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.

该项目的目标是开发一套工具，能够执行真正的局部傅里叶分析和合成的目标感兴趣的对象在给定的图像，具有光滑的边界，但一般形状。为此，我们将引入传统不同领域的工具，如边界检测和描绘算法（计算机视觉），图像压缩和去噪方案（图像处理），椭圆边界值问题解决方案和势函数计算（科学计算），傅立叶分析和快速算法（计算谐波分析）。我们主要关注的是分析（例如，空间频率特征的提取和表征）和合成(例如，在检测到物体的边界后，从物体的压缩表示中进行重建，并由人类口译员使用指向设备手动分割，或由其他研究人员提出的算法自动分割。一方面，物体的边界提供了重要的信息：物体的几何形状。另一方面，它成为其他任务的麻烦，如物体的内部信息（如纹理）的傅里叶分析，因为它产生了虚假的干涉模式，由于吉布斯现象，掩盖了物体的重要的内部信息。我们将解耦对象的几何和内部信息，通过解决椭圆边值问题的领域上的对象的支持。更精确地说，对于目标的分析，我们将检测和分割的目标嵌入到一个空白的矩形域中，并通过在覆盖矩形边缘求解具有齐次Dirichlet边界条件的泊松方程，将目标平滑地扩展到目标边界的外部。由于覆盖矩形边缘上的值消失，因此可以将该平滑扩展的分量展开为具有快速衰减系数的傅立叶正弦级数，从而使我们能够有效地表征和压缩对象的内部信息。最后，我们从支持域上的原始对象中减去该分量，得到负责对象几何信息的分量，即该域上拉普拉斯方程的解。对于目标的合成或重建，我们需要在分析阶段存储扩展分量的傅立叶正弦系数和目标在这些点的边界坐标和原始值。然后，通过添加几何分量（通过评估域上的单层和双层电位来恢复）和平滑扩展分量（很容易从傅里叶正弦系数中恢复）来恢复原始对象。为了求解拉普拉斯/泊松方程，我们将充分利用基于快速多极方法的先进拉普拉斯/泊松求解器。我们还将把我们的分析和综合范例扩展到有孔的对象（即多连通域），研究噪声对边界形状和值的影响，并研究边界信息压缩对重建图像质量的影响。此外，我们将开发一种具有正则化（高频衰减）能力的梯度和方向导数估计算法，这将提供良好的边界条件，从而提高边界检测和描绘算法的性能以及拉普拉斯/泊松方程解的准确性。我们的方法的潜在应用包括生物识别和基于图像的医学诊断以及其他领域，如地质和材料科学。由于国土安全的原因，生物识别技术最近已经成为一门非常重要的学科。由于我们的范例为各种传感器和成像模式获得的图像以单独的方式提供感兴趣对象的几何/形状信息和内部纹理信息，因此与仅使用形状或纹理信息的方法相比，它可以允许数据检查人员更可靠地表征感兴趣对象的特征。例如，使用我们将在项目中开发的工具（例如，基于对象的存储、编目、压缩和分析），可以表征和诊断包括妇科巴氏涂片检查图像在内的各种图像模式中的癌细胞。同样，从岩芯样品切片光学图像中提取定量信息（如某些化石的尺寸和内部纹理/空间频率信息）在包括油气勘探行业在内的地球科学中也很重要，也可能受益于我们的研究。我们设想，在完全不同的学科，如医学，生物学和地质学的科学家，将开始注意到计算谐波分析和某些偏微分方程（PDEs）的重要性，如果他们使用我们在这个项目中开发的工具，并觉得这些对他们自己的任务很有用。这是我们作为应用数学家和计算数学家所希望的一件伟大的事情。就教育影响而言，这个项目将为不同领域的学生创造一个共同的聚会场所：应用数学、计算机科学、电子工程、统计学和神经科学。我们期待这些将参与本研究项目或参加我们正在开发的相关课程和研讨会的学生之间进行活跃的思想交流。参与我们项目的学生还将学习计算谐波分析，某些偏微分方程的基本理论和快速计算算法，以及图像分析，这将成为未来在成像科学领域工作的应用数学家和科学家不可或缺的，这必将对他们未来的职业生涯有所帮助，无论是在学术界还是工业界。