- •Preface
- •Biological Vision Systems
- •Visual Representations from Paintings to Photographs
- •Computer Vision
- •The Limitations of Standard 2D Images
- •3D Imaging, Analysis and Applications
- •Book Objective and Content
- •Acknowledgements
- •Contents
- •Contributors
- •2.1 Introduction
- •Chapter Outline
- •2.2 An Overview of Passive 3D Imaging Systems
- •2.2.1 Multiple View Approaches
- •2.2.2 Single View Approaches
- •2.3 Camera Modeling
- •2.3.1 Homogeneous Coordinates
- •2.3.2 Perspective Projection Camera Model
- •2.3.2.1 Camera Modeling: The Coordinate Transformation
- •2.3.2.2 Camera Modeling: Perspective Projection
- •2.3.2.3 Camera Modeling: Image Sampling
- •2.3.2.4 Camera Modeling: Concatenating the Projective Mappings
- •2.3.3 Radial Distortion
- •2.4 Camera Calibration
- •2.4.1 Estimation of a Scene-to-Image Planar Homography
- •2.4.2 Basic Calibration
- •2.4.3 Refined Calibration
- •2.4.4 Calibration of a Stereo Rig
- •2.5 Two-View Geometry
- •2.5.1 Epipolar Geometry
- •2.5.2 Essential and Fundamental Matrices
- •2.5.3 The Fundamental Matrix for Pure Translation
- •2.5.4 Computation of the Fundamental Matrix
- •2.5.5 Two Views Separated by a Pure Rotation
- •2.5.6 Two Views of a Planar Scene
- •2.6 Rectification
- •2.6.1 Rectification with Calibration Information
- •2.6.2 Rectification Without Calibration Information
- •2.7 Finding Correspondences
- •2.7.1 Correlation-Based Methods
- •2.7.2 Feature-Based Methods
- •2.8 3D Reconstruction
- •2.8.1 Stereo
- •2.8.1.1 Dense Stereo Matching
- •2.8.1.2 Triangulation
- •2.8.2 Structure from Motion
- •2.9 Passive Multiple-View 3D Imaging Systems
- •2.9.1 Stereo Cameras
- •2.9.2 3D Modeling
- •2.9.3 Mobile Robot Localization and Mapping
- •2.10 Passive Versus Active 3D Imaging Systems
- •2.11 Concluding Remarks
- •2.12 Further Reading
- •2.13 Questions
- •2.14 Exercises
- •References
- •3.1 Introduction
- •3.1.1 Historical Context
- •3.1.2 Basic Measurement Principles
- •3.1.3 Active Triangulation-Based Methods
- •3.1.4 Chapter Outline
- •3.2 Spot Scanners
- •3.2.1 Spot Position Detection
- •3.3 Stripe Scanners
- •3.3.1 Camera Model
- •3.3.2 Sheet-of-Light Projector Model
- •3.3.3 Triangulation for Stripe Scanners
- •3.4 Area-Based Structured Light Systems
- •3.4.1 Gray Code Methods
- •3.4.1.1 Decoding of Binary Fringe-Based Codes
- •3.4.1.2 Advantage of the Gray Code
- •3.4.2 Phase Shift Methods
- •3.4.2.1 Removing the Phase Ambiguity
- •3.4.3 Triangulation for a Structured Light System
- •3.5 System Calibration
- •3.6 Measurement Uncertainty
- •3.6.1 Uncertainty Related to the Phase Shift Algorithm
- •3.6.2 Uncertainty Related to Intrinsic Parameters
- •3.6.3 Uncertainty Related to Extrinsic Parameters
- •3.6.4 Uncertainty as a Design Tool
- •3.7 Experimental Characterization of 3D Imaging Systems
- •3.7.1 Low-Level Characterization
- •3.7.2 System-Level Characterization
- •3.7.3 Characterization of Errors Caused by Surface Properties
- •3.7.4 Application-Based Characterization
- •3.8 Selected Advanced Topics
- •3.8.1 Thin Lens Equation
- •3.8.2 Depth of Field
- •3.8.3 Scheimpflug Condition
- •3.8.4 Speckle and Uncertainty
- •3.8.5 Laser Depth of Field
- •3.8.6 Lateral Resolution
- •3.9 Research Challenges
- •3.10 Concluding Remarks
- •3.11 Further Reading
- •3.12 Questions
- •3.13 Exercises
- •References
- •4.1 Introduction
- •Chapter Outline
- •4.2 Representation of 3D Data
- •4.2.1 Raw Data
- •4.2.1.1 Point Cloud
- •4.2.1.2 Structured Point Cloud
- •4.2.1.3 Depth Maps and Range Images
- •4.2.1.4 Needle map
- •4.2.1.5 Polygon Soup
- •4.2.2 Surface Representations
- •4.2.2.1 Triangular Mesh
- •4.2.2.2 Quadrilateral Mesh
- •4.2.2.3 Subdivision Surfaces
- •4.2.2.4 Morphable Model
- •4.2.2.5 Implicit Surface
- •4.2.2.6 Parametric Surface
- •4.2.2.7 Comparison of Surface Representations
- •4.2.3 Solid-Based Representations
- •4.2.3.1 Voxels
- •4.2.3.3 Binary Space Partitioning
- •4.2.3.4 Constructive Solid Geometry
- •4.2.3.5 Boundary Representations
- •4.2.4 Summary of Solid-Based Representations
- •4.3 Polygon Meshes
- •4.3.1 Mesh Storage
- •4.3.2 Mesh Data Structures
- •4.3.2.1 Halfedge Structure
- •4.4 Subdivision Surfaces
- •4.4.1 Doo-Sabin Scheme
- •4.4.2 Catmull-Clark Scheme
- •4.4.3 Loop Scheme
- •4.5 Local Differential Properties
- •4.5.1 Surface Normals
- •4.5.2 Differential Coordinates and the Mesh Laplacian
- •4.6 Compression and Levels of Detail
- •4.6.1 Mesh Simplification
- •4.6.1.1 Edge Collapse
- •4.6.1.2 Quadric Error Metric
- •4.6.2 QEM Simplification Summary
- •4.6.3 Surface Simplification Results
- •4.7 Visualization
- •4.8 Research Challenges
- •4.9 Concluding Remarks
- •4.10 Further Reading
- •4.11 Questions
- •4.12 Exercises
- •References
- •1.1 Introduction
- •Chapter Outline
- •1.2 A Historical Perspective on 3D Imaging
- •1.2.1 Image Formation and Image Capture
- •1.2.2 Binocular Perception of Depth
- •1.2.3 Stereoscopic Displays
- •1.3 The Development of Computer Vision
- •1.3.1 Further Reading in Computer Vision
- •1.4 Acquisition Techniques for 3D Imaging
- •1.4.1 Passive 3D Imaging
- •1.4.2 Active 3D Imaging
- •1.4.3 Passive Stereo Versus Active Stereo Imaging
- •1.5 Twelve Milestones in 3D Imaging and Shape Analysis
- •1.5.1 Active 3D Imaging: An Early Optical Triangulation System
- •1.5.2 Passive 3D Imaging: An Early Stereo System
- •1.5.3 Passive 3D Imaging: The Essential Matrix
- •1.5.4 Model Fitting: The RANSAC Approach to Feature Correspondence Analysis
- •1.5.5 Active 3D Imaging: Advances in Scanning Geometries
- •1.5.6 3D Registration: Rigid Transformation Estimation from 3D Correspondences
- •1.5.7 3D Registration: Iterative Closest Points
- •1.5.9 3D Local Shape Descriptors: Spin Images
- •1.5.10 Passive 3D Imaging: Flexible Camera Calibration
- •1.5.11 3D Shape Matching: Heat Kernel Signatures
- •1.6 Applications of 3D Imaging
- •1.7 Book Outline
- •1.7.1 Part I: 3D Imaging and Shape Representation
- •1.7.2 Part II: 3D Shape Analysis and Processing
- •1.7.3 Part III: 3D Imaging Applications
- •References
- •5.1 Introduction
- •5.1.1 Applications
- •5.1.2 Chapter Outline
- •5.2 Mathematical Background
- •5.2.1 Differential Geometry
- •5.2.2 Curvature of Two-Dimensional Surfaces
- •5.2.3 Discrete Differential Geometry
- •5.2.4 Diffusion Geometry
- •5.2.5 Discrete Diffusion Geometry
- •5.3 Feature Detectors
- •5.3.1 A Taxonomy
- •5.3.2 Harris 3D
- •5.3.3 Mesh DOG
- •5.3.4 Salient Features
- •5.3.5 Heat Kernel Features
- •5.3.6 Topological Features
- •5.3.7 Maximally Stable Components
- •5.3.8 Benchmarks
- •5.4 Feature Descriptors
- •5.4.1 A Taxonomy
- •5.4.2 Curvature-Based Descriptors (HK and SC)
- •5.4.3 Spin Images
- •5.4.4 Shape Context
- •5.4.5 Integral Volume Descriptor
- •5.4.6 Mesh Histogram of Gradients (HOG)
- •5.4.7 Heat Kernel Signature (HKS)
- •5.4.8 Scale-Invariant Heat Kernel Signature (SI-HKS)
- •5.4.9 Color Heat Kernel Signature (CHKS)
- •5.4.10 Volumetric Heat Kernel Signature (VHKS)
- •5.5 Research Challenges
- •5.6 Conclusions
- •5.7 Further Reading
- •5.8 Questions
- •5.9 Exercises
- •References
- •6.1 Introduction
- •Chapter Outline
- •6.2 Registration of Two Views
- •6.2.1 Problem Statement
- •6.2.2 The Iterative Closest Points (ICP) Algorithm
- •6.2.3 ICP Extensions
- •6.2.3.1 Techniques for Pre-alignment
- •Global Approaches
- •Local Approaches
- •6.2.3.2 Techniques for Improving Speed
- •Subsampling
- •Closest Point Computation
- •Distance Formulation
- •6.2.3.3 Techniques for Improving Accuracy
- •Outlier Rejection
- •Additional Information
- •Probabilistic Methods
- •6.3 Advanced Techniques
- •6.3.1 Registration of More than Two Views
- •Reducing Error Accumulation
- •Automating Registration
- •6.3.2 Registration in Cluttered Scenes
- •Point Signatures
- •Matching Methods
- •6.3.3 Deformable Registration
- •Methods Based on General Optimization Techniques
- •Probabilistic Methods
- •6.3.4 Machine Learning Techniques
- •Improving the Matching
- •Object Detection
- •6.4 Quantitative Performance Evaluation
- •6.5 Case Study 1: Pairwise Alignment with Outlier Rejection
- •6.6 Case Study 2: ICP with Levenberg-Marquardt
- •6.6.1 The LM-ICP Method
- •6.6.2 Computing the Derivatives
- •6.6.3 The Case of Quaternions
- •6.6.4 Summary of the LM-ICP Algorithm
- •6.6.5 Results and Discussion
- •6.7 Case Study 3: Deformable ICP with Levenberg-Marquardt
- •6.7.1 Surface Representation
- •6.7.2 Cost Function
- •Data Term: Global Surface Attraction
- •Data Term: Boundary Attraction
- •Penalty Term: Spatial Smoothness
- •Penalty Term: Temporal Smoothness
- •6.7.3 Minimization Procedure
- •6.7.4 Summary of the Algorithm
- •6.7.5 Experiments
- •6.8 Research Challenges
- •6.9 Concluding Remarks
- •6.10 Further Reading
- •6.11 Questions
- •6.12 Exercises
- •References
- •7.1 Introduction
- •7.1.1 Retrieval and Recognition Evaluation
- •7.1.2 Chapter Outline
- •7.2 Literature Review
- •7.3 3D Shape Retrieval Techniques
- •7.3.1 Depth-Buffer Descriptor
- •7.3.1.1 Computing the 2D Projections
- •7.3.1.2 Obtaining the Feature Vector
- •7.3.1.3 Evaluation
- •7.3.1.4 Complexity Analysis
- •7.3.2 Spin Images for Object Recognition
- •7.3.2.1 Matching
- •7.3.2.2 Evaluation
- •7.3.2.3 Complexity Analysis
- •7.3.3 Salient Spectral Geometric Features
- •7.3.3.1 Feature Points Detection
- •7.3.3.2 Local Descriptors
- •7.3.3.3 Shape Matching
- •7.3.3.4 Evaluation
- •7.3.3.5 Complexity Analysis
- •7.3.4 Heat Kernel Signatures
- •7.3.4.1 Evaluation
- •7.3.4.2 Complexity Analysis
- •7.4 Research Challenges
- •7.5 Concluding Remarks
- •7.6 Further Reading
- •7.7 Questions
- •7.8 Exercises
- •References
- •8.1 Introduction
- •Chapter Outline
- •8.2 3D Face Scan Representation and Visualization
- •8.3 3D Face Datasets
- •8.3.1 FRGC v2 3D Face Dataset
- •8.3.2 The Bosphorus Dataset
- •8.4 3D Face Recognition Evaluation
- •8.4.1 Face Verification
- •8.4.2 Face Identification
- •8.5 Processing Stages in 3D Face Recognition
- •8.5.1 Face Detection and Segmentation
- •8.5.2 Removal of Spikes
- •8.5.3 Filling of Holes and Missing Data
- •8.5.4 Removal of Noise
- •8.5.5 Fiducial Point Localization and Pose Correction
- •8.5.6 Spatial Resampling
- •8.5.7 Feature Extraction on Facial Surfaces
- •8.5.8 Classifiers for 3D Face Matching
- •8.6 ICP-Based 3D Face Recognition
- •8.6.1 ICP Outline
- •8.6.2 A Critical Discussion of ICP
- •8.6.3 A Typical ICP-Based 3D Face Recognition Implementation
- •8.6.4 ICP Variants and Other Surface Registration Approaches
- •8.7 PCA-Based 3D Face Recognition
- •8.7.1 PCA System Training
- •8.7.2 PCA Training Using Singular Value Decomposition
- •8.7.3 PCA Testing
- •8.7.4 PCA Performance
- •8.8 LDA-Based 3D Face Recognition
- •8.8.1 Two-Class LDA
- •8.8.2 LDA with More than Two Classes
- •8.8.3 LDA in High Dimensional 3D Face Spaces
- •8.8.4 LDA Performance
- •8.9 Normals and Curvature in 3D Face Recognition
- •8.9.1 Computing Curvature on a 3D Face Scan
- •8.10 Recent Techniques in 3D Face Recognition
- •8.10.1 3D Face Recognition Using Annotated Face Models (AFM)
- •8.10.2 Local Feature-Based 3D Face Recognition
- •8.10.2.1 Keypoint Detection and Local Feature Matching
- •8.10.2.2 Other Local Feature-Based Methods
- •8.10.3 Expression Modeling for Invariant 3D Face Recognition
- •8.10.3.1 Other Expression Modeling Approaches
- •8.11 Research Challenges
- •8.12 Concluding Remarks
- •8.13 Further Reading
- •8.14 Questions
- •8.15 Exercises
- •References
- •9.1 Introduction
- •Chapter Outline
- •9.2 DEM Generation from Stereoscopic Imagery
- •9.2.1 Stereoscopic DEM Generation: Literature Review
- •9.2.2 Accuracy Evaluation of DEMs
- •9.2.3 An Example of DEM Generation from SPOT-5 Imagery
- •9.3 DEM Generation from InSAR
- •9.3.1 Techniques for DEM Generation from InSAR
- •9.3.1.1 Basic Principle of InSAR in Elevation Measurement
- •9.3.1.2 Processing Stages of DEM Generation from InSAR
- •The Branch-Cut Method of Phase Unwrapping
- •The Least Squares (LS) Method of Phase Unwrapping
- •9.3.2 Accuracy Analysis of DEMs Generated from InSAR
- •9.3.3 Examples of DEM Generation from InSAR
- •9.4 DEM Generation from LIDAR
- •9.4.1 LIDAR Data Acquisition
- •9.4.2 Accuracy, Error Types and Countermeasures
- •9.4.3 LIDAR Interpolation
- •9.4.4 LIDAR Filtering
- •9.4.5 DTM from Statistical Properties of the Point Cloud
- •9.5 Research Challenges
- •9.6 Concluding Remarks
- •9.7 Further Reading
- •9.8 Questions
- •9.9 Exercises
- •References
- •10.1 Introduction
- •10.1.1 Allometric Modeling of Biomass
- •10.1.2 Chapter Outline
- •10.2 Aerial Photo Mensuration
- •10.2.1 Principles of Aerial Photogrammetry
- •10.2.1.1 Geometric Basis of Photogrammetric Measurement
- •10.2.1.2 Ground Control and Direct Georeferencing
- •10.2.2 Tree Height Measurement Using Forest Photogrammetry
- •10.2.2.2 Automated Methods in Forest Photogrammetry
- •10.3 Airborne Laser Scanning
- •10.3.1 Principles of Airborne Laser Scanning
- •10.3.1.1 Lidar-Based Measurement of Terrain and Canopy Surfaces
- •10.3.2 Individual Tree-Level Measurement Using Lidar
- •10.3.2.1 Automated Individual Tree Measurement Using Lidar
- •10.3.3 Area-Based Approach to Estimating Biomass with Lidar
- •10.4 Future Developments
- •10.5 Concluding Remarks
- •10.6 Further Reading
- •10.7 Questions
- •References
- •11.1 Introduction
- •Chapter Outline
- •11.2 Volumetric Data Acquisition
- •11.2.1 Computed Tomography
- •11.2.1.1 Characteristics of 3D CT Data
- •11.2.2 Positron Emission Tomography (PET)
- •11.2.2.1 Characteristics of 3D PET Data
- •Relaxation
- •11.2.3.1 Characteristics of the 3D MRI Data
- •Image Quality and Artifacts
- •11.2.4 Summary
- •11.3 Surface Extraction and Volumetric Visualization
- •11.3.1 Surface Extraction
- •Example: Curvatures and Geometric Tools
- •11.3.2 Volume Rendering
- •11.3.3 Summary
- •11.4 Volumetric Image Registration
- •11.4.1 A Hierarchy of Transformations
- •11.4.1.1 Rigid Body Transformation
- •11.4.1.2 Similarity Transformations and Anisotropic Scaling
- •11.4.1.3 Affine Transformations
- •11.4.1.4 Perspective Transformations
- •11.4.1.5 Non-rigid Transformations
- •11.4.2 Points and Features Used for the Registration
- •11.4.2.1 Landmark Features
- •11.4.2.2 Surface-Based Registration
- •11.4.2.3 Intensity-Based Registration
- •11.4.3 Registration Optimization
- •11.4.3.1 Estimation of Registration Errors
- •11.4.4 Summary
- •11.5 Segmentation
- •11.5.1 Semi-automatic Methods
- •11.5.1.1 Thresholding
- •11.5.1.2 Region Growing
- •11.5.1.3 Deformable Models
- •Snakes
- •Balloons
- •11.5.2 Fully Automatic Methods
- •11.5.2.1 Atlas-Based Segmentation
- •11.5.2.2 Statistical Shape Modeling and Analysis
- •11.5.3 Summary
- •11.6 Diffusion Imaging: An Illustration of a Full Pipeline
- •11.6.1 From Scalar Images to Tensors
- •11.6.2 From Tensor Image to Information
- •11.6.3 Summary
- •11.7 Applications
- •11.7.1 Diagnosis and Morphometry
- •11.7.2 Simulation and Training
- •11.7.3 Surgical Planning and Guidance
- •11.7.4 Summary
- •11.8 Concluding Remarks
- •11.9 Research Challenges
- •11.10 Further Reading
- •Data Acquisition
- •Surface Extraction
- •Volume Registration
- •Segmentation
- •Diffusion Imaging
- •Software
- •11.11 Questions
- •11.12 Exercises
- •References
- •Index
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form. Two examples of fully automatic approaches are atlas-based segmentation and statistical shape models. No automatic methods are perfect, but given the laborious nature of manual segmentation and the pressures on clinicians’ time, segmentation is a very active research field.
11.6 Diffusion Imaging: An Illustration of a Full Pipeline
Although the physics of the 3D imaging modalities can be considered quite complex, involving different wavelengths, different fields, radioactivity and so on, it could be argued that the main output is fairly simple in terms of the techniques described in this chapter. Essentially, we get grayscale images. But this is misleading, medical imaging can also provide dynamic frame images from ultrasound, dynamic MRI or CT of the heart, contrast enhanced images showing the evolution of a contrast agent (MRI, CT, PET), flow images, and even tensor matrix images. All of this requires sophisticated post-processing and data analysis. Clearly, it is not possible to describe every technique, rather, in this section, we concentrate on tensor imaging. This relates to the most complex objects (tensors) and illustrates issues with other techniques, thus we use it as a case study of data processing in medical imaging. In particular, we use this to illustrate that nice images do not come for free, but only after processing steps that all require care. Specifically, we see the steps:
1.From raw images to the object which is a tensor field; this requires spatial alignment of different images, the input is a set of scalar images; the output consists of tensor images;
2.Data processing from the tensor field to a field of glyphs: input is a tensor image; the output is a visualization (glyphs), or vector images (showing principal direction of diffusion);
3.Information extraction from the glyph shapes, first pointwise: the input is a tensor image; the output consists of different quantitative scalar images;
4.We can also extract global information by connectivity: the input is a vector image, the output consists of a collection of 3D streamlines.
Processing requires different steps mentioned in previous sections, such as segmentation to delineate white matter, smoothing and thresholding to stop lines with unrealistic shapes and so on.
11.6.1 From Scalar Images to Tensors
This entire chapter tries to convey the message that the underlying physics of imaging cannot be ignored. Understanding where the images come from, and why they look like how they do is important. In some cases one might waste months of work trying to perform an imaging operation that would have just required a radiographer
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Fig. 11.19 Diffusion weighted images with corresponding gradient directions below. Note that differences are subtle—for each gradient direction (red line in lower row), the image has darker voxels when it contains structures forcing diffusion in that direction. For this reason, such images are rarely used directly, but replaced by the FA, MD, etc. below, which are orientation independent
to change scan settings or inject a different contrast agent. In diffusion MRI, we can and do manipulate magnetic field gradients in different directions, g. As a second order diffusion tensor can be represented by a symmetric 3 × 3 matrix, it contains six independent components. Thus, provided we have a model of the physics of how these gradients affect images, we should be able to reconstruct the matrix. The images in Fig. 11.19 show the diffusion weighted images corresponding to the gradient directions below.
One model, the default for diffusion imaging, is that each of these 6 images obeys the equation S(g) = S0 exp(−bgt Dg) where g = [x, y, z]T is a unit vector in the gradient direction, the scalar b comprises several acquisition parameters, such as gradient strength and duration, and D is the unknown tensor matrix with unknowns [Dxx , Dyy , Dzz, Dxy , Dxz, Dyz]. A more compact way to describe them is via a matrix, or tensor D, which is a table of numbers called diffusion coefficients:
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Fig. 11.20 Components of the diffusion tensor image. In other words, this is a matrix of images, or the image of a matrix (tensor). This provides very rich information, but MD shown in Fig. 11.21 and FA shown in Fig. 11.22 are easier to interpret directly, although they have lost information
By solving the equations using least squares, we can get tensor images as output, as shown in Fig. 11.20.
11.6.2 From Tensor Image to Information
Although, in theory, the diffusion tensor image in Fig. 11.20 contains all available information on diffusion, it is hard to interpret. For this reason, we construct images which display the information in a more intuitive way. The main tool for this is that the 3 × 3 diffusion tensor can be decomposed in orientation and anisotropy information. Shape information is encoded by three numbers describing the specific diffusivities in three orthogonal directions.
The Mean Diffusivity (MD) describes how ‘strong’ the diffusion is, or how unhindered, or how far in average water molecules diffuse. It is a mean, over all directions, and thus contains no directional information. To interpret it, higher MD, i.e. with a light to white contrast, correspond to regions where there are few barriers. For example, a region with higher MD than normal could describe a degradation of the
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Fig. 11.21 An image of Mean Diffusivity (MD). Bright areas correspond to voxels containing water able to diffuse relatively freely.
Dark regions contain more restricted water molecules
cellular structure. On the other hand, dark regions indicate that the water molecules find it harder to move around. Macroscopically it would be as if the medium has become more viscous, or microscopically cells could have swollen. In units, MD has the same units as the diffusion tensor, mm2/s, see Fig. 11.21.
The Fractional Anisotropy (FA) measures, as its name suggests, how anisotropic the diffusion is. It is an index, thus has no units, and ranges between 0–1, where 0 indicates fully isotropic diffusion and 1 indicates that diffusion happens in a single direction. The definition of FA is the standard deviation of the specific diffusivities, suitably normalized. Typically, cerebro-spinal fluid (CSF) has a low FA as it is essentially a liquid, so should not have a preferred direction. FA is high in white matter, in particular in the Corpus Callosum, or in any fibre like structure such as muscle fibres. (See Fig. 11.22a.)
Both MD and FA indicate nothing about direction. A useful way to do this is to use colors. The three components in RGB are matched to spatial directions. Note the blue large balls in the ‘smarties’ picture above a FA region. They indicate strong diffusion, but not very directional. The smaller ellipsoids show that in regions where diffusion is directional it is also much smaller (see Fig. 11.22b).
Fig. 11.22 An image of Fractional Anisotropy (FA). Bright regions indicate high anisotropy; note that this information complements the MD, also note that ventricles, for example, contain fully isotropic voxels (dark in FA) as they contain Cerebro-Spinal Fluid (CSF), but the corresponding voxels are bright in MD, as the water molecules are essentially free. Regions of the Corpus Callosum, on the other hand, contain lots of white matter fibres pointing in a single direction thus they appear dark in MD, as molecules cannot move much, but bright in FA as, when they move, they do so preferably in a single direction. (b) Colored FA, with colors indicating direction