CN107463708B - A Method for Joint Visualization of UKF Fiber Tracking Data - Google Patents

Abstract

The invention relates to a method for joint visualization of UKF fiber tracking data, which belongs to the field of fiber tracking data visualization in magnetic resonance imaging. The method combines the fiber tracking data (comprising fiber track data, diffusion tensor data and tensor measurement data) obtained by the UKF fiber tracking algorithm into a unified framework and performs visual processing simultaneously; the present invention establishes a new prism Volume tensor icon, which visualizes the diffusion tensor data through the shape, size and orientation of the prism tensor icon, and at the same time visualizes multiple tensor measurement data through the color of the prism surface, and then the prism tensor Quantity icons are concatenated on the fiber trajectory curve, thereby enabling joint visualization of UKF fiber tracking data in the same view. The problem of switching and alignment between different data views is avoided, the efficiency of data visualization analysis can be effectively improved, and the analysis depth and breadth of UKF fiber tracking data can be improved.

Description

A Method for Joint Visualization of UKF Fiber Tracking Data

technical field

The invention relates to a multi-attribute data visualization method, in particular to a visualization method for nerve fiber tracking data including one tensor attribute and multiple scalar attributes, and belongs to the field of fiber tracking data visualization in magnetic resonance imaging.

Background technique

Tractography based on magnetic resonance diffusion tensor imaging (DTI) and diffusion weighted imaging (DWI) is an important method for exploring the structure of living nerve fibers in the field of neuroscience and clinical application , It is widely used in the study of central nervous system histomorphology and pathology. For this reason, researchers in related fields have proposed many fiber tracking algorithms, such as the streamline tracking algorithm (S.Mori, B.Crain, V.Chacko, P.Van Zijl. Three-dimensional tracking of axonal projections in the brain by magnetic resonance imaging .Annals of Neurology,45(2),265-269,1999) and probabilistic pursuit algorithm (J.Berman,S.Chung,P.Mukherjee,C.Hess,E.Han,R.Henry.Probabilistic streamline q-ball tractography using the residual bootstrap.NeuroImage, 39(1), 215-222, 2008) and so on; among them, the streamline tracking algorithm is more common, but the streamline tracking algorithm cannot handle complex situations such as fiber crossing and fiber bifurcation (D.C.Alexander, G . Barker, S. Arridge. Detection and modeling of non-Gaussian apparent diffusion coefficient profiles in human brain data. Magnetic Resonance in Medicine, 48(2), 331-340, 2002).

In order to deal with fiber crossing and bifurcation, Malcolm et al. proposed a fiber tracking algorithm based on multi-tensor model and UKF (unscented Kalman filter) technology (J.G.Malcolm, M.E.Shenton, Y.Rathi.Filtered multi-tensor tractography .IEEE Trans.on Medical Imaging,29(9),1664–1675,2010), referred to as the UKF fiber tracking algorithm. Recently, the research work of Chen et al. showed that the UKF fiber tracking algorithm has obvious advantages in dealing with complex situations such as crossing fibers and edema regions (Z. Chen, Y. Tie, O. Olubiyi, L. Rigolo, A. Mehrtash, I. Norton, O. Pasternak, Y. Rathi, A.J. Golby, L.J. O'Donnell. Reconstruction of the arcuate fasciculus for surgical planning in the setting of peritumoral edema using two-tensor unscented Kalman filter tractography. NeuroImage: Clinical, 7, 815-822, 2015).

The result obtained by the fiber tracking algorithm is a data set describing the fiber structure and its attribute characteristics, which usually consists of three parts: (1) fiber tracts, (2) the diffusion tensors corresponding to the fiber tracts, (3) and various derived measures derived from the diffusion tensor. In order to analyze and understand fiber tracking results, this data needs to be visualized.

Currently, fiber trajectories are generally visualized as curves in three-dimensional space. Diffusion tensors are represented as geometric symbols (glyphs) in three-dimensional space. The common diffusion tensor symbol is ellipsoids, because the 6 degrees of freedom of ellipsoids in size, shape and direction can be compared with the 6 parameters describing the diffusion tensor (eigenvectors e ₁ , e ₂ and e ₃ , and the corresponding eigenvalues λ ₁ , λ ₂ and λ ₃ ) in one-to-one correspondence (C. Pierpaoli, PJ Basser. Toward a quantitative assessment of diffusion anisotropy. Magnetic Resonance in Medicine, 36(6), 893-906, 1996). Fiber trajectories and diffusion tensors can be visualized individually or combined. In the fiber tracking algorithm, the diffusion tensor data is generally stored as the attribute data of the fiber track sample points. Therefore, during visualization, the icon representing the diffusion tensor is usually attached to the curve representing the fiber trajectory, so as to combine local attributes with global features for association analysis.

However, from the perspective of data analysis, it is not enough to only visualize the fiber trajectory and the diffusion tensor, because the measurement data derived from the diffusion tensor, such as Volume Ratio (VR), Mean Diffusivity (MD), FractionalAnisotropy (FA), etc., It is also an important way to comprehensively analyze and understand the results of fiber tracking, and some are even more important than fiber track and diffusion tensor itself. These measurement data are generally scalars, which measure the attribute characteristics of the diffusion tensor from different angles. For these tensor measurement data, the commonly used visualization methods are: (1) Separate visualization as independent data, that is, different views are established for different measurement data. This method is simple and easy to implement, but during comprehensive analysis, alignment and switching operations between different views are required. This is neither convenient nor accurate, and seriously affects the efficiency of data analysis. (2) As additional data of the fiber trajectory or diffusion tensor, it is combined with it and visualized at the same time, for example, the FA value is mapped to the tensor icon or the color of the fiber trajectory curve. In this way, the fiber trajectory, diffusion tensor and measurement data can be combined for visual display, which improves the efficiency of data analysis. Unfortunately, the additional data that current tensor graphs or fiber trajectories can carry is extremely limited. For example, the fiber track can only express one kind of measurement data through the color attribute; the tensor ellipsoid also has only the color attribute that can be used to encode one kind of additional data. However, in practical applications, there are as many as ten kinds of measurement data derived from the diffusion tensor. This situation severely limits the depth and breadth of analysis of fiber tracking data.

As one of the fiber tracking algorithms, the UKF fiber tracking algorithm also faces the same problem, that is, how to perform efficient and comprehensive visual analysis on fiber tracking data. In order to comprehensively analyze UKF fiber tracking data and gain an in-depth understanding of the underlying fiber structure and histomorphological characteristics, it is necessary to develop a new visualization method for UKF fiber tracking data.

In practical applications, the UKF fiber tracking algorithm generally uses a simplified double tensor model, which assumes that the diffusion-weighted imaging signal

where _s0 is the reference signal, b is the signal acquisition constant, u _i is the gradient vector, For the transpose of u _i , D1 and D2 are simplified _{second -} order tensors. For regular second-order tensors, there are generally three distinct eigenvalues λ ₁ , λ ₂ and λ ₃ . But D ₁ and D ₂ here are simplified tensors with only two distinct eigenvalues λ ₁ and λ ₂ (another λ ₃ =λ ₂ ). The result is: D ₁ can be described by the main eigenvector v ₁ and the main eigenvalue λ ₁₁ and the secondary eigenvalue λ ₁₂ , D ₂ can be described by the main eigenvector v ₂ and the main eigenvalue λ ₂₁ and the secondary eigenvalue λ ₂₂ ( JG Malcolm, ME Shenton, Y. Rathi. Filtered multi-tensortractography. IEEE Trans. on Medical Imaging, 29(9), 1664–1675, 2010).

Based on the above model, the fiber tracking data output by the UKF fiber tracking algorithm is: a fiber track data set L={l _k }, where each fiber track l _k is composed of a series of sampling points, that is, l _k ={p _j }, Each sample point p _j has two associated diffusion tensors D ₁ and D ₂ , and each diffusion tensor can derive the following measurement data: Mean Diffusivity (MD), Apparent Diffusion Coefficient (ADC), Volume Ratio (VR) , Fractional Anisotropy (FA), Rational Anisotropy (RA), General Anisotropy (GA), Linear anisotropy (CL), Planar anisotropy (CP), Spherical anisotropy (CS) and Trace (TR), etc. (A.Vilanova, S. Zhang, G. Kindlmann, D. Laidlaw. An Introduction to Visualization of DiffusionTensor Imaging and Its Applications. In: J. Weickert, H. Hagen (eds.), Visualization and Processing of Tensor Fields, pp.121-153, Springer, Heidelberg ,2006).

Different from the general fiber tracking data, the diffusion tensors D ₁ and D ₂ in the UKF fiber tracking data have only two different eigenvalues: the main eigenvalue and the secondary eigenvalue. Based on this particularity, the present invention provides a method for joint visualization of fiber trajectories, diffusion tensor and metric data in UKF fiber tracking data.

Contents of the invention

The purpose of the present invention is to provide a method for joint visualization of UKF fiber tracking data, which integrates the fiber tracking data (including fiber track data, diffusion tensor data and tensor measurement data) obtained by the UKF fiber tracking algorithm into one Visualization processing is performed simultaneously under a unified framework to improve the efficiency of data analysis as well as the depth and breadth of data analysis.

The purpose of the present invention is achieved through the following technical solutions.

A method for joint visualization of UKF fiber tracking data comprising the steps of:

Step 1. Input the UKF fiber tracking data set, which includes fiber track data, diffusion tensor data and tensor measurement data.

Step 2. Select 3 or more tensor measures that need to be visualized from the tensor measure data set, and mark the selected tensor measures as m ₁ , m ₂ , ..., m _N , where N is The number of measures selected.

Step 3, according to the parameter N, establish the basic tensor icon G, the method is: construct a regular prism with a height of 1, the number of sides is N, and the diameter of the top circumscribed circle is 1; the center of the prism is placed in Cartesian The origin of the coordinate system XYZ, so that its axis coincides with the Z axis, and the normal vector of one of the sides of the rectangle is consistent with the positive direction of the X axis. Without loss of generality, let the side of the rectangle whose normal vector coincides with the positive direction of the X axis be f ₁ , and the other sides are marked as f ₂ , f ₃ , ..., f _N in turn, where the order of marking can be clockwise or counterclockwise, Other custom orders are also possible. At the same time, on the end face of the prism pointed by the positive direction of the Z axis, connect the center point with each apex, divide the end face into N triangles, and follow the same principle as the side faces f ₁ , f ₂ ,..., f _N Sequentially label the triangles as t ₁ , t ₂ , . . . , t _N .

Step 4: Establish a color mapping scheme for the tensor measurement data m ₁ , m ₂ , . . . , m _N , so as to map the measurement data into specific colors. The color mapping scheme may be a rule-based color mapping scheme or a color lookup table-based color mapping scheme, but is not limited thereto. The preferred color mapping scheme of the present invention is based on rules, and the specific mapping rules are: based on the HSV color model, the color saturation of _each measurement is set to be fully saturated, and then respectively specify m1, _m2 , ..., _mN Different color hues, and finally make the color brightness of each measure proportional to the specific value of the measure.

Step 5: Select part or all of the fiber trajectories from the input data set, and perform steps 6 and 7 in sequence for each selected fiber trajectories F. The method for selecting fiber tracks is (but not limited to): select fiber tracks with specified numbers at equal intervals according to the serial numbers of the fiber tracks from small to large, and the serial number intervals are set according to actual needs.

Step 6. According to the coordinate values of the sampling points of the fiber trajectory F, draw a three-dimensional curve representing the fiber trajectory, wherein the color of the curve can be a certain fixed color, or a color that can change with the direction of the tangent of the curve, depending on the actual needs depends.

Step 7. According to a certain sampling interval, a series of sampling points are sequentially selected from the fiber track F, and for each sampling point p, step 8 and step 9 are executed in sequence. The sampling interval of the sampling points is preset and can be adjusted according to actual needs.

Step 8. Visualize the diffusion tensor data D ₁ of sampling point p and the derived tensor measurement data, the method is: copy the basic tensor icon G as G′, and then translate G′ to the position of point p, And carry out rotation and telescopic transformation, so that the positive direction of the Z axis of G′ is consistent with the direction of the main eigenvector v ₁ of D ₁ , the height is equal to the main eigenvalue λ ₁₁ of D ₁ , and the diameter of the circumscribed circle at the top is equal to the secondary eigenvalue λ of D _{1 12} ; then according to the color mapping scheme of measures m ₁ , m ₂ , ..., m _N and the specific values of D ₁ on each measure, the color values c ₁ , c ₂ , ..., c corresponding to each measure data of D ₁ are obtained _N ; use color values c ₁ , c ₂ , ..., c _N to color the sides f ₁ , f ₂ , ..., f _N and the top triangles t ₁ , t ₂ , ..., t _N of G′ in sequence.

Step 9. Visualize the diffusion tensor data D ₂ of sampling point p and the derived tensor measurement data, the method is: copy the basic tensor icon G as G′, and then translate G′ to the position of point p, And carry out rotation and telescopic transformation, so that the positive direction of the Z axis of G′ is consistent with the direction of the main eigenvector v ₂ of D ₂ , the height is equal to the main eigenvalue λ ₂₁ of D ₂ , and the diameter of the circumscribed circle at the top is equal to the secondary eigenvalue λ of D _{2 22} ; then according to the color mapping scheme of measures m ₁ , m ₂ , ..., m _N and the specific values of D ₂ on each measure, the color values c ₁ , c ₂ , ..., c corresponding to each measure data of D ₂ are obtained _N ; use color values c ₁ , c ₂ , ..., c _N to color the sides f ₁ , f ₂ , ..., f _N and the top triangles t ₁ , t ₂ , ..., t _N of G′ in sequence.

Beneficial effect

A method of joint visualization of UKF fiber tracking data described in the present invention is actually a new prism tensor icon invented to visualize the dispersion through the shape, size and orientation of the prism tensor icon Tensor data, at the same time, visualize multiple tensor measurement data through the color of the prism surface, and then concatenate the prism tensor icon on the fiber trajectory curve, thereby realizing UKF fiber tracking data (including fiber trajectory data, diffusion tensor data, and tensor measure data) in the same view.

Compared with the existing fiber tracking data visualization method, the method of the present invention has the following advantages:

(1) The method of the present invention can simultaneously visualize UKF fiber tracking data (including fiber track data, diffusion tensor data and tensor measure data) in the same view.

(2) The prism tensor icon in the method of the present invention can simultaneously visualize multiple tensor measurement data through color mapping.

(3) The method of the present invention visualizes a variety of UKF fiber tracking data with different properties in the same view, avoids switching and alignment problems between different data views, and can effectively improve the efficiency of data visualization analysis.

(4) The method of the present invention can improve the analysis depth and breadth of UKF fiber tracking data through joint visualization of different data.

Description of drawings

Figure 1 Basic tensor icon;

Figure 2 Basic tensor notation in color;

Drawing results of fiber track 1 in Fig. 3;

Fig. 4 Joint visualization results of fiber track 1;

Fig.5 Visualization results of diffusion tensor data D ₁ and its measurement data on fiber track 1;

Fig. 6 Visualization results of diffusion tensor data D ₂ and its measurement data on fiber track 1;

Figure 7 is the final visualization result of the implementation example.

Detailed ways

The present invention will be described in detail below in conjunction with the accompanying drawings and embodiments.

Example

A method for joint visualization of UKF fiber tracking data comprising the steps of:

Step 1. Input the UKF fiber tracking dataset, which includes fiber track data, diffusion tensor data and tensor measure data.

The present invention uses a simple corpus callosum nerve fiber data set as an implementation example. The data set is calculated according to the UKF fiber tracing algorithm, including 7 nerve fiber trajectories passing through the corpus callosum of the brain, and the diffusion tensor data and tensor measurement data associated with the sample points on these fiber trajectories, where each sample Each point has two simplified diffusion tensors D ₁ and D ₂ , and the tensor measurement data associated with D ₁ and D ₂ are: Fractional Anisotropy (FA), Trace (TR), Major Eigenvalue (Major), Minor Eigenvalue (Minor), Volume Ratio (VR), General Anisotropy (GA), Linear anisotropy (CL), Planar anisotropy (CP), Spherical anisotropy (CS), etc.

Step 2. From the tensor measurement data set, select 6 tensor measurements that need to be visualized: FractionalAnisotropy (FA), Trace (TR), Minor Eigenvalue (Minor), Volume Ratio (VR), Linearanisotropy (CL), Spherical anisotropy (CS), these measures are denoted m ₁ , m ₂ , m ₃ , m ₄ , m ₅ , m ₆ , respectively.

Step 3. Based on the tensor measure determined in step 2, a basic tensor graph G is established.

The specific method is: construct a regular prism with a height of 1, the number of sides is 6, and the diameter of the circumscribed circle at the top is 1; the center of the prism is placed at the origin of the Cartesian coordinate system XYZ, so that its axis coincides with the Z axis, And the normal vector of one of the sides of the rectangle is consistent with the positive direction of the X axis. Without loss of generality, let the side of the rectangle whose normal vector coincides with the positive direction of the X axis be f ₁ , and the other sides are marked as f ₂ , f ₃ , . . . , f ₆ in counterclockwise order. At the same time, on the end face of the prism pointed by the positive direction of the Z axis, connect the central point with each apex, divide the end face into six triangles, and follow the same principle as the side f ₁ , f ₂ ,..., f ₆ In order, the six triangles are marked as t ₁ , t ₂ , . . . , t ₆ in sequence.

The basic tensor graph G established according to the above method is shown in Figure 1.

Step 4. Establish a color mapping scheme for the tensor measurement data m ₁ , m ₂ , . . . , m ₆ so as to map the measurement data into specific colors.

The preferred color mapping scheme in this implementation example is: based on the HSV color model, set the color saturation of each measure to be fully saturated, and then specify a fixed color hue value of 0° for the measures m ₁ , m ₂ , ..., m ₆ , 60°, 120°, 180°, 240°, 300°, and finally make the color brightness of each measure proportional to the specific value of the measure.

According to the above color mapping scheme, when the specific values of each measure take the maximum value, the colors corresponding to each measure m ₁ , m ₂ , ..., m ₆ are: red, yellow, green, cyan, blue, purple . Use the colors corresponding to the measures m ₁ , m ₂ , ..., m ₆ to color f ₁ , f ₂ , ..., f ₆ and t ₁ , t ₂ , ..., t ₆ of the basic tensor graph G in sequence, and we can get Colored basic tensor icons as shown in Figure 2.

Step 5. Select part or all of the fiber trajectories from the input data set, and perform steps 6 and 7 in sequence for each selected fiber trajectories F.

Without loss of generality, in this implementation example, all fiber trajectories are sequentially selected according to the numbers of the fiber trajectories, and then step 6 and step 7 are sequentially performed for each fiber trajectory F.

Step 6. Draw a three-dimensional curve representing the fiber trajectory according to the coordinate values of the sampling points of the fiber trajectory F. The color of the curve can be a certain fixed color or a color that can change with the change of the tangent direction of the curve, depending on actual needs.

In this implementation example, the color of the fiber track is set to black. For the fiber track numbered 1, the drawing results are shown in Figure 3.

Step 7. On the fiber trajectory F, select a sampling point at intervals of 50 sampling points to obtain a series of sampling points, and perform steps 8 and 9 in sequence for each sampling point p.

In this implementation example, the fiber track numbered 1 is formed by sequentially connecting 535 sample points, and a sampling point is selected at intervals of 50 sample points, and a total of 10 sampling points are selected.

Step 8. Visualize the diffusion tensor data D ₁ of the sampling point p and the derived tensor measure data.

The specific method is: copy the basic tensor icon G as G', then translate G' to the position of point p, and perform rotation and stretch transformation, so that the Z axis of G' is positively aligned with the main eigenvector v of D _{1 1} direction is the same, the height is equal to the main eigenvalue λ ₁₁ of D ₁ , and the diameter of the top circumscribed circle is equal to the minor eigenvalue λ ₁₂ of D ₁ ; then according to the color mapping scheme of measures m ₁ , m ₂ ,..., m ₆ and D ₁ in The specific numerical values on each measure can obtain the color values c ₁ , c ₂ , ..., c ₆ corresponding to each measure data of D ₁ ; use the color values c ₁ , c ₂ , ..., c ₆ to compare the side f ₁ of G′ in turn , f ₂ , ..., f ₆ and top triangles t ₁ , t ₂ , ..., t ₆ are colored.

Step 9. Visualize the diffusion tensor data D ₂ of the sampling point p and the derived tensor measure data.

The specific method is: copy the basic tensor icon G as G', then translate G' to the position of point p, and perform rotation and stretch transformation, so that the Z axis of G' is positively aligned with the main eigenvector v of D _{2 2} have the same direction, the height is equal to the main eigenvalue λ ₂₁ of D ₂ , the diameter of the top circumscribed circle is equal to the secondary eigenvalue λ ₂₂ of D _{2 ;} then according to the color mapping scheme of the measures m ₁ , m ₂ ,..., m ₆ and the The specific numerical values on each measure can obtain the color values c ₁ , c ₂ , ..., c ₆ corresponding to each measure data of D ₂ ; use the color values c ₁ , c ₂ , ..., c ₆ to sequentially compare the side f of G′ ₁ , f ₂ , . . . , f ₆ and top triangles t ₁ , t ₂ , . . . , t ₆ are colored.

For the fiber track numbered 1, repeat steps 8 to 9 until each sampling point selected in step 7 is processed, and the joint visualization result shown in Figure 4 can be obtained. The visualization result in Fig. 4 includes the visualization result of the diffusion tensor data D ₁ and its measurement data in step 8 (as shown in Fig. 5), and the visualization result of the diffusion tensor data D ₂ and its measurement data in step 9 ( As shown in Figure 6). Through comparative analysis, it can be seen that the relationship between tensor D ₁ and D ₂ is clear at a glance; the direction of tensor D ₁ (the direction of the main eigenvalue) is consistent with the tangent direction of the fiber trajectory, while tensor D ₂ is different The angles intersect the fiber tracks; each tensor glyph varies in size, orientation, shape, and color, revealing attribute characteristics that vary at different locations on the fiber track.

For the fiber track data set used in this implementation example, since all fiber tracks are selected in step 5, after performing step 6 and step 7 (including step 8 and step 9) in turn for each fiber track, it will be obtained The final visualization result of the implementation example is shown in Figure 7.

The above steps illustrate the whole process of the method for joint visualization of UKF fiber tracking data according to the present invention.

It should be understood that this embodiment is only a specific example of the implementation of the present invention, and should not limit the protection scope of the present invention. Without departing from the spirit and scope of the present invention, equivalent modifications or changes to the above contents shall be included in the scope of protection claimed by the present invention.

Claims (1)

1. a kind of pair of UKF Fiber track data carry out joint visualization method, which comprises the following steps:

Step 1: input UKF Fiber track data set, estimates including fiber track data, dispersion tensor data and tensor Data；

Step 2: estimating in data set from tensor, chooses 3 or 3 or more and visual tensor is needed to estimate, that will be chosen Measurement degree is respectively labeled as m₁、m₂、…、m_N, wherein N be choose estimate number；

Step 3: establishing basic tensor icon G according to parameter N；

Method is: construction one height be 1, the regular prism that side quantity is N, top circumscribed circle diameter is 1；By the prism Center be placed in the origin of cartesian coordinate system XYZ, be overlapped it axially with Z axis, and the normal vector of one of rectangle sides It is positive consistent with X-axis；Enabling normal vector and the positive consistent rectangle sides of X-axis is f₁, other sides are successively labeled as f₂、f₃、…、 f_N, wherein the sequence marked can be customized；At the same time, the prism end face positive signified in Z axis, central point with it is each Vertex connects, which is split into N number of triangle, and according to side f₁、f₂、…、f_NConsistent sequence is by triangle Successively it is labeled as t₁、t₂、…、t_N；

Step 4: establishing tensor estimates data m₁、m₂、…、m_NColor mapping scheme, be mapped as specifically so that data will be estimated Color；

Step 5: being concentrated from input data, selected part or whole fiber tracks, the fiber track F chosen for every, successively Execute step 6 and step 7；

Step 6: drawing the three-dimensional curve for indicating fiber track according to the coordinate value of the sampled point of fiber track F；

Step 7: a series of sampled points are successively chosen from fiber track F according to certain sampling interval, for each sampling Point p successively executes step 8 and step 9；Wherein the sampling interval of sampled point is previously set；

Step 8: the dispersion tensor data D of visualization sampled point p₁And its derived tensor estimates data, method is: will open substantially Spirogram symbol G copies as G ', G ' is then moved to the position of point p, and carry out rotation and stretching, make the Z axis forward direction of G ' with D₁Main feature vector v₁Direction is consistent, is highly equal to D₁Dominant eigenvalue λ₁₁, top circumscribed circle diameter be equal to D₁Sub-eigenvalue λ₁₂；Then basis estimates m₁、m₂、…、m_NColor mapping scheme and D₁It is each estimate on specific value, obtain D₁'s It is each to estimate the corresponding color value c of data₁、c₂、…、c_N；Using color value c₁、c₂、…、c_NSuccessively to the side f of G '₁、f₂、…、 f_NWith tip triangular shape t₁、t₂、…、t_NColoring；

Step 9: the dispersion tensor data D of visualization sampled point p₂And its derived tensor estimates data, method is: will open substantially Spirogram symbol G copies as G ', G ' is then moved to the position of point p, and carry out rotation and stretching, make the Z axis forward direction of G ' with D₂Main feature vector v₂Direction is consistent, is highly equal to D₂Dominant eigenvalue λ₂₁, top circumscribed circle diameter be equal to D₂Sub-eigenvalue λ₂₂；Then basis estimates m₁、m₂、…、m_NColor mapping scheme and D₂It is each estimate on specific value, obtain D₂'s It is each to estimate the corresponding color value c of data₁、c₂、…、c_N；Using color value c₁、c₂、…、c_NSuccessively to the side f of G '₁、f₂、…、 f_NWith tip triangular shape t₁、t₂、…、t_NColoring.