into account both the scale and orientation compatibility).It is to be noted that the compatible zerotrees differ from the spatial orientation trees of[7] in the sense that nodes of a compatible zerotree rep-resent a full subband,rather than one or more coeffi-cients.We utilize the compatible zerotrees tofind the coefficients which are zerotree roots,for a given thresh-old.A set of rules is defined to construct the compati-ble zerotrees so that the overall zerotree structure can be constructed for any arbitrary wavelet packet basis. The only assumption made is that the basis has its lowest frequency bands at the coarsest scale.This is based upon the fact that a significant amount of the signal energy is concentrated in the lowest frequency subbands,which cannot be merged by any of the tree pruning algorithms.Let R denotes the node representing the lowest fre-quency subband,situated in the top-left corner of a conventional subband decomposed image.Then R represents the root node of overall compatible ze-rotree.It has three children(T1,T2,and T3as shown in Figure1)which represent the coarsest scale high-frequency subbands.These children are themselves root nodes of three compatible zerotrees with different global orientations;horizontal,vertical,and diagonal respectively.These zerotrees are generated in a re-cursive manner(only once for a given wavelet packet basis)using the following set of rules,each of them applied to nodes with similar global orientation.a)If a node P is followed by four nodes C1,C2,C3,and C4(at the same scale),then P is declared to be the parent of all these four nodes.b)If four subbands P1,P2,P3,and P4at a coarserscale are followed by four subbands C1,C2,C3, and C4at immediately nextfiner scale,then node P i is declared to be the parent of node C i(for i=1,2,3,4).c)If a node P at a coarser scale is followed only bya node C at the immediately nextfiner scale(likein a wavelet transform),the node P is declared as a parent of C.d)If a node P is at afiner scale than four of itschildren,say C1,C2,C3,and C4,then P is disre-garded as being the parent of all these nodes and all this bunch is moved up in the tree.Let us consider the segmentation shown in Figure1to explain these rules.The construction offinal compat-ible zerotrees proceeds in two steps.In thefirst step, the primary compatible zerotrees T1,T2,and T3are constructed using rules a,b,and c,as shown in Figure 2.In the second step,the overall tree is re-organized (as shown in Figure3)using rule d,in order to resolve the parenting conflict under node P.This rulefirst identifies nodes with the parenting conflicts and when such nodes are found,the whole bunch(consisting of C1,C2,C3,and C4)is plucked from its current po-sition in the tree.The algorithm then climbs up thetree to look for an appropriate node,a compatible an-cestor at the nearest scale and having the nearest ori-entation,and glues the bunch under this newly found compatible parent.The parent-offspring dependencies for the primary compatible zerotree T1are shown in Figure1.For more details of the algorithm to con-struct the compatible zerotrees,please refer to[5].Given an arbitrarily segmented wavelet packet ba-sis,the compatible zerotrees are constructed using the set of rules described as above.In order to be able to make predictions for wavelet packet coefficients at finer scale subbands,the compatible zerotree hypothe-sis is defined as follows:If a wavelet packet coefficient of a node from the compatible zerotree is insignificant, it is more likely that the wavelet packet coefficients at similar spatial locations in all the descendent nodes of the same compatible zerotree will be insignificant as well.We tested the success of compatible zerotree hy-pothesis,as defined above,for wavelet packets.The plots of wavelet packet coefficients’amplitude against the coefficient indices for the wavelet packet decompo-sition,using the entropy-based basis selection criterion as in[4],of a512×512Barbara image are shown in Figures4and5.The amplitude axis in both plots was restricted to±1000to facilitate the display of smaller coefficients.While the subbands were arranged in an increasing frequency order for the plot in Figure4,the plot in Figure5refers to the subbands as organized in the compatible zerotrees.The three families of pri-mary compatible zerotrees T1,T2,and T3are clearly visible in the latter of these plots showing that the new arrangement was successful in isolating the coef-ficients of similar orientation in a hierarchy to be used for prediction in a top-down order.3The Coder AlgorithmOnce the compatible zerotrees have been generated, based upon knowledge of the best basis,the encod-ing takes place by repeatedly running the detection stage and thefine-tuning stage until the bit budget is expired.Our algorithm requires only one list of detected coefficients(LDC)to be maintained,as op-posed to two(three)lists kept by the EZW(SPIHT). The decoderfirst reads geometry of the best basis and generates the compatible zerotrees,and then proceeds by entropy decoding and interpreting the codewords (see[5]for more details of the algorithm).4Experiments and Conclusions The idea of compatible zerotree quantization was coupled with the wavelet packet transform for pro-gressive image coding and its performance was tested against the fast adaptive wavelet packet(FAWP)[4] image codec algorithm.A comparison of the peak sig-nal to noise ratio(PSNR)achieved by CZQ and FAWP for three standard512×512images encoded at tar-get bit rates of0.25and0.10bits per pixel(bpp)is presented in Table1.While being relatively faster,by a factor ranging from two to four,than the bitplane coding of FAWP,the CZQ performs comparably well. We note that the effectiveness of zerotree quantization depends strongly upon the scale-invariance of edges among subbands of similar global orientation.The success of prediction offiner scale coefficients(set of all descendents D i at similar spatial location)based upon a given coefficient i at a coarser scale is a func-tion of the conditional probability of D i being insignifi-cant,given i is insignificant.We compared the number of zerotree root nodes detected,for various values of threshold,both for wavelet and wavelet packet bases. It was observed that the probability of detection of the zerotree root nodes in a wavelet decomposition is far greater than that in a wavelet packet decomposition.It is important to note here that the wavelet packet basis was selected by simply usingfirst-order entropy as a criterion for the split/merge decisions.The coeffi-cients of basis were,therefore,imposed to be zerotree quantized.As opposed to wavelets,the wavelet packet basis might not necessarily produce coefficients that help towards success of the zerotree hypothesis(that is,produce more zerotrees).Figure6shows the geom-etry of wavelet packet basis selected for the512×512 Barbara image.In fact,we found that using the en-tropy as a criterion for the best basis selection might result into many coarse scale high frequency subbands (whose coefficients represent oscillatory patterns in some parts of the image),as is clear from basis ge-ometry in Figure6.It is the encoding of coefficients in these types of subbands which takes considerable amount of the bit budget for the following reason.If a coefficient somewhere at the bottom of a compati-ble zerotree is found to be significant,the significance of all the corresponding coefficients in sibling and the parent nodes needs to be encoded as well,even if some or most of them are insignificant.Our future work aims to look at the methods of optimizing the selec-tion of best basis using a criterion which results into optimal number of zerotrees and optimal distortion value,given the bit budget.Moreover,the ordering of wavelet packet coefficients with respect to increas-ing frequency(that is,decaying magnitude)within the compatible zerotrees will provide a gain in the coding performance,by increasing the efficiency of arithmetic coder to encode the zerotree codewords.Figure 2:The overall compatible zerotree structure com-prising of the three primary compatible zerotrees T 1,T 2,and T 3;each node represents a whole wavelet packet sub-band,and the node radius is directly proportional to the support of wavelet packets at that transformlevel.Figure 3:The overall compatible zerotree structure,afterre-organization to resolve the parenting conflicts;the edge labels depict the rules used to generate the links between parent and the childrennodes.Figure 4:Plot of wavelet packet coefficients’amplitudeversus coefficient indices for the wavelet packet decompo-sition of 512×512Barbara image;the subbands were ar-ranged in an increasing frequencyorder.Figure 5:Plot of wavelet packet coefficients’amplitudeversus coefficient indices for the wavelet packet decom-position of 512×512Barbara image;the subbands were organized in compatible zerotrees with the help of rules mentioned in Section2.Figure 6:Wavelet packet basis selected for the 512×512Barbara image using first-order entropy as selection crite-rion.Rate (bpp)FA WPFingerprint26.680.1023.81Goldhill29.190.1027.54Lena31.880.1029.33AcknowledgementThis work was supported in part by DARPA/AFOSR under Grant F49620-97-1-0011.References[1]R.R.Coifman and Y.Meyer,Othornormal wavepacket bases,preprint,Dept.of Mathematics, Yale University,New Haven(1990).[2]R.R.Coifman and M.V.Wickerhauser,Entropy-based algorithms for best basis selection,IEEE Trans.on Information Theory38(1992),no.2, 713–718.[3]M.Effros,Zerotree design for image compression:Toward weighted universal zerotree coding,IEEE Intl.Conf.on Image Processing,November1997.[4]F.G.Meyer, A.Z.Averbuch,J-O.Str¨o mberg,and R.R.Coifman,Fast wavelet packet image compression,Data Compression Conference-DCC’98,1998.[5]N.M.Rajpoot,F.G.Meyer,and R.R.Coifman,Wavelet packet image coding using compatible ze-rotree quantization,Technical Report,Dept.of Computer Science,Yale University,New Haven (1999).[6]K.Ramchandran and M.Vetterli,Best waveletpacket bases in a rate-distortion sense,IEEE Trans.on Image Processing(1993),160–175. 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