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Chapter 10

Hessian-Based Multiscale Enhancement, Description, and Quantification of Second-Order 3-D Local Structures from Medical Volume Data

Yoshinobu Sato1

10.1 Introduction

With high-resolution three-dimensional (3-D) imaging modalities becoming commonly available in medical imaging, a strong need has arisen for a means of accurate extraction and 3D quantification of the anatomical structures of interest from acquired volume data. Three-dimensional local structures have been shown to be useful for 3-D modeling of anatomical structures to improve their extraction and quantification [1-16]. In this chapter, we describe an approach to enhancement, description, and quantification of the anatomical structures characterized by second-order 3D local structures, that is, line, sheet, and blob structures.

The human body contains various types of line, sheet, and blob structures. For example, bloodvessels, bone cortices, and nodules are characterized by line, sheet, and blob structures, respectively. We present a theoretical framework for systematic analysis of second-order local structures in volume data. A set of volume data is typically represented as a discrete set of samples on a regular grid. The basic approach is to analyze the continuous volume intensity function

1 Division of Interdisciplinary Image Analysis, Osaka University Graduate School of Medicine, 2-2-D11 Yamada-oka, Suita, Osaka 565-0871, Japan that underlies the discrete sample data. Second-order local structures around a point of interest in the underlying continuous function can be fully represented using up to second derivatives at the point, that is, the gradient vector and Hessian matrix. In order to reduce noise as well as deal with second-order local structures of "various sizes," isotropic Gaussian smoothing with different standard deviation (SD) values is combined with derivative computation. Combining Gaussian smoothing has another effect that accurate derivative computation of the Gaussian smoothed version of the underlying "continuous" function is possible by convolution operations within a size-limited local window.

In this chapter, the following topics are discussed:

• Multiscale enhancement filtering of second-order local structures, that is, line, sheet, and blob structures [5,7,11] in volume data.

• Analysis of filter responses for line structures using mathematical line models [7].

• Description and quantification (width and orientation measurement) of these local structures [10,12].

• Analysis of sheet width quantification accuracy restricted by imaging resolution [17,18].

For the multiscale enhancement, we design 3-D enhancement filters, which selectively respond to the specific type of local structures with specific size, based on the eigenvalues of the Hessian matrix of the Gaussian smoothed volume intensity function. The conditions that the eigenvalues need to satisfy for the local structures are analyzed to derive similarity measures to the local structures. We also design a multiscale integration scheme of the filter responses at different Gaussian SD values. The condition for the scale interval is analyzed to enhance equally over a specific range of structure sizes.

For the analysis of filter responses, mathematical line models with a non-circular cross section are used. The basic characteristics of the filter responses and their multiscale integration are analyzed simulationally. Although the line structure is considered here, the presented basic approach is applicable to filter responses to other local structures.

For the detection and quantification, we formulate a method using a second-order polynomial describing the local structures. We focus on line and sheet structures, which are basically characterized by their medial elements (medial surfaces and axes, respectively) and widths associated with the medial elements. A second-order polynomial around a point of interest is defined by the gradient vector and Hessian matrix at the point. The medial elements are detected based on subvoxel localization of local maximum of the second-order polynomial within the voxel territory of a point of interest. The widths are measured along normal directions of the detected medical elements.

For the analysis of quantification accuracy, a theoretical approach is presented based on mathematical models of imaged local structures, imaging scanners, and quantification processes. Although the sheet structure imaged by MR scanners is focused here, the presented basic approach is applicable to accuracy analysis for different local structures imaged by either MR or CT scanners.

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