Object-Oriented Image Analysis and Synthesis via Computational Harmonic Analysis and Boundary Value Problems

通过计算调和分析和边值问题进行面向对象的图像分析和合成

• 批准号: 0410406
• 负责人: Naoki Saito
• 金额: $ 28.25万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-08-01 至 2008-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0410406&HistoricalAwards=false
• 关键词: Object; Oriented; Image; Analysis; Synthesis

The objective of this project is to develop a set of tools capable of performingtruly localized Fourier analysis and synthesis of objects of interest in a givenimage that have smooth boundaries but of general shape. To do so, we will bringin tools in the traditionally different fields such as boundary detection and delineation algorithms (computer vision), image compression and denoising schemes (image processing), elliptic boundary value problem solvers and potential function computations (scientific computing), and Fourier analysis andfast algorithms (computational harmonic analysis). Our primary concern is analysis (e.g., extraction and characterization of spatial frequency features)and synthesis (e.g., reconstruction from the compressed representation of objects after their boundaries are detected and they are segmented either manually by a human interpreter using a pointing device or automatically by thealgorithms proposed by other researchers. On the one hand, the boundary of anobject provides important information: geometry and shape of the object. On theother hand, it becomes a nuisance for other tasks such as the Fourier analysisof the internal information (e.g., textures) of the object because it createsspurious interference patterns due to the Gibbs phenomenon that masks theimportant internal information of the object. We will decouple the geometry andinternal information of the object by solving the elliptic boundary value problem on the domain where the object is supported. More precisely, for the analysis of the object, we embed the detected and segmented object in an otherwise empty rectangular domain, and smoothly extend the object to the outside of the object boundary by solving the Poisson equation with the homogeneous Dirichlet boundary condition at the edges of the covering rectangle.Since the values on the edges of the covering rectangle vanish, this smoothlyextended component can be expanded into the Fourier sine series with quickly decaying coefficients, which enable us to effectively characterize and compressthe internal information of the object. Finally, we subtract this componentfrom the original object on the supported domain to obtain the component responsible for geometric information of the object, which turns out to be a solution of the Laplace equation on that domain. For the synthesis or reconstruction of the object, we need to store the Fourier sine coefficients ofthe extended component and the boundary coordinates and the original values of the object at those points in the analysis stage. Then, the original object is recovered by adding the geometric component (which is recovered by evaluating the single and double layer potentials on the domain) and the smoothly extendedcomponent (which is easily recovered from the Fourier sine coefficients). In order to solve the Laplace/Poisson equations, we will fully utilize the advancedLaplace/Poisson solvers based on Fast Multipole Methods. We will also extend our analysis and synthesis paradigm to an object with holes (i.e., multiply-connected domains), investigate the effect of noise on the boundary shapes and values, and investigate the effect of compression of the boundary information to the quality of the reconstructed images. Furthermore, we willdevelop a gradient and directional derivative estimation algorithm equipped withregularization (high-frequency attenuation) capabilities, which will provide good boundary conditions and consequently improve the performance of the boundary detection and delineation algorithms as well as the accuracy of the solutions of the Laplace/Poisson equations.Potential applications of our methodology include biometrics and image-based diagnostics in medicine and other fields such as geology and material sciences.Biometrics has recently become a tremendously important subject for homeland security reasons. Since our paradigm provides both geometric/shape informationand internal texture information of an object of interest in an separate mannerfor images obtained by various sensors and imaging modalities, it may allow dataexaminers to characterize the features of the objects of interest much more reliably compared to the methods which solely use either shape or texture information. For example, characterizing and diagnosing cancerous cells in various image modalities including Pap smear test images in gynecology may benefit from using the tools we will develop in our project (e.g., object-based storage, cataloging, compression, and analysis). Similarly, extracting quantitative information from optical images of sections of rock core samples (e.g., the size and internal texture/spatial frequency information of some fossils), which is important in earth science including oil and gas exploration industry, may also benefit from our research. We envision that scientists in completely different disciplines such as medicine, biology, and geology, will start noticing the importance of computational harmonic analysis and certain partial differential equations (PDEs) if they use our tools to be developed inthis project and feel that these are useful for their own tasks. This is a great thing we, as applied and computational mathematicians, can hope for.In terms of the educational impact, this project will create a common meetingground among students in the different fields: applied mathematics, computer science, electrical engineering, statistics, and neuroscience. We expect livelyinterchanges of ideas among such students who will participate in this researchproject or attend the associated courses and seminars we are developing. Students participating in our project will also learn computational harmonic analysis, the basic theory and fast computational algorithms of certain PDEs, and image analysis, which will become indispensable for the future applied mathematicians and scientists working in the area of imaging science, and whichwill be surely helpful for their future career, either in academia or in industry.

该项目的目的是开发一组能够执行具有平稳边界但具有一般形状的给定图的对象的局部傅立叶分析和综合感兴趣对象的工具。为此，我们将将工具带入传统上不同的领域，例如边界检测和描述算法（计算机视觉），图像压缩和降解方案（图像处理），椭圆边界值问题求解器以及潜在的函数计算（科学计算）以及傅立叶分析和Fourier Analysis andfast算法（计算和谐分析）。 我们的主要关注点是分析（例如，提取和表征空间频率特征）和合成（例如，检测到对象的界限后的压缩表示，从其界限后的压缩表示，并通过对其他研究人员的thealgorithms向其他研究人员提出的thealgorithms向对象进行了对象，将其自动提出的thealgorithms构建，并由人工解释器手动分段。它成为其他任务的滋扰，例如对象的内部信息（例如，纹理），因为它会由于gibbs现象而引起的，从而掩盖了对象的重要内部信息，我们将通过求解对象的几何学和内部信息。将检测到的和分段的物体嵌入原本空的矩形域中，并通过求解泊松方程，并在覆盖矩形的边缘上求解泊松方程，将对象顺利扩展到对象边界的外部，以便在覆盖矩形的边缘上的价值。为了有效地表征和压缩对象的内部信息。然后，在分析阶段，通过添加几何组件（通过评估域上的单个和双层电位）恢复原始对象，并平滑地扩展了组件（可以轻松地从傅立叶系数中恢复。我们的分析和合成范式范围（即，倍增域），研究噪声对边界形状和值的影响，并研究边界信息的压缩效果，以及我们将提供重建图像的质量，我们将提供良好的衍生估计值，并提供良好的量级量级（且高且高且富度量）。条件并改善了边界检测和描述算法的性能，以及Laplace/Poisson方程的解决方案的准确性。我们的方法论的潜在应用包括医学和基于图像的诊断和基于图像的诊断和其他领域，例如地质和物质诊所，例如最近都非常重要的是室内的。与各种传感器和成像模式获得的图像相比，与单独的图像相比，与单独的图像获得的纹理信息可以更可靠地表征感兴趣对象的特征，而这些方法与仅使用形状或纹理信息的方法，例如，在各种图像模态中表征和诊断出对涂抹图像的图像，我们将使用gynec cape nime the Gynec compution compution（gynec compution of the Gynec中）。类似地，从岩石核心样本的光学图像中提取定量信息（例如，某些化石的大小和内部纹理/空间频率信息）在地球科学（包括石油和天然气探索行业）中很重要，我们可能会从我们的研究中受益。某些部分差分方程式（PDE）如果他们在此项目中使用我们的工具来开发我们的任务，这是一个很棒的事情。参加我们正在开发的学生的学生将参加我们的项目。