AVIZO曲率计算

AVIZO曲率计算

参考视频教程：

通过分形维数和表面曲率衡量样品的粗糙度

Avizo中利用auto refresh功能半自动化进行REV（体元表征）

计算流程

原始数据
↓
数据体分割提取
↓
分割后感兴趣的区域（孔隙）
↓
生成表面网格
↓
曲率计算

如果要实现批量计算需要写脚本代码实现

Image Curvature 3D-原理

3维会生成两个曲率：
第一个默认是高斯
第二个默认是平均

2维生成一个曲率→高斯

Description:

This module computes for each voxel of an input image local texture curvature. Computed curvature(s) takes voxel size into account.
More informations on these algorithms can be found in:

该模块计算输入图像中每个体素的局部纹理曲率。计算出的曲率考虑了体素的大小。
有关这些算法的更多信息，请参见：

该模块接收两个或多个输入数据对象（例如两个表面或两个四面体网格），并通过线性插值计算输出对象。插值过程可以在线性插值顶点位置的同时插值附加到表面或网格上的数据字段。

• On Curvature Estimation of ISO Surfaces in 3D Gray-Value Images and the Computation of Shape Descriptors, Bernd Rieger, Frederik J. Timmermans, Lucas J. van Vliet and Piet W. Verbeek, IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 26, NO. 8, AUGUST 2004.

This module compute local gray level 2D curvature in image. For each pixel, the curvature parameter K is linked to radius of curvature R by formula:
该模块计算图像中的局部灰度二维曲率。对于每个像素，曲率参数 K 与曲率半径 R 的关系如下：

$$k = \frac{1}{R}$$

This module computes local gray level 3D curvatures in image. Each voxel is then associated with two principal curvatures: Kmax and Kmin. The Kmax stands for maximal local curvature whereas stands for minimal curvature.

此模块计算图像中的局部灰度 3D 曲率。每个体素随后与两个主曲率相关联：Kmax 和 Kmin。Kmax 代表最大局部曲率，而 Kmin 代表最小曲率。

Two important indicators can be extracted from these values:
从这些数值中可以提取两个重要指标：

Gaussian curvature:

$$K = K_{max} × K_{min}$$

Mean curvature:

$$K = \frac{1}{2} (K_{max} + K_{min})$$

Gaussian curvature can be used to characterize local surface comportment:

• If Gaussian curvature is positive, both principal curvatures have same sign and local surface will be dome like (which mean that it is locally lying on one side of its tangent plane. In this case, a positive curvature stand for convex surface whereas a negative value stand for a concave surface.
• If Gaussian curvature is negative, principal curvatures have opposite signs and local surface will be saddle shaped (see Figure 1).
• If Gaussian curvature is null, surface will be locally parabolic like. If Kmax = 0, surface profil will locally be valley like. If Kmin = 0, surface profil will locally be crest like.
  See also: Flow Inpainting.

高斯曲率可用于表征局部表面行为：

• 如果高斯曲率为正，则两个主曲率具有相同的符号，局部表面将呈圆顶状（这意味着它局部位于其切平面的一侧。在这种情况下，正曲率代表凸面，而负值代表凹面。
• 如果高斯曲率为负，则主曲率具有相反的符号，局部表面将呈马鞍形（见上图 ）。
• 如果高斯曲率为零，则表面将局部呈抛物线状。如果Kmax = 0 ，表面轮廓将局部呈山谷状。如果 Kmin = 0 ，表面轮廓将局部呈波峰状。
  另请参阅：流修复。

Connections:

Input Image [required] 输入图像 [必需]

The image to be analyzed. Supported types include: grayscale image (Uniform Scalar Field), binary (Uniform Label Field with 2 labels) and label (Uniform Label Field) images.

要分析的图像。支持的类型包括：灰度图像（统一标量场）、二进制（具有 2 个标签的统一标签场）和标签（统一标签场）图像。

Input Image Mask [optional] 输入图像蒙版 [可选]

The masking image. Supported types include: binary images (Uniform Label Field with 2 labels).

蒙版图像。支持的类型包括：二进制图像（具有 2 个标签的统一标签场）。

Ports:

Interpretation 解释

This port specifies whether the input will be interpreted as a 3D volume or a stack of 2D images for processing.
“3D”: the module configuration is set to 3D. The image will be processed as a whole in 3D.
“XY planes”: the module configuration is set to 2D. The image will be processed slice per slice.

此端口指定输入是否将被解释为 3D 体积或 2D 图像堆栈以供处理。
“3D”：模块配置设置为 3D。图像将以 3D 形式整体处理。
“XY 平面”：模块配置设置为 2D。图像将逐片处理。

Standard Deviation

This port defines the standard deviation used for Gaussian smoothing of image Tensor.
该端口定义用于图像张量高斯平滑的标准偏差。

Curvature Integrals -原理

Description:

For an introduction, see section Analysis.
有关介绍，请参阅分析部分。

This module computes the integral of mean curvature and integral of total curvature of objects in a binary image. Intuitively, “curvature” is the amount by which a geometric object deviates from being “flat”.
此模块计算二值图像中物体的平均曲率积分和总曲率积分。直观地说，“曲率”是几何物体偏离“平面”的量。

This module computes a local measure. It is obtained as the sum of measures in local 2x2x2 neighborhoods (a cube), for 13 planes associated with different normal directions and hitting three or four vertices of the cells (in the cubical lattice).
此模块计算局部测量值。它是局部 2x2x2 邻域（立方体）中测量值的总和，针对与不同法线方向相关的 13 个平面，并击中立方体格子中的三个或四个顶点。

In the case of very elongated objects (needles or fibers) the integral of mean curvature M can be used to measure the length L of the object: L= M / π
对于非常细长的物体（针或纤维），平均曲率积分 M 可用于测量物体的长度 L：L= M / π

For convex object, the integral of mean curvature M is (up to a constant) equivalent to the mean diameter, i.e.
对于凸物体，平均曲率积分 M（最多一个常数）等于平均直径，即

$$M = 2 \pi d where d = \frac{1}{13} (\sum^1_{i=0} 3 d_i)$$

The Euler number and the Integral of Total Curvature carry the same information about the object. They differ by the constant factor 4π. If we consider a set X of the 3-dimensional space and X（X） being its Euler number then the integral total curvature of X will be: K(X) = 4πX(X)
欧拉数和总曲率积分都包含相同的物体信息，它们相差一个常数因子 4π。如果我们考虑三维空间中的集合 X，X（X）是它的欧拉数，那么 X 的积分总曲率将是：K(X) = 4πX(X)

See Also: Euler Number

For more information on Integral Curvatures you can refer to

• C .Lang, J. Ohser, R.Hilfer (1999) On the Analysis of Spatial Binary Images

See also: Area 3D, Euler Number 3D, Fractal Dimension, Moments of Inertia 3D, Volume Fraction.

Connections:

Input Image [required]
The image to be analyzed. Supported types include: binary images (Uniform Label Field with 2 labels).
需要分析的图像。支持的类型包括：二值图像（具有 2 个标签的统一标签字段）。

Ports:

Interpretation

This port specifies whether the input will be interpreted as a 3D volume or a stack of 2D images for processing. The port is grayed out if alternate interpretation is not available.
此端口指定输入是否将被解释为 3D 体积或 2D 图像堆栈以供处理。如果没有其他解释，则端口将变灰。

curvature-原理

需要生成表面

Description:

This module computes curvature information for a discrete triangular surface of type Surface.
该模块计算表面类型的离散三角表面的曲率信息。

Either the maximum principal curvature value, the reciprocal curvature value, or the direction of maximum principal curvature can be computed. The algorithm works by approximating the surface locally by a quadric form. The eigenvalues and eigenvectors of the quadric form correspond to the principal curvature values and to the directions of principal curvature. Note that the algorithm does not produce meaningful results near locations where the input surface is not topologically flat, i.e., where it has non-manifold structure.

可以计算最大主曲率值、倒数曲率值或最大主曲率方向。该算法通过用二次型局部近似表面来工作。二次型的特征值和特征向量对应于主曲率值和主曲率方向。请注意，该算法在输入表面不是拓扑平坦的位置附近（即具有非流形结构的位置）不会产生有意义的结果。

Press the Apply button to start the computation.

按下“应用”按钮开始计算。

Connections:

Data [required]
The surface for which curvature information should be computed.

数据 [必需]
应计算曲率信息的表面。

Ports:

Method

Radio box allowing the user to select between two different computational algorithms. Choice on triangles produces a surface field with curvature values or curvature vectors being defined on the surface’s triangles. Alternatively, by selecting on vertices a surface field with data being defined on the vertices can be generated.

单选框允许用户在两种不同的计算算法之间进行选择。选择三角形on triangles会产生一个表面场，其曲率值或曲率向量在表面的三角形上定义。或者，选择顶点on vertices可以生成一个表面场，其数据在顶点上定义。

Parameters

The first input, denoted nLayers, determines which triangles are considered to be neighbors of a given triangle and which points are considered to be neighbors of a given point. If the value of this input is 1, then only triangles sharing a common edge with a given triangle are considered to be neighbors of this triangle and only points directly connected to a given point are considered to be neighbors of this point. For larger values of nLayers successively larger neighborhoods are taken into account.

第一个输入表示为 nLayers，它确定哪些三角形被视为给定三角形的邻居，哪些点被视为给定点的邻居。如果此输入的值为 1，则只有与给定三角形共享公共边的三角形才被视为此三角形的邻居，只有直接连接到给定点的点才被视为此点的邻居。对于较大的 nLayers 值，会依次考虑较大的邻域。

The second input, denoted nAverage, determines how many times the initial curvature values computed for a triangle or for a point are being averaged with the curvature values of direct (first-order) neighbor triangles or points. The larger the value of nAverage the smoother the curvature data being obtained. Note that averaging only applies to the scalar curvature values, not to the directional curvature vectors which are computed when port output is set to max direction.

第二个输入表示为 nAverage，它决定了为三角形或点计算的初始曲率值与直接（一阶）相邻三角形或点的曲率值取平均值的次数。nAverage 的值越大，获得的曲率数据越平滑。请注意，平均仅适用于标量曲率值，而不适用于当端口输出设置为最大方向时计算的方向曲率向量。

Output

This menu controls the output of the curvature module.
此菜单控制曲率模块的输出。

Max curvature

a surface scalar field is generated containing the maximal principal curvature of a triangle or of a point.
生成一个包含三角形或点的最大主曲率的表面标量场。

1/Max curvature

the curvature values are inverted. In this case, the output values have the dimension of a length, indicating the radius of a sphere locally fitting the surface.

曲率值被反转。在这种情况下，输出值具有长度维度，表示局部拟合表面的球体的半径。

Mean curvature

is similar to Max curvature. Here, however, the mean value of the two principal curvature values is computed. This quantity will be negative in strictly concave regions and positive in strictly convex regions. It can be zero in regions where a positive and negative principal curvature cancel each other.
与最大曲率类似。不过，这里计算的是两个主曲率值的平均值。这个量在严格凹陷区域为负，在严格凸陷区域为正。在正负主曲率相互抵消的区域，它可以为零。

1/Mean curvature

the mean curvature values are inverted.
平均曲率值被反转。

Gauss curvature

is the product of the two principal curvatures. It is negative in surface areas with hyperbolic geometry (convex-concave, like near saddle points) and positive in areas with elliptic geometry (strictly convex or strictly concave).
是两个主曲率的乘积。在具有双曲几何（凸凹，如鞍点附近）的表面区域，它是负值；在具有椭圆几何（严格凸或严格凹）的区域，它是正值。

1/Gauss curvature

Gauss curvature values are inverted.
高斯曲率值被反转。

Both curvature values

returns the two principal curvature values as a surface complex scalar field that should be interpreted as a field of pairs of scalar values (use Arithmetic to retrieve each value).
将两个主曲率值作为表面复标量场返回，该标量场应解释为标量值对的场（使用算术来检索每个值）。

Both 1/curvature values

returns the inverted two principal curvature values.
返回两个主曲率值的倒数。

Max curvature vector

a surface vector field is computed indicating the direction of maximum principal curvature. The length of a directional vector is equal to the corresponding curvature value.
计算一个表面矢量场来指示最大主曲率的方向。方向矢量的长度等于相应的曲率值。

Both curvature vector

a surface complex vector field (that should be interpreted as a field of pairs of vectors) is computed indicating the direction of the two principal curvatures. The length of a directional vector is equal to the corresponding curvature value.
计算表面复矢量场（应理解为矢量对的场），指示两个主曲率的方向。方向矢量的长度等于相应的曲率值。

Shape index

computes the surface scalar field which values are equal to
计算表面标量场，其值等于

$$\frac{2}{\pi} atan \frac{C_1 + C_2}{C_1 - C_2}$$

where $C_1$ and $C_2$ are the two principal curvatures.
其中 $C_1$ 和 $C_2$ 是两个主曲率。

Curvedness

computes the surface scalar field which values are equal to
计算表面标量场，其值等于

$$\frac{1}{2} \sqrt{C_1^2 + C_2^2}$$

where $C_1$ and $C_2$ are the two principal curvatures.
其中 $C_1$ 和 $C_2$ 是两个主曲率。