Patent Application
20200117838
The invention regards a method for generating a set of unified efficient shape descriptors which is helpful in order to enable engineers to efficiently optimize a shape of an objector, to retrieve a shape, or classify a plurality of shapes.
During the engineering design process of complex shapes, such as shapes of cars or turbo-fan engine blades, the crucial question exists how to represent all possible shapes for development and evaluation. In the context of shape optimization many different types of representations exist, prominent examples are direct parameterizations or various deformation methods. During a design process for a certain shape, many different geometries are created and their performance is evaluated in various disciplines, such as aerodynamic efficiency, crashworthiness, structural mechanical properties, thermal properties, noise, etc. In the course of the design process the type as well as the details of the representation usually changes frequently.
The complete set of geometries along with the evaluated performance data constitutes a very valuable dataset which enables the engineers to generate novel insights and knowledge. In such a context machine learning offers very powerful tools for automatic data analysis and knowledge generation, such as prediction and classification models, or dimensionality reduction techniques. Usually, the parameters of the shape representation are used as the input values for such machine learning techniques. However, if representations of the shapes vary between different datasets, the direct application of machine learning techniques to the complete dataset using the representation parameters as inputs is not possible, even though they all encode the same type of geometric shape which are in principle comparable to each other. The reason is that the meaning of the input parameters changes, as for example, in one representation the parameters encode the shape of the frontal part of a car, whereas in another representation the rear is encoded.
The alternative of directly using the coordinates of the entire shape geometry as input parameters for the machine learning has the major drawback that a huge training dataset with geometry and performance data is necessary, due to the very high dimensionality of the input parameter space. But in typical engineering applications, the generation of data is quite time and resource consuming and therefore the amount of data is usually rather limited in comparison to modern learning approaches, such as deep learning for 2D image data.
In order to tackle this problem, usually shape descriptors are used which encode the properties of a shape in a lower dimensional representation.
There exist many shape descriptors for two and three dimensional shapes in the literature, as reviewed, for example, in “Survey on 3D Shape Descriptors” (L. Zhang, M. João da Fonseca, A. Ferreira, Internal Report Univ. Lisboa, POSC/EIA/59938/2004 (2004)). More recent approaches also utilize deep learning, see, for example, “Deepshape: Deep learned shape descriptor for 3D shape matching and retrieval” (J. Xie, Y. Fang, F. Zhu, E. Wong, IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 1275-1283 (2015) doi: 10.1109/CVPR.2015.7298732).
Shape descriptors can be grouped into two fundamental categories: local and global descriptors. Local descriptors work by determining some salient points on the shape and calculating features based on the geometry in the neighborhood of those key points. Global descriptors calculate features based on the complete shape or large parts of it.
Examples of global shape descriptors are transformation-based descriptors, such as Fourier descriptors as disclosed for example in “Efficient feature extraction for 2D/3D objects in mesh representation.” (Zhang C, Chen T, IEEE International conference on Image Processing (2001). doi: 10.1109/ICIP.2001.958278), spherical harmonics descriptors and 3D Zernike moments as disclosed for example in “Survey on 3D Shape Descriptors” (L. Zhang, M. João da Fonseca, A. Ferreira, Internal Report Univ. Lisboa, POSC/EIA/59938/2004 (2004)). All these global shape descriptors sum over all points of the shape and transform them into a different representation by using some sort of kernel function. As a drawback, the known global descriptors suffer from a lack of local geometric information. Firstly, as a pre-processing step, the shape is either voxelized or some shape signatures such as centroid distances are derived which incorporates a loss of information about the local structure of the shape. Secondly, the individual entries in the actual feature vectors derived by using the actual transform are really global as each feature encodes one specific type of statistics of the complete shape without directly encoding local geometric information (of the already coarse-grained representation) as well. For example, Fourier shape descriptors encode the statistics of the wavelengths, that is, they encode which spatial variations occur when moving along the shape and how strong the signature of each wavelength is, i.e. a spectrum. The information about absolute positions in space and the correlated relative changes in a local neighborhood of the shape is distributed over the complete spectrum, i.e. in the complete feature vector. When truncating the spectrum by selecting only a small set of feature values (wavelengths), which is a necessary step to arrive at a manageable feature vector, some portion of such information is lost and it is a priori not clear what the implications are for the global and local representation of the shape. The equivalent insights apply to other transformation-based descriptors, such as Zernike moments, where the Zernike polynomials are used to decomposed the shape, or spherical harmonics descriptors.
It is thus an object of the present invention to provide a method for generating a set of shape descriptors for a set of two or three-dimensional geometric shapes as an efficient unified low-dimensional representation of a set of shapes that allows cost-efficient processing in an engineering process without losing information deriving from an entire set of shapes of objects.
This objective is achieved by a method according to claim 1.
Before aspects and details of the present invention are explained with respect to the figures, the basic idea underlying the set of feature descriptors shall be explained. With the method according to the present invention, discriminative features are calculated from 2D and 3D shapes. The basic idea for such calculation is that each point of the shape is assumed to emit one or more spherical wave, which are propagated throughout space with a spherical wave kernel
or some slight modification of it. Each point of the shape emitting such spherical wave generates a complex-valued field which decays with the distance from the point of the shape. At every position in space around the shape, the contributions of all the spherical waves emitted from each point of the shape add up. Due to their complex-valuedness they will generate an interference pattern throughout space which is characteristic of the shape.
Such a resulting interference pattern is used in order to represent a given shape. For such a given shape, feature descriptors are calculated for a set of N feature locations positioned some distance away from the shape and distributed around the shape. Since contributions with different wave numbers can be associated with each point of the shape and each contribution decays with distance, the interference pattern at those feature locations will be very characteristic of the shape and the relative positon of the shape and the feature location. Generally, the feature descriptors are calculated for a set of M wave numbers, so for each feature location M feature descriptors are calculated resulting in M·N feature descriptors for one shape, i.e. a M·N-dimensional feature vector which is the shape descriptor {right arrow over (F)}.
According to the inventive method a set of global shape descriptors for a set of two or three-dimensional geometric shapes is generated. At first, a set {{right arrow over (R)}n} of N feature locations is determined by an engineer for example and read in from a user interface or read from a memory. The feature locations have a distance from the shapes in the set of shapes and {right arrow over (R)}n is the position vector of the feature location having the length Rn=|{right arrow over (R)}n| and n=1, . . . , N. Additionally a set {km} of M wave numbers with m=1, . . . , M is determined by an engineer and read in via a user interface or also read from a memory. Further, a parameter γ∈ is read in, which is a parameter controlling a degree of locality of the features. This parameter may be read in directly from an interface where the respective value is input by an engineer, but may of course also be read in from a memory where for example a default value for the parameter is stored.
For each shape s of the set of shapes {Ss, s=1, . . . , Ns} a feature descriptor is then calculated for each of the N feature locations and M wave numbers, according to the rule
where the integral is summing all contributions from each point of the shape s. Here, {right arrow over (s)} is a position vector of points of the shape s, i is the imaginary unit, ∥{right arrow over (x)}∥α=[Σi(xi)α]1/α is the Lα norm of the vector {right arrow over (x)}. The now calculated feature descriptors are the elements of an M·N dimensional vector of features which is the shape descriptor for shape s. Optionally, the feature descriptors for each shape can be normalized to the volume or surface area of the respective shape, i.e. a normalization factor of 1/Volshape s (volume normalization) or 1/Areashape s (area normalization) can be used for each feature descriptor. The feature descriptors are associated with the shape descriptor of shape s according to
{right arrow over (F)}
s=(fs(n=1,m=1),fs(n=1,m=2), . . . ,fs(n=N,m=M))T
Finally, the set of shape descriptors for all shapes is output for further processing.
Each of the shape descriptors of the novel set of shape descriptors which is generated by the inventive method is represented by a set of feature descriptors, that may also be called diffraction features, which can be derived from any geometry in a well-defined and straight-forward manner, and which encodes relevant local and global geometric aspects of each shape. The number of feature descriptors, i.e. the dimensionality of the shape descriptor, some qualitative aspects of the features, and the global properties of the set of shape descriptors can be adjusted freely and therefore enables the engineer to adjust it to the type and amount of available data, as well as to the requirements from the data management side for storage in memory and on disk, and to the future applications to be performed on the set of shape descriptors. The advantageous aspect of the inventive method is to enable effective storage, post-processing and information extraction for a possibly large set of large shapes in an improved and effective manner.
The shape descriptor as calculated by the inventive method is a global descriptors, where for each shape descriptor of each shape and for each entry in the vector, i.e. each feature descriptor the complete shape is taken into account. However, due to the fact that the distance between the feature location and the contribution of each part of each shape enters explicitly and, in particular, it's scaling with the inverse distance to the power of γ, each feature descriptor is rather sensitive to local changes. In particular, each entry in each shape descriptor is most sensitive to local changes of that region of the shape which is spatially closest to its feature location.
The inventive method thus provides a low dimensional representation for a set of shapes that allows efficient computational processing as while at the same time does not require omission of information due to voxelization or extracting statistics from the shapes like state of the art approaches, but instead always uses the complete information of the complete set of shapes.
In usual engineering applications the shape data are provided as volume or surface meshes and an advantageous aspect of the present invention is that it can easily be adapted to such situations. For each feature descriptor of each shape s in the set of shapes {Ss, s=1, . . . , Ns} the integration over the shape is performed by summing over all mesh cells c of each shape where the contribution of each mesh cell c is calculated using the center-of-mass coordinate of each cell, {right arrow over (s)}c, and the volume- or area-weighted sum is taken over all mesh cells, i.e.,
where Ac is the volume or area of the mesh cell c. Optionally, the entries of the feature vectors can also be normalized to the total volume or surface area of each shape, i.e. a factor of 1/Volshape s or 1/Areashape s can be introduced.
The positions of the feature locations are chosen preferably to lie on a surface around the complete set of shapes. Preferred surfaces are for example a sphere with radius D, a cuboid with edge lengths A, B, and C, or a right circular cylinder with radius D and height H. The distribution of the points on the surface could be either deterministic, where the positions on the surface are calculated by a deterministic algorithm to follow a desired pattern or randomly, where the positions on the surface are determined by a randomized sampling technique in order to follow a desired distribution. The value for the center and the dimensions of such surfaces (radius D of the sphere, edge lengths A, B, and C of the cuboid, radius D and height H of the right circular cylinder) can be either given directly as absolute numbers or determined relative to the shapes by an algorithm, where, for example, the maximal linear length of each shape is calculated and then the parameters of the surface are set to a multiple of maximal length found for the complete set of shapes. In this case the feature locations are chosen relative to the complete set of shapes and the variations of the positions of individual shapes within the set of shapes is encoded in the shape descriptors. Alternatively, the location of the shape features could be chosen relative to each shape in the set of shapes individually, which amounts to a normalization of the position of the feature location to each shape individually and removes the dependency of relative shifts between the shapes in the set of shapes from the shape descriptors.
The M wave numbers may be chosen to range from kmin to kmax and the spacing between the values may be chosen according to a desired behavior, for example, constant, linearly increasing or decreasing, exponentially increasing or decreasing, or user-defined values. It is additionally possible to add random noise of a defined strength to the values of the wave numbers. The values for kmin and kmax can be either specified directly as absolute numbers or calculated relative to each shape with an algorithm, where, for example, kmin and kmax are multiples of
and L is a length scale extracted from the set of shapes by determining the smallest sphere or cuboid which fully contains the set of shapes and then setting L to the value of the radius or the largest edge length of such a minimal sphere or cuboid, respectively.
It is in particular preferred that the feature locations as well as the values for the wave numbers are determined by an optimization procedure which optimizes a desired property of the complete set of shape descriptors. For example, for a given set of shapes, the position vectors of the feature locations as well as the values for the wave numbers are optimized in order to maximize the mutual distances between the shape descriptors. Alternatively, for a given training set of shapes along with some class labels, the position vectors as well as the values for the wave numbers are optimized in order to maximize classification performance when applying a classification algorithm to the set of shape descriptors of the training set which tries to determine the proper class label for each shape.
Advantageously, a pose-normalization procedure is applied to the shape descriptor of each shape in the set of shapes {right arrow over (F)}s in order to arrive at a normalized shape descriptor {right arrow over (F)}s, final which is then used for further processing. For example, a set of symmetry operations {b} is included which map the set of feature locations {{right arrow over (R)}n} onto itself. The shape descriptors from one shape {right arrow over (F)}s of all those symmetry operations are considered equivalent and thus define the identity mapping on the final feature vector {right arrow over (F)}s, final.
In order to minimize the computational cost for applications of the diffraction feature shape descriptors, for a given set of shapes the corresponding set of shape descriptors can be further processed by applying dimensionality reduction and manifold embedding techniques such as principal component analysis (PCA), independent component analysis (ICA), locally linear embedding (LLE), multi-dimensional scaling (MDS), Isomaps, or other linear or non-linear techniques. In this process the shape descriptor is reorganized and the dimensionality of each shape descriptor is possibly reduced from M·N to D≤M·N, which allows for an even more useful and compact low-dimensional representation of shapes for a given set of shapes.
The inventive method is further developed when the calculated set of shape descriptors (set of diffraction feature shape descriptors) are used as an input in a classification algorithm which is thus run based on the respective shape descriptor. For a given set of shapes which are organized in several categories the diffraction feature shape descriptor of one specific new shape is used to determine the category to which this newly observed shape belongs. The classification can either be done in a supervised manner, where a labeled training data set with shapes and their respective categories is used to train a classifier. Depending on the data set and the number of calculated feature descriptors, the classification algorithm can be any of the established algorithms such as linear classifiers, support vector machines, kernel estimation, decision trees, neural networks, learning vector quantization or deep learning convolutional neural network approaches to name a few. The classification can also be organized in an unsupervised manner, where the categories for the shapes as well as the classification of the newly observed shape are determined by a clustering algorithm, for example a density-based, distribution-based, centroid-based, or hierarchical clustering algorithm.
Alternatively, the set of diffraction feature shape descriptors may be used in a shape retrieval algorithm where for a given query shape the feature descriptors of that shape are used to find a set of similar shapes from the given set of shapes. The similarity of two shapes is evaluated by assessing the similarity of the two corresponding shape descriptors. For example, the absolute value (L2-norm) of a difference vector of the two shape descriptors of the two shapes can serve as a distance measure for the two shapes and shapes with smaller distances are considers more similar.
The set of diffraction feature shape descriptors may also be used in a surrogate-assisted shape optimization process. In such a context, a set of shape descriptors as described by this invention is calculated from the data from many shape optimization runs, possibly from different application fields (e.g., crash, aero-dynamics, structural mechanics, thermal analysis, noise-vibration-harshness, etc. . . . ), where quality criteria for many different shapes are determined, and this set is used to learn one or more surrogate models which given the diffraction feature shape descriptor of a new shape as input will predict the performance values of the shape in one or more disciplines.
Explanations on an embodiment of the present invention will now be given with respect to the annexed drawings in which
For a given set of shapes {Ss, s=1, . . . , Ns} the set of shape descriptors {{right arrow over (F)}s, s=1, . . . , Ns} (i.e. the set of diffraction feature vectors) can be calculated in the following way which is explained with reference to
Then, a set of wave numbers is determined by the engineer, for example
where L is the characteristic length scale of the set of shapes {Ss} determined above. Now, the complete set of diffraction feature shape descriptors {{right arrow over (F)}s} can be generated where each individual shape descriptor
{right arrow over (F)}
s=(fs(1,1),fs(1,2), . . . ,fs(1,M),fs(2,1) . . . ,fs(N,M))T
of the shape Ss is composed of feature descriptor fs(n, m) which are calculated according to the formula
where Rn=|{right arrow over (R)}n|=5L is the length (L2 norm) of the position vector of the feature location, {right arrow over (s)} is a position vector of the points on the shape Ss and the integral is summing all contributions from each point of the shape Ss. i is the imaginary unit and km the wave number. The pre-factor (Rn)γe−ik
In typical engineering applications where the shapes are usually evaluated with respect to some performance using simulations such as computational fluid dynamics (CFD) or computations structural dynamics (CSD), the shape data is naturally provided as a volume or surface mesh. In such a case the integral over the shape can be represented as a sum over all mesh cells c and the contribution of each mesh cell c is calculated using the center-of-mass coordinate of each cell, {right arrow over (s)}c, and the volume- or area-weighted sum is taken over all mesh cells, i.e.,
where Ac is the volume or area of the mesh cell c and the last factor realizes the normalization of the diffraction features to the volume or surface area of the shape.
It is to be noted, that scale-invariance with respect to the overall scale of each shape S from the set of shapes is achieved if desired by using the normalization of the shape to the absolute surface area or volume of the shape, as indicated by the last factor in the above equation, and by calculating the maximal linear length L of each shape s and placing the feature locations on a sphere with a radius given in terms of this length L and also selecting the wave numbers dependent on this length scale.
Further, invariance of the features with respect to orientation of the shape can be achieved if desired by determining a set of symmetry transformations {b} which map the set of feature locations onto itself, {
b: {Rn}→{
b(Rn)=Rn′}}. A symmetry-transformation just re-names the feature locations, i.e. permutes the ordering of the set of feature locations. All shape descriptors which are produced under such symmetry transformation just have permuted entries and are considered as equivalent.
In combination, both measures realize pose normalization of the 3D shape.
The set of shape descriptors {{right arrow over (F)}s, s=1, . . . , Ns} of a set of shapes serves as a low-dimensional representation of the complete set of shapes and is, according to a preferred embodiment of the invention, used to build models which are trained to predict a performance of a new shape. Thus, one preferred embodiment regards multi-disciplinary shape optimization starting from an initial shape. For example, such optimization could be an optimization of the car body shape s for aerodynamic efficiency where drag should be minimized while certain other aspects of the flow should be maintained such as downforce in the rear part of the car. In addition to the aerodynamics, the structural mechanical properties of the car should also be optimized for supporting various given static loads cases. For such application, the car shape is parameterized by one or more methods most convenient for the development engineer of each discipline and such paramterization usually changes during such a process as different parts of the car are optimized separately in each discipline. The evaluation of the aerodynamic properties is done with computational fluid dynamics (CFD) and finite element method (FEM) simulations, for example.
A significant gain and speed-up in the individual disciplines, CFD or FEM, can already be achieved by utilizing surrogate models which replace part of the actual CFD or FEM simulations and—after being trained on a set of simulation data obtained for a set of shapes—predict the aerodynamics and structural mechanics for new car shapes only using the efficient low-dimensional representation of the car shape as input. In order to utilize all available data, the efficient low-dimensional representation must represent the complete car geometry. Even further gain is possible by using a unified efficient low-dimensional representation which can be utilized in multiple disciplines such as CFD of FEA simultaneously.
The proposed set of diffraction feature shape descriptors for a set of shapes is a very promising realization of such unified efficient low dimensional representations for a set of shapes. In such a scenario, the set of diffraction feature shape descriptors is calculated for a set of shapes generated during some initial phase of the design procedure. The dimensionality of each diffraction feature shape descriptor can be reduced by applying a dimensionality reduction technique such as principal component analysis or locally linear embedding to the complete set of shape descriptors. Then surrogate models, such as Gaussian process kriging models, support vector regressors, or random forest models are trained on the set of dimensionally reduced shape descriptors to predict the aerodynamic and structural mechanics performance given the dimensionality reduced diffraction feature shape descriptors of a new shape as input. Then, such models can be used in one of the many surrogate-assisted single-objective and multi-objective optimization approaches that are already known in the art. In such optimization approaches, for a newly proposed design the diffraction feature shape descriptor is calculated, the dimensionality reduction transformation is applied and then the surrogate models are used to estimate the aerodynamic and structural mechanics performance of the new shape, without running the resource consuming CFD or FEM simulations. Thus, a new car shape with improved aerodynamic and structural mechanics performance can be achieved with a numerical optimization approach, where depending on the details of the surrogate-assisted optimization algorithm, the number of necessary CFD and FEM simulations can be drastically reduced. Similar approaches can be taken for other and arbitrary number of disciplines, where this embodiment of the invention serves as the one unified efficient low-dimensional representation for a set of shapes.
One significant advantage of the shape descriptors generated by the invention used for such applications described above using CFD simulations in particular, is that the interference pattern encoded in the diffraction feature shape descriptors is sensitive to small changes of the geometry which might have strong impact on the aerodynamics. But at the same time, the features are global in nature, where large scale changes are captured as well. And additionally, when used without any pose normalization, the features are sensitive to the absolute positioning and orientation of the shapes which is a very good aspect when used for modelling fluid flow around shapes or the effect static or dynamic directed forces have in a shape since in both applications the absolute orientation of the shapes is important.
According to another advantageous embodiment of the invention, the set of diffraction feature shape descriptors are used in shape matching or shape retrieval algorithms where similarities between shapes need to be assessed. This application is, for example, relevant in the engineering design process when an engineer develops a new shape and then tries to find similar shapes in the existing database of shapes which were already evaluated in prior design processes. In a typical application we are given a set of shapes, {Ss, s= . . . , Ns}, which is for example the archive of already evaluated shapes, which we would like to partition into different categories and then allow for shape retrieval applications, where the most similar shape to a novel shape to arrive later is sought. First, the feature locations are chosen where the same feature locations and wave vectors will be used for all shapes. Then, for each shape Ss the diffraction feature shape descriptor {right arrow over (F)}s is calculated. The set of shapes {Ss} can then be partitioned into different categories by applying a clustering algorithm on the set of respective shape descriptors {{right arrow over (F)}s, s=1, . . . , Ns}. For the subsequent shape retrieval application the query shape is evaluated with the clustering algorithm which was used to determine the categories and all shapes with the same class label are determined and considers as similar.
The major advantage of the proposed set of diffraction feature shape descriptors calculated in the above described manner is that one unified set of shape descriptors can be used for all applications described above. For properly chosen feature locations, degree of locality and wave numbers, the one set of shape descriptor {{right arrow over (F)}s, s=1, . . . , Ns} calculated once for the associated set of shapes {Ss, s=1, . . . , Ns} can be simultaneously used for all types of applications, such as shape classification, shape retrieval, performance prediction of new shapes, surrogate assisted shape optimization, and more. This is enabled by the global nature of the shape descriptors which are at the same time sensitive to local changes of shapes due to the interference and complex valuedness of the shape descriptors.
In
Alternatively, the definition of the feature location, the wave numbers and the degree of locality could be input into the computer system directly by an engineer via an interface.
After the feature descriptors have been calculated for each shape s, they are assigned to elements of feature vector, in order to generate the set of shape descriptors {{right arrow over (F)}s} in step S5. The set of shape descriptors {{right arrow over (F)}s} is then output either directly into an algorithm where, based on the set of shape descriptors {{right arrow over (F)}s}, further processing is performed, for example in order to perform shape matching or retrieval of the shape, or to perform surrogate-assisted optimization is already mentioned above in greater detail. Of course, the set of shape descriptors {{right arrow over (F)}s} may also be stored in the memory of the computer system.
| Number | Date | Country | Kind |
|---|---|---|---|
| 18 199 841.0 | Oct 2018 | EP | regional |
