US 20050200623 A1 Abstract A system and method for designing, on a display device, a new design. The method includes receiving a plurality of at least two predefined models defined as parameters in a common coordinate system, extracting patterns and relationships from the predefined models and capturing the patterns and relationships in a processor memory to form a statistical model providing plural, selectable vehicle shapes, and generating a new design based on a selectable shape of the statistical model.
Claims(19) 1. A method for generating a geometric design, comprising:
receiving input of first components of a geometric design of a high dimensional space; a first mapping of the first components of a high dimensional space into first components of a low dimensional space of principal components; applying the first components of a low dimensional space of principal components to an optimization computation to generate second components of a low dimensional space of principal components; a second mapping of the second components of a low dimensional space of principal components to second components of a new geometric design of a high dimensional space. 2. A method as recited in on a display, displaying a representation of the first components of a geometric design of a high dimension; on a display, displaying a representations of the second components of a geometric design of high dimension. 3. A method as recited in on a display screen, a user manipulating a representation of the first components of a geometric design of high dimension; in sequence the first mapping occurs; in sequence, the optimization computation applies constraints to the first components of a low dimensional space of principal components to generate second components of a low dimensional space of principal components; in sequence the second mapping occurs; on the display screen, displaying a representation of the second components of a geometric design of high dimension. 4. A method as recited in 5. A method as recited in 6. A method as recited in 7. A method as recited in 8. A system for generating a geometric design, comprising:
an input module for receiving of first components of a geometric design of a high dimensional space; a first mapping module for mapping the first components of a high dimensional space into first components of a low dimensional space of principal components; an optimization computation module for applying the first components of a low dimensional space of principal components to an optimization computation to generate second components of a low dimensional space of principal components; a second mapping module for mapping the second components of a low dimensional space of principal components to second components of a new geometric design of a high dimensional space. 9. A system as recited in a display module for displaying, on a display screen, a representation of the first components of a geometric design of a high dimension and on the display, displaying a representations of the second components of a geometric design of high dimension. 10. A system as recited in a manipulation module for a user manipulating on a display screen, a representation of a first component of the first components of a geometric design of high dimension. 12. A system as recited in 13. A system as recited in 14. A method as recited in 15. A method as recited in 16. A method for design, comprising:
defining a first space for a first design to occupy; defining a second space for a second design to occupy; manipulating the second design within the first space. 17. A method as recited in receiving a plurality of at least two predefined models defined as parameters in a common coordinate system; calculating means from correspondences between parameters of the predefined models; calculating a covariance based on the means. 18. A method as recited in receiving a plurality of at least two predefined models defined as parameters in a common coordinate system; calculating means from correspondences between parameters of the predefined models; calculating a covariance based on the means. 19. A method as recited in receiving the second design of a high dimensional space; a first mapping of the first components of the high dimensional space into first components of a low dimensional space of principal components; applying the first components of a low dimensional space of principal components to an optimization computation to generate second components of a low dimensional space of principal components; a second mapping of the second components of a low dimensional space of principal components to second components of a new geometric design of a high dimensional space. 20. A method as recited in on a display screen, a user manipulating the second design of high dimensional space. Description This application claims priority to U.S. Provisional Application Ser. No. 60/598,290, titled, “A SYSTEM AND METHOD FOR GEOMETRIC SHAPE DESIGN,” filed Jul. 30, 2004, which itself claims priority to U.S. Provisional Application Ser. No. 60/552,975, titled, “CAPTURING AND MANIPULATING AUTOMOTIVE DESIGN CHARACTERISTICS IN A STATISTICAL SHAPE MODEL,” filed Mar. 12, 2004, both of which are incorporated by reference herein in their entirety. A design tool for use in manufacture is disclosed. More particularly, this disclosure relates to computer-aided geometric shape design tools. Computer design tools useful for industrial design are most commonly “computer aided design” (CAD) based. Users of CAD programs often undergo training and have much experience in the use of CAD before their proficiency reaches a level high enough for complex design and engineering. In the automotive industry, car body designers typically sketch their designs. Car body designers are creative artists who produce styling sketches and typically do not use CAD programs. From sketches and discussion with the car body designer, a CAD designer will rework the sketch onto the computer. Accordingly, the sketch is engineered into three-dimensional detail. There are often many instances of refinement discussed between the artist and CAD user. Oftentimes, for example, the designer's sketches are not in correct perspective and details may need to be added and changed, and therefore, the process to completion by a CAD user may become tedious and repetitive. The CAD tool requires construction of a shape, piece by piece. The overall shape may not emerge until a significant amount of work has been done. It would be advantageous for a designer to have available a design tool that is simple to operate and can readily replace hand sketching, and whose resultant design is captured in a computer file. Therefore, the time consuming, step of refinement between the artist and a CAD designer may be substantially eliminated. In automobile design, a designer most likely must keep the design within certain style parameters. For example, the task at hand for the designer may be to design a new Cadillac. In this way, it may be advantageous for the designer to have available Cadillac designs and then to change some aspect or another to create a new look in keeping with the brand character of the Cadillac. Alternatively, a designer may want to create a design that is an intermediate between two designs, or is a blend of three or more designs. In any of these events, the process currently depends on the designer's strong familiarity with the various automobiles' designs. In this way, the ability to use a computer to maintain data on designs and create automobile designs from any number of combinations would be particularly advantageous for the design process. Complex design shapes such as automobiles may have topologies that vary greatly. The list of automobiles, even for just one automobile manufacturer, is extensive and the styling is diverse and includes many discrete variations. The functional categories of automobiles for a single manufacturer may include coupes, sedans, SUVs, sports cars, and trucks. Secondary categories may include minivans, wagons and convertibles. A computer based design tool that would allow a designer to combine any number of models to form a resultant new style or model would be advantageous. It would be advantageous if the design tool visually offered to the artist a plurality of automobiles to choose from and provided the ability to combine them into a combined resultant automobile design. If the designer desires a sportier car, or, for example, a Buick to be more Cadillac-like, or to use the grill of one car on another car, it would be advantageous to provide in a design tool the flexibility to the user to reach his design goals or otherwise explore options. Once a user has created a resultant combined design by combining as many models as desired, an additional benefit would come from the ability to change or morph that resultant design. A design tool that would be useful for morphing automobile designs is preferably flexible enough to allow a designer to explore different combinations and then provide the ability to morph the resultant design into many possible designs. Furthermore, a design tool that provides many options for morphable features, and allows the addition of constraints on the features in the morphing process, would be advantageous as well. In this manner it would be also advantageous that a computer operable design tool provide the output so that after the vehicle designer's initial design is complete, a CAD programmer would then be able to work from the output. This invention is a software package, a system, a method, and an apparatus for designing geometric shapes for automobiles or any other manufactured objects. On a display screen a first set of exemplar designs-hereinafter referred to as a catalog—is provided for selecting a second set of exemplars to create a resultant design space or mixture. A design space includes space defined by features of a mixture. Once a user has created a resultant combined design, also available is the ability for the designer to explore the design space and therefore to change or morph the resultant design. Automobile designs may be embodied as the exemplars. The term exemplar includes a model that is a registered model as defined below. Once the resultant combined design is chosen then manipulated, altered, or morphed according to one or more statistical models, the result may be a new design. A statistical model includes a probabilistic object derived from a design space. The mathematical space and statistical manipulations of the mathematical space allow the user to explore the space allowing for stylistic and functional interpretations. A mathematical algorithm described herein allows the user to select one or more feature constraints and apply the constraints to the morphable model. Accordingly, a CAD user may input the model generated by the described design tool and begin the process of model making therefrom or use the model directly. The system and method, apparatus and product includes central system modules This invention may be embodied in the form of any number of computer-implemented processes and apparatuses for practicing those processes. Embodiments of the invention may be in the form of computer program code containing instructions embodied in tangible media, such as floppy diskettes, CD-ROMs, hard drives, or any other computer-readable storage medium, wherein, when the computer program code is loaded into and executed by a computer, the computer becomes an apparatus for practicing the invention. The present invention may also be embodied in the form of computer program code, for example, whether stored in a storage medium, loaded into and/or executed by a computer, or transmitted over some transmission medium, such as over electrical wiring or cabling, through fiber optics, or via electromagnetic radiation, wherein, when the computer program code is loaded into and executed by a computer, the computer becomes an apparatus for practicing the invention. When implemented on a general-purpose microprocessor, the computer program code segments configure the microprocessor to create specific logic circuits. The user Exemplars that correspond to existing models may inherently incorporate engineering constraints. The term constraint used in this disclosure is a mathematical term and used in the sense of a mathematical operation, and is not intended to add limitations to the meaning of the features with which it is used in conjunction. Packaging and criteria information exists in the data stored in datastore The design tool therefore provides the ability to morph models in many ways. Basic geometric shape features such as points can be constrained to lie in certain positions. Other features that are selectable for constraint during morphing may include primary criteria such as vehicle height, wheelbase, H-point, and steering wheel position. A feature may include a simple attribute of the reference model, for example, a point. Therefore, the catalog provides the designer exemplars to incorporate into a resultant object that may meet target design requirements. Accordingly, certain criteria including engineering constraints may be built into the resultant average vehicle as exemplars are existing models and inherently include engineering constraints. Many engineering criteria are manipulable, changeable or morphable in accordance with this process, method and apparatus. Engineering constraints available for application in the morphing process may be displayed as, for example, options in drop down menus. There may be a style that the designer may want to emulate, e.g., a Buick—having a particular brand flavor or identity that in this example should remain in the mix. Alternatively, favorite models may be mixed together and new design elements may be added. It will be appreciated that there are many ways to approach design with the design tool described herein, including such considerations as aesthetics, brand identity, and function. Selectable morphable features include features named herein as well as others that may be recognized by persons skilled in the art of vehicle design. In a first embodiment, the user's interaction Returning again to A morphable model summarizes the space defined by exemplars. Any point in the space can be considered a “morph” of the exemplars, i.e. a weighted linear combination of their features. However, constraints are also applied to features of the morphable model to produce updates of the shape based on optimal estimation. These actions create a new design (a single entity in space), and the process would not typically be called morphing, although the result could still be considered a combination of the exemplars. Under the ADD button Exemplars in the catalog have been through a registration process that may be automatic or manual. Exemplars have been fit so that there is a substantially one-to-one correspondence between them. For example, a particular curve, the front piece of a hood on a car, is numbered Still referring to Initially, by selecting a mix as described above, the initial probabilities and initial mathematical space are set. An average or mean of these chosen cars is taken to represent the morphable model. A morphable model is a statistical model derived from the cars chosen to populate the design space. A design includes a model in the design space. The morphable model can be changed by the user in the manner that the user chooses. The user is able to drive the design tool's output to satisfy aesthetics, and impose functional and engineering constraints on the desired design. Once a morphable model is generated, the user can select a particular place, space, or feature to modify, for example, a windshield or window. As an example shown in On the display screen On the display In this example the user then may pick a point or a curve on the average vehicle to pursue changes to the morphable model. For example, the user may pick a point at the rear window A drop-down menu or any other user interface may provide a list of points or options for the designer to pick from to alter the resultant vehicle. Or the user may simply click near or on the rear window to select the point The average vehicle The designer may drag point The catalog of designs includes geometric design data, coded into a representation such that different, but related, designs may be put into correspondence. The coding may, for example, entail segmenting a geometric design into points, curves or surfaces. Some or all designs in the catalog may be represented as an array of values that characterize the vertices and curves for that design. Corresponding values in different arrays map to corresponding vertices and curves in different designs. On the other hand, one or more catalogs of any number of objects may be included. One or more catalogs may provide a statistical basis for the class of shapes that are generated by the method, system and apparatus described herein. To build a morphable model from the selected set of exemplars, the set of mathematical arrays, one array representing selected designs Morphing with statistics supports consistency of the resulting design. In the statistical methods as described in detail below, the more models used, the more accurately the model class is described by the statistics. Other statistical models may be used that operate on less data. Mathematical methods including statistics are described for their ability to reach results, and any other methods used to reach the results are considered within the scope of this disclosure. One skilled in the art of vehicle design will recognize that the vehicle design process typically begins with vehicle criteria. Some of such criteria include points called “hard points” which in certain circumstances should not be changed. Hard points may be categorized according to vehicle architecture class, for example. A resulting set of dimensions for hard points is therefore developed and communicated to a designer. The designer may then typically design the vehicle exterior with constraints given on the location of at least some of the vehicle hard points. The designer input The designer may then manipulate the features of the statistical model corresponding to the hard points to conform with standard dimensions. The model is drawn on the display The process, system and apparatus described herein may be applied by vehicle designers in additional ways. For example the vehicle models may be used to design a vehicle interior from a plurality of vehicle interior exemplars. Also, the interior and exterior may be designed simultaneously. Additionally, plural vehicle interior exemplars and plural exterior exemplars may be used independently to form separate exterior and interior resultant statistical models. These separate models may then be combined into a common coordinate system by, for example, causing the hard points of the models to conform to one another. The designer may then adjust the interior and exterior models simultaneously with a single selection or with multiple selections. Cluster points similar to those as described above in reference to As described above, the method, system and apparatus described herein provide the designer the ability to be creative and to drive the output. For example, the designer may be working in a space that includes angular exemplars, but may wish to add some curvature. The designer may add a previously registered round shape to the mix for that result. In this way, design qualities or features may be added into the mix. Also they may be weighted—such as 50 cars and 1 round shape, or weighted as 25 round shapes and 25 cars. The average and the statistics can be changed by the designer's choice. A slide bar or other similar interface may provide the user the ability to specify weights. As noted with the previous example, exemplars may include other items than existing vehicles. New designs, not yet built into production models, may also be included. Different shapes having other purposes than vehicle bodies may also be used as exemplars. The design space may be large. The designer can use the mixer to select a smaller set of models appropriate to his design goal. For example, if only Cadillac vehicles are chosen for the mixture, the design space will reflect the characteristics of Cadillacs. Referring now to A catalog of exemplar designs is provided at As an aside to illustrate the correspondence, It may also be part of the definition of a design topology that two curves meet at a common endpoint, i.e., no two curves cross at an interior point of either curve. Compliance with this part of the definition of a design topology may be checked for compliance after coordinates have been provided for the points p Note that this definition of design topology is not limited to points and curves. As one example, this definition of design topology may be extended to include, for example, a finite set T of triangles t In the case where a topology includes a set of triangles T, it may further be a part of the definition of the topology that two triangles intersect at a common endpoint or common edge, so that no two triangles cross in the interior of one or the other. As with the part of the definition of design topology discussed earlier that two curves meet at a common endpoint, this further part of the definition of the topology may be checked for compliance after coordinates have been provided for the points p It is to be noted that neither non-intersection requirement discussed above is a requirement or restriction on the design topology itself, but rather a stipulation of what it means for a design to conform to the topology. Returning now to discussion of the method shown in the flowchart of A selection of exemplar designs from the catalog is chosen at Once a mix of designs has been chosen in step More precisely, the design space is defined by the statistics of the feature vectors x for the set of designs selected for the mix. This definition may be augmented by associating with the set of chosen designs a joint Gaussian probability distribution for the values of the coordinates of points of the design topology, i.e., a joint distribution for the feature vector components. This joint distribution is constructed from the set of selected designs so as to have the same values for the first and second moments as would be obtained from the statistics of the set of selected designs. It will be appreciated that the use of a Gaussian distribution is not necessary; another probability distribution may be more appropriate in particular applications. In fact, it may be desirable to provide a choice of probability modeling functions through buttons or menu pick lists on the user interface. Preferably, a Gaussian distribution may be used as a default probability modeling function. However, use of a Gaussian distribution in this discussion is not intended to limit this disclosure. The basic properties of the design space defined at In particular, these statistics provide for determining the average value for the coordinates of the points p Again referring to Now that a morphable model is generated, a user may opt at Feature constraints are functions of the model features, referenced as “H” in the mathematical detail later. For example, a wheelbase length constraint could be defined as in the following. A vehicle may be aligned with a coordinate axis—for example, the X axis—so that the distance between two points along this axis is just the difference of their x-coordinates. Suppose the features of the model are kept in a column vector of numbers called ‘x’, that includes the points for the centers of the front and rear wheels on the driver's side. One of the rows of the matrix H—say the i Different ‘H’ matrices or functions multiplied by the features ‘x’ compute different lengths, positions, or other quantities, on 1 or more features, by varying the numbers in H. Examples of H that are considered important to constrain the model may be defined in advance, and may be stored in a menu or list for selection (see The particular value of ‘z’ for any particular function H (e.g. wheelbase length) depends on the current value of the features in the model ‘x’. If the designer now wishes to change the design so that the wheelbase length is different than the current value of ‘z’, he provides a new value, for example ‘z0’ through a slider, or other means, indicating that the constraint must be applied with the new value (see This system uses a method to update the current design (see The constraint functions do not need to be linear functions. They may also be non-linear—referenced as ‘h( )’ in the mathematical detail below. In that case the “optimal” solution, or the solution that has minimum total error is not guaranteed to be optimal or minimum over the entire design space, but only in the neighborhood of the current design values. Once the feature for manipulation is selected the next step in the flowchart of Selectable feature constraints may be provided in a drop down menu, having a pick list for, e.g., H-point or wheelbase or any other suitable user interface. The step at The update step is shown in the flowchart of If the selected point has been moved outside the extent of the usable design space, the user may see intersection of one or more curves or triangles at an interior point of one of the curves or triangles, signaling nonconformance to the design topology, and breakdown of the morphable model. This caricaturing phenomenon provides feedback to the user on the usability of the current design. A user may choose to pick a new point or feature constraint, returning again to Another option, among many other possible options such as those described and contemplated herein, is for the user after the update step to add an additional design to the mix, or remove a selected design from the mix, as shown at The current design as shown on a display screen The input As described above, the general statistical model provides a variety of selectable shapes defined in terms of combinatorial relationships between parameters. These relationships are well-defined within the bounds of the extremes given by the input designs as shown in The selected shape is stored in the datastore It will be understood that extraction module Constraint module Constrained model editing provides the power to change the design globally, and consistently within the model space, with only a few interactions by the designer. As shown in Optimal estimation module As explained in detail below, z may be assumed to have a probability distribution characterized by additive noise v with covariance R. The magnitude of the covariance (R) for v compared to the magnitude of the system covariance used by H determines how precisely the new value is adopted. In the case of moving a point, the designer may want to specify the location exactly (R→0). However, it is possible to let the design “pull back” and settle into a state that is influenced less strongly by the manipulation (in that case R is larger, allowing the dragged point position, for example, some variability in subsequent steps). When a range of values is permissible, e.g. a range of wheelbases, increasing the value of R allows the design to find an optimum balance of this and other constraints. This relative weighting is also a designer-specified value, perhaps through a slider. Turning now to a more detailed discussion of the three steps The abstraction of the design process—in terms of the mathematical model—may be given by the following steps: (1) specify the model space for the new creation, which was discussed above; (2) review the current shape, or read out geometric values (e.g. dimensions); (3) apply constraints to parts of the geometric shape, reducing freedom in the design space; and (4) add innovation, adding freedom back into the design space. In the first step, exemplars are chosen to populate the object space with the right character. This is like the pre-determination of a market segment for a new design, resulting in a population of past, present, and concept vehicles as context for the designer. The choice of exemplars for the object space is part of the creative process. The discussion below shows how the exemplars are processed to produce a usable low dimensional space (with a compressed representation characterized by u and Λ, to be defined and discussed below). The remaining steps are used iteratively, although at any moment one will apply. Step To draw the current shape, or to query the model, mapping to the original feature space is performed. For example, the points at the center of the wheels may be defined in the original feature vector (x), so wheelbase can be defined as the distance between the points. However, the same points may not appear in the compressed representation (u), although they can be reproduced from it. The wheelbase calculation is automatically redefined to use u. An example of a constraint is the specification that wheelbase be changed to a particular value. The constraint uses the same mapping as the query function. Another example is the specification of the position of a particular model point (during shape editing). In either case, the rest of the shape preferably will assume some plausible values based on the specification. Finally, it may be the case that the designer wants to isolate part of the design for change, without globally affecting the rest of the design. In that case, the features being manipulated would preferably have their statistical correlations to the rest of the model weakened so the impact on the rest of the design is removed or lessened to a chosen degree. A model space (x, Ω) for a design will be defined as a topology Ω, and an associated Gaussian-distributed vector-valued random variable x with given means and covariance matrix Ξ P(x)˜N(υ,Ξ). Estimates of the parameters υ and Ξ will be written {circumflex over (x)} and C In general, the covariance matrix C Within Ω a particular model—x If a probabilistic interpretation is not warranted or desired, the same mathematics can be developed from recursive, weighted least-squares solutions to the problem. The “weights” may be chosen to be the inverse of the “covariances” calculated in the following. The definition of the topology has been discussed above. The current state of the design is defined by the topology, and the current estimates of the shape vector and shape covariance matrix, {circumflex over (x)} and C Referring to step (3)as described in a paragraph above, that is, reducing the freedom in the design space, Principal Component Analysis (PCA) may be used as a data dimensionality reduction technique that seeks to maximize the retained variance of the data in a (lower dimensional) projected space. The PCA space includes a design space with a reduced number of dimensions via PCA. The projection model can be derived in closed form from the data in a set of examples. The following discussion illustrates the technique. The projection to principal components is implemented with the eigenvectors and eigenvalues of the sample covariance of the data, S {circumflex over (D)}=[x The x The eigenvectors of S The eigenvectors and eigenvalues of {tilde over (D)}{tilde over (D)} Using SVD, factor the d×n matrix {tilde over (D)}: {tilde over (D)}=UΣV Here, U is a d×n matrix, and Σ and V are n×n matrices. By the definition of SVD, Σ is a diagonal matrix. Further, the matrix U is column orthonormal, and the matrix V U and V From Σ and its transpose, define a new matrix Λ=ΣΣ′. The columns of U and the diagonal elements of Λ are the eigenvectors and corresponding eigenvalues of {tilde over (D)}{tilde over (D)} Next, in PCA the eigenvalues are sorted by size (largest first). The columns of U and rows of V {tilde over (D)}=U′L′V′ After reordering of the matrices, any zero eigenvalues will be in the last diagonal elements of A (and thus last in Σ′). If there are m≦n−1 non-zero eigenvalues from the n exemplars, then rewriting the last equation in tableau
exposes the relevant sub-matrices and their dimensions. The result after multiplying the sub-matrices is
The remaining columns of U″ and rows of V″ The columns of U are the eigenvectors of {tilde over (D)}{tilde over (D)} There are at most m=n−1 non-zero eigenvalues in the solution of {tilde over (D)}{tilde over (D)} Assuming no small eigenvalues are discarded (i.e., if m=n−1) the (unbiased) estimate of the sample covariance (S S using the definition
(this has been scaled from the earlier definition of Λ, above). This defines a transformation from the small (m×m) diagonal matrix Λ to S Λ=U PCA defines the model covariance (estimated as C where {tilde over (x)}=x−{overscore (x)}, and E is the operator for taking the statistical expectation value. Thus u≡U and E[u]=û=0. Further, the expression for C The vector u is the projection of x in the m-dimensional principal component space. This PCA projection maximizes the retained variance of S The design state may be defined in terms of the low-dimensional space as P(x)˜N({tilde over (x)},C P(u)˜N(û, Λ), with initial values b={overscore (x)}, û Λ This discussion of the reformulation of the statistics of the feature vectors of the selected exemplars in the low-dimensional space characterized by u and A provides further detail of step Constraints may be incorporated through introduction of a random vector z, with r components, taken to be a linear function of the d-component random vector x, with constant r×d coefficient matrix H, and r-component random noise vector v, with mean 0 and covariance R. z=Hx+v with components of v uncorrelated with components of the deviation of x from its mean υ: E[(x−u)v The choice of a value for z, and the choice of H, is completely equivalent, in this context, to choosing a feature constraint. This is step The mean and covariance of z, and the cross-covariances of z with x, regardless of the distributions of x and v, are: {circumflex over (z)}=H{circumflex over (x)} C C C If x is Gaussian-distributed, so are the marginal (P(z)) and conditional (P(z|x)) densities of z: P(z)˜N({circumflex over (z)},C P(z|x)˜N(Hx,R). Using Bayes' Rule
a posteriori density of x given a particular z (both Gaussian) is: P(x|z)=N({circumflex over (x)}+C With z given as the particular function of x above, the conditional mean and covariance are: {overscore (x|z)}={circumflex over (x)}+C C These last two equations are the Kalman Filter Measurement Update equations. As mentioned, deterministic arguments can be used instead of probabilistic ones, and an equivalent recursive weighted least squares solution can be obtained if desired. To briefly summarize the above description, the reduction to principal components by the Principal Component Analysis (PCA) allows the design space to be explored by morphing and other changes in a generally stable, computationally economical way. Again referring to With the definition, G=HU the Kalman Filter Measurement Update formulae can be re-written
can be substituted above, showing the relationship of principal components (u) to the original space, and its re-projection: {circumflex over (x)} The projection to principal components space is then û In the above set of equations, the update is driven by the user's choice of a new value z It should be noted that when C As presented, the update equations are the optimal, minimum variance Bayesian estimate, which is equal to the a posteriori conditional density of x given the prior statistics of x and the statistics of the measurement z. A non-linear estimator may not produce estimates with smaller mean-square errors. If the noise does not have a Gaussian distribution, then the update is not optimal, but produces the optimal linear estimate (no linear estimator does better, but a non-linear estimator may). If the measurement function is non-linear in x, then H is a partial derivative matrix (not constant) and will have to be evaluated. A one-step evaluation of H on {circumflex over (x)} Thus, assuming a linear function of the state with mean zero, additive noise (v): z=Hx+v P(v)˜N(0,R) The marginal density of z is then P(z)˜N({circumflex over (z)},C Given the current estimate of the model ({circumflex over (x)}) the value z is expected. Forcing the model to any other given value changes the current model estimate. The minimum variance estimate seeks to minimize the variance of (z−H{circumflex over (x)}) jointly; i.e., both the estimate of x, and the independent noise v of z are adjusted so that (z Using the linear composition z(u)=z(x(u)) the marginal density for z(u) is P(z)˜N({circumflex over (z)} and the desired conditional distribution for u is P(u|z=z K The model estimate is then updated as P(x|z)=N({circumflex over (x)} shown in Constrained model editing provides the power to change the design globally, and consistently within the model space, with a few interactions by the designer. The effects of editing are not local to the feature being edited. If the wheel centers are features in the design, then a different matrix H can calculate the difference of those features along the vehicle length axis, and z The magnitude of the covariance (R) for v compared to the magnitude of the system covariance used by H determines how precisely the new value is adopted. In the case of moving a point, the designer may want to specify the location exactly (R→ To recapitulate the update discussion in brief, in general, the covariance matrix may be ill-conditioned. A singular value decomposition (SVD) may be applied to obtain the principal components characterizing the mix of designs. The reduction to principal components may then allow the design space to be explored in a generally stable, computationally economical way. The average array and covariance matrix completely characterize data, in the case where the data were drawn from an underlying Gaussian distribution. Linear constraints imposed on the data then result in different Gaussian distributions, as can be determined by evaluating the conditional probabilities for the data array elements in the presence of a linear constraint. Even in the case of non-linear constraints, however, this approach may be used to advantage in exploring the design space. As discussed above, in the probability calculation from the populated design space, one gets an average or mean array and a covariance matrix. The user can choose to constrain the geometric shape delivered by the system and method, thereby obtaining a new geometric shape. The user may choose to continue to use the original, unconstrained covariance matrix. That is, when the method is implemented, the average array is updated, and not the covariance matrix. In this way, some information about the original mix of designs is still available to the designer for further use in exploring the design space from a new starting design. Another manipulation possible is shifting. As illustrated in Accordingly, a user can choose a covariance matrix derived from one mix of designs from the catalog This is shifting—without rotation—of the Gaussian distribution derived from the selected mix, away from the average of the selected mix of designs, to instead be centered at the particular design. In summary, and with all of the qualifications and those that may be inferred by the preceding disclosure incorporated herein, a segmentation tool method and system may be incorporated in or separate from the design tool apparatus and product. Adjustments to the methods described herein are within the scope of this disclosure so that the output results are substantially achieved accordingly. The output generated by the use of the general statistical and mathematical models described herein provides a substantial variety of derivable shapes defined in terms of continuously variable parameters. Referring to FIGS. While the invention has been described with reference to exemplary embodiments, it will be understood by those skilled in the art that various changes may be made and equivalents may be substituted for elements thereof without departing from the scope of the invention. In addition, many modifications may be made to adapt a particular situation or material to the teachings of the invention without departing from the essential scope thereof. Therefore, it is intended that the invention not be limited to the particular embodiment disclosed as the best mode contemplated for carrying out this invention, but that the invention wilt include all embodiments falling within the scope of the appended claims. Moreover, the use of the terms first, second, etc. do not denote any order or importance, but rather the terms first, second, etc. are used to distinguish one element from another. Referenced by
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