Abstract:
In one embodiment of the present invention, a position-based dynamics (PBD) framework provides realistic modeling and simulation for elastic rods. In particular, the twisting and bending physics of elastic rods is incorporated into the PBD framework. In operation, an elastic rod model generator represents the center line of an elastic rod as a polyline of points connected via edges. For each edge, the elastic rod model generator adds a ghost point to define the orientation of a material frame that encodes the twist of the edge. Subsequently, a PBD simulator solves for positions of both points and ghost points that, together, represent the evolving position and torsion of the elastic rod. Advantageously, the ghost points enable more realistic animation of deformable objects (e.g., curly hair) than conventional PBD frameworks. Further, unlike force based methods, elastic rod simulation in the PBD framework performs acceptably in environments where speed is critical.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application claims benefit of the U.S. Provisional Patent Application having Ser. No. 61/911,303 (Attorney Docket Number AUTO/1305USL) and filed on Dec. 3, 2013. This related application is hereby incorporated herein by reference. 
     
    
     BACKGROUND OF THE INVENTION 
       [0002]    1. Field of the Invention 
         [0003]    Embodiments of the present invention relate generally to computer science and, more specifically, to techniques for modelling elastic rods in position-based dynamics frameworks. 
         [0004]    2. Description of the Related Art 
         [0005]    Generating realistic simulation results in real-time is essential for animating three-dimensional objects in time-sensitive animation applications, such as game engines. Position-based dynamics (PBD) frameworks are well-suited to many such animation applications. Notably, PBD frameworks employ techniques that sacrifice some quantitative accuracy to generate visually plausible results in real-time. For example, in PBD frameworks, all deformations of objects are characterized with discrete positions of points. 
         [0006]    While the tradeoffs exhibited by PBD frameworks are often acceptable, some types of animation are not adequately represented in PBD frameworks. In particular, because PBD frameworks specify positions but not angles, PBD frameworks are typically unable to simulate complex bending and twisting of objects modeled as rods. Such a restriction limits the applicability of PBD frameworks to animation. For example, while PBD frameworks successfully animate much of a human body, PBD frameworks are typically unable to realistically simulate the twisting of curly hair. 
         [0007]    Accordingly, elastic rods are commonly simulated using force-based methods, such as the finite element method, that are not optimized to represent deformations as only positions. While such force-based methods are capable of effectively emulating the natural motion of curly hair, the time required to simulate the thousands of hair strands included in a typical head of hair using force-based methods is prohibitive and unsuitable for animation-oriented applications. 
         [0008]    As the foregoing illustrates, what is needed in the art are more effective techniques for simulating bending and twisting of elastic rods. 
       SUMMARY OF THE INVENTION 
       [0009]    One embodiment of the present invention sets forth a computer-implemented method for simulating an elastic rod in a graphics application. The method includes generating a polyline that represents a center line associated with the elastic rod as a series of edges and points; for each edge of the polyline, associating a ghost point with the edge, where the ghost point has coordinates that define an orientation of a material frame that encodes a torsion associated with the edge; and computing new positions of the points and the ghost points after a time interval. 
         [0010]    One advantage of the disclosed elastic rod modelling techniques is that these techniques enable the real-time simulation of bending and twisting behavior of elements modelled as rods, such as strands of hair. In particular, simulations may be performed in a position-based dynamics environment, coupling the efficiency inherent in such environments with the realistic modeling of torsion. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0011]    So that the manner in which the above recited features of the present invention can be understood in detail, a more particular description of the invention, briefly summarized above, may be had by reference to embodiments, some of which are illustrated in the appended drawings. It is to be noted, however, that the appended drawings illustrate only typical embodiments of this invention and are therefore not to be considered limiting of its scope, for the invention may admit to other equally effective embodiments. The patent or application file contains at least one drawing executed in color. Copies of this patent or patent application publication with color drawing(s) will be provided by the Office upon request and payment of the necessary fee. 
           [0012]      FIG. 1  is a block diagram illustrating a computer system configured to implement one or more aspects of the present invention; 
           [0013]      FIG. 2  is a conceptual diagram of an elastic rod model generated and maintained within the position-based dynamics (PBD) simulation framework of  FIG. 1 , according to embodiments of the present invention; 
           [0014]      FIG. 3  depicts a modified discrete Darboux vector implemented within the position-based dynamics (PBD) simulation framework of  FIG. 1 , according to one embodiment of the present invention; 
           [0015]      FIG. 4  is a conceptual illustration of a bilateral interleaving ordering enforced by the constraint tool of  FIG. 1 , according to one embodiment of the present invention; 
           [0016]      FIG. 5  is a conceptual diagram of a curly haired bunny that is based on a 3D model in which the hair strands are represented using the elastic rod model of  FIG. 2 , according to one embodiment of the present invention; and 
           [0017]      FIG. 6  is a flow diagram of method steps for establishing a position-based dynamics simulation (PBD) framework that supports elastic rod modeling, according to one embodiment of the present invention. 
       
    
    
     DETAILED DESCRIPTION 
       [0018]    In the following description, numerous specific details are set forth to provide a more thorough understanding of the present invention. However, it will be apparent to one of skilled in the art that the present invention may be practiced without one or more of these specific details. 
       System Overview 
       [0019]      FIG. 1  is a block diagram illustrating a computer system  100  configured to implement one or more aspects of the present invention. As shown, the computer system  100  includes, without limitation, a central processing unit (CPU)  170 , a system memory  174 , a graphics processing unit (GPU)  172 , input devices  112 , and a display device  114 . 
         [0020]    The CPU  170  receives input user information from the input devices  112 , such as a keyboard or a mouse. In operation, the CPU  170  is the master processor of the computer system  100 , controlling and coordinating operations of other system components. In particular, the CPU  170  issues commands that control the operation of the GPU  172 . The GPU  172  incorporates circuitry optimized for graphics and video processing, including, for example, video output circuitry. The GPU  172  delivers pixels to the display device  114  that may be any conventional cathode ray tube, liquid crystal display, light-emitting diode display, or the like. In various embodiments, GPU  172  may be integrated with one or more of other elements of  FIG. 1  to form a single system. For example, the GPU  172  may be integrated with the CPU  170  and other connection circuitry on a single chip to form a system on chip (SoC). 
         [0021]    The system memory  174  stores content, such as software applications and data, for use by the CPU  170  and the GPU  172 . As shown, the system memory  174  includes a 3D modelling graphical user interface (GUI)  120 , a position-based dynamics (PBD) simulation framework  110 , and an animation engine  130 . The 3D modelling GUI  120 , the PBD simulation framework  110 , and the animation engine  130  are software applications that execute on the CPU  170 , the GPU  172 , or any combination of the CPU  170  and the GPU  172 . 
         [0022]    In operation, the 3D modelling GUI  120  enables specification and modification of a 3D model that describes a 3D object. The 3D modelling GUI  120  may be implemented in any technically feasible fashion and include a variety of functionality. For instance, the 3D modelling GUI  120  may include an interface that converts designer input such as symbols and brush stroke operations to geometries in the 3D model coupled to computer aided design (CAD) software that provides interactive feedback. Alternatively the 3D model GUI  120  may be configured to receive the 3D model via a 3D scanner that analyzes an existing 3D solid object to create the 3D model as a digital template. 
         [0023]    The 3D model may conform to any 3D format as known in the art. For instance, in some embodiments the 3D model may capture unit normal and vertices that define the 3D solid object in the stereolithograpy format. In alternate embodiments, the 3D model may capture a 3D mesh of interconnected triangles that define the 3D solid object in the collaborative design activity (COLLADA) format. In alternate embodiments, the 3D model is created manually and the 3D modelling GUI  120  is not included in the computer system  100 . 
         [0024]    The 3D modelling GUI  120  is configured to receive designer input information from the input devices  112 . After the 3D modelling GUI  120  processes the designer input information in conjunction with the 3D model, the 3D modelling GUI  120  delivers pixels to the display device  114 . The 3D modelling GUI  20  is configured to continuously repeat this cycle, enabling the designer to dynamically interact with the 3D model based on real-time images on the display device  110 . 
         [0025]    The 3D modelling GUI  120  is coupled to the PBD simulation framework  110 . This coupling may be implemented in any technically feasible fashion, such as exporting the 3D model from the 3D modelling GUI  20  and then importing the 3D model to the PBD simulation framework  120 . 
         [0026]    The PBD simulation framework  110  enables simulation of 3D models. Notably, the PBD simulation framework  110  implements position-based simulation techniques to efficiently simulate movement of 3D models for time-sensitive applications such as movies, computer games, etc. As shown, the PBD simulation framework  110  includes an elastic rod model generator  112 , a constraint tool  114 , and a simulation engine  116 . To optimize performance within the PBD simulation framework  110 , the shape and associated deformations of a 3D model are specified as discrete position of points. Consequently, conventional PBD simulation frameworks are unable to represent twisting of the 3D model (directly described with angular information as opposed to positional information). 
         [0027]    For this reason, the PBD simulation framework  110  includes the elastic rod model generator  112 . The elastic rod model generator  112  describes twist around the center line of a 3D model indirectly—using positional information to encode the angular information. While the PBD simulation framework  110  includes features that extend and optimize the processing of 3D models to enable realistic bending and twisting of elastic rods, the PBD simulation framework  110  leverages and extends many existing PBD processes, such as time-integration, collision, and constraint handling. 
         [0028]    In particular, the constraint tool  114  formulates elasticity via constraints. The values of these constraints are zero for the rest shape of the 3D object and increase as strain is applied. At each time step, the simulation engine  116  updates the affected positional information in the gradient direction of the constraint, representing the internal force attempting to restore the 3D object to the rest shape. In general, the constraint tool  114  and the simulation engine  116  implement heuristics that mimic the real-world behavior of objects, such as the bending and twisting a rod. 
         [0029]    The system memory  174  also includes an animation engine  130  that leverages the PBD simulation framework  110  to represent real-time motion of 3D objects. In alternate embodiments, the system memory  174  may include any number of applications that exploit features included in the PBD simulation framework  110 . 
         [0030]    In alternate embodiments, the 3D modelling GUI  120 , the PBD simulation framework  110 , and/or the animation engine  130  are integrated into any number (including one) of software applications. In other embodiments, the system memory  174  may not include the 3D modelling GUI  120  and/or the animation engine  130 . In some embodiments, the 3D modelling GUI  120 , the PBD simulation framework  110 , and/or the animation engine  130  may be provided as an application program (or programs) stored on computer readable media such as a CD-ROM, DVD-ROM, flash memory module, or other tangible storage media. 
         [0031]    The components illustrated in the computer system  100  may be included in any type of computer system  100 , e.g., desktop computers, server computers, laptop computers, tablet computers, and the like. Additionally, software applications illustrated in computer system  100  may execute on distributed systems communicating over computer networks including local area networks or large, wide area networks, such as the Internet. Notably, the elastic rod model generator  112  described herein is not limited to any particular computing system and may be adapted to take advantage of new computing systems as they become available. 
         [0032]    It will be appreciated that the computer system  100  shown herein is illustrative and that variations and modifications are possible. The number of CPUs  170 , the number of GPUs  172 , the number of system memories  174 , and the number of applications included in the system memory  174  may be modified as desired. Further, the connection topology between the various units in  FIG. 1  may be modified as desired. 
       Modelling Elastic Rods 
       [0033]      FIG. 2  is a conceptual diagram of an elastic rod model  200  generated and maintained within the position-based dynamics (PBD) simulation framework  110  of  FIG. 1 , according to embodiments of the present invention. In conventional PBD simulation frameworks, the center line of a rod is represented by a 3D polyline that can model such behaviors as bending and stretching, but is unable to represent torsion. To handle torsion, the PBD simulation framework  110  uses material frames  250  that are defined on the center line on the elastic rod model  200  and track the motion of the center line over time. 
         [0034]    For each material frame  250  at a point on the center line, torsion is represented as the angle between the current orientation of the material frame  250  and the orientation of the material frame  250  when the rod is at rest (i.e., an untwisted state of zero energy). Since the PBD simulation framework  110  does not allow direct specification of angular information, the elastic rod model generator  112  is configured to represent the angular information that defines the orientation of the material frames  250  indirectly. More specifically, the elastic rod model generator  112  encodes the orientation of the material frames  250  via positional information of ghost points  230 . Conceptually, the ghost points  230  represent material distributed around the center line. 
         [0035]    In operation, the elastic rod model generator  112  initially creates a 3D polyline that represents the center line of the elastic rod. As shown, the 3D polyline of the elastic rod model  200  includes (N+1) points and N edges  220 . The points  210  represent the end points of the interconnected line segments, and the edges  220  represent the line segments. 
         [0036]    After establishing the 3D polyline, the elastic rod model generator  112  adds N ghost points  230  to the 3D polyline. Notably, for each of the N edges  220 , the elastic rod model generator  112  adds the ghost point  230  that defines the material frame  250  for the line segment corresponding to the edge  220 . In general, the elastic rod model generator  112  places each of the ghost points  230  on a perpendicular bisector of the corresponding edge  220 . The angle of the perpendicular bisector encodes the twist about the center line relative to the rest state, and the distance along the perpendicular bisector reflects the rotational inertia around the corresponding edge  220 . 
         [0037]    As shown for material frame  250   e , each material frame  250  is defined based on an orthonormal material bases d 1 , d 2 , d 3 . The material frame  250  is a three-by-three orthonormal matrix D=|d 1 , d 2 , d 3 | that includes the orthonormal material bases as columns. The material frame axes  255   e  depicts the calculation of the material frame  250  based on the associated ghost point  230   e  and the points  210   e-1  and  210e . Using interpolation techniques in conjunction with the discrete points  210  and ghost points  230 , components within the PBD framework  110  efficiently access both the orientation of the material frames  250  and the resistance of the elastic rod to twisting in a continuous fashion across the length of the elastic rod. 
         [0038]      FIG. 3  depicts a modified discrete Darboux vector  350  implemented within the position-based dynamics (PBD) simulation framework  110  of  FIG. 1 , according to one embodiment of the present invention. 
         [0039]    In general, the simulation engine  116  models bending and twisting of the elastic rod model  200  using a discretization of Cosserat theory. Cosserat theory describes the bending and twisting energy of a rod from the rate-of-change of the material frame  250 . Accordingly, a key component of Cosserat theory is the Darboux vector  310 —an axial vector of frame rotation with respect to position along the rod that is defined using the bases of the material frames  250 . 
         [0040]    In Cosserat theory, the strain energy  330  is the Darboux vector difference between the deformed shape and the rest shape of a rod. To increase both the speed and robustness of the energy calculations, instead of implementing Cosserat formulas based on the conventional Darboux vector  350 , the simulation engine  116  substitutes a modified discrete Darboux vector  350 . As shown, the modified discrete Darboux vector  350  is a relative simple formula that does not use trigonometry. Advantageously, the modified discrete Darboux vector  350  enables the simulation engine  116  to apply the discretization of Cosserat theory without computing relatively expensive trigonometric functions. 
         [0041]    In various embodiments, the PBD framework  110  may implement any type equations to simulate motion based on the elastic rod model  200 . Further, components within the PBD framework  110  may simplify and/or discretize formulas to optimize performance in any technically feasible manner. 
         [0042]      FIG. 4  is a conceptual illustration of a bidirectional interleaving ordering  410  enforced by the constraint tool  114  of  FIG. 1 , according to one embodiment of the present invention. As part of representing and simulating the elastic rod model  200 , the PBD simulation framework  110  handles elasticity in the form of constraints, such as coupling between a point  210 , a material frame  250 , a triangle, and/or a rigid body. 
         [0043]    In general, the simulation engine  116  iteratively updates the locations of the points  210  and the ghost points  230  such that linearized constraints are locally satisfied. In the PBD simulation framework  110 , the simulation engine  116  enforces each constraint in the form of an internal force. Enforcing the twist constraint of the elastic rod model  220  in the conventional sequential order is similar to an elastic wave propagating inside the elastic rod. The elastic rod has many harmonic vibration modes where the elastic wave goes back and force in the strand, and the sequential constraint enforcement may excite such vibrations. 
         [0044]    For this reason, the constraint tool  114  configures the simulation engine  116  to apply the constraints based on the bidirectional interleaving order  410 . As shown, the constraint tool  114  interleaves the constraints for both directions of the elastic rod model  200 . Advantageously, applying constraints simultaneously from two directions cancels the vibration modes, producing stable results without excessive energy gain. 
         [0045]      FIG. 5  is a conceptual diagram of a curly haired bunny  510  that is based on a 3D model in which the hair strands are represented using the elastic rod model  200  of  FIG. 2 , according to one embodiment of the present invention. 
         [0046]    Because the PBD simulation framework  100  is both efficient and robust, the PBD framework  110  may be used to provide advanced functionality in a variety of interactive tools. In some embodiments, the 3D modelling GUI  120  includes functionality that enables the user to create hair volumes. In particular, the 3D modelling GUI  120  provides mechanisms to interactive manipulate parameters of the hair, such as twisting and scaling individual strands. This hair design functionality may also be extended to create solid hair for 3D printing purposes. For explanatory purposes,  FIG. 5  illustrates the model of the curly haired bunny  510  designed and simulated in the PBD simulation framework  100 , and the resulting 3D-printed bunny object. 
         [0047]    Since there are thousands of hairs on a typical scalp, without the efficiency exhibited by the PBD simulation framework  100 , such constructive interactive design would be prohibitively slow. For example, the complex calculations employed by finite element methods require unreasonable amounts of time for time-sensitive applications. Further, because of the high number of PBD-based calculations necessary to realistically simulate 3D objects, manual approaches are unable to produce useful real-time results. 
         [0048]      FIG. 6  is a flow diagram of method steps for establishing a position-based dynamics simulation (PBD) framework that supports elastic rods modelling, according to one embodiment of the present invention. Although the method steps are described with reference to the systems of  FIGS. 1-5 , persons skilled in the art will understand that any system configured to implement the method steps, in any order, falls within the scope of the present invention. 
         [0049]    As shown, a method  600  begins at step  604 , where the elastic rod model generator  112  initializes the elastic rod model  200  for a rod. More specifically, the elastic rod model generator  112  creates a polyline that resents the center line of the rod and includes (N+1) points  210  and (N) edges  220 . At step  606 , the elastic rod model generator  112  sets a current segment to the portion of the center line segment that is represented by the first edge  220 . 
         [0050]    At step  610 , the elastic rod model generator  112  determines a “rest” perpendicular bisector as the perpendicular bisector of the current line segment when the current segment is at rest (state of zero energy). The elastic rod model generator  112  may determine the rest perpendicular bisector in any technically feasible fashion that is consistent across the points  210  and the ghost points  230 . 
         [0051]    At step  612 , the elastic rod model generator  112  computes a “current” perpendicular bisector of the current segment, encoding the twist of the current segment as the angle between the current perpendicular bisector and the rest perpendicular bisector. Notably, the elastic rod model generator  112  sets the current perpendicular bisector such that the cross product between the current edge  220  and the current perpendicular bisector represents the rotation of the current edge  220 . In alternate embodiments, the elastic rod model generator  112  may encode the twist in any technically feasible and deterministic fashion. 
         [0052]    At step  614 , the elastic rod model generator  112  computes a strength distance based on the bending resistance of the current segment. The elastic rod model generator  112  may encode the bending resistance in any technically feasible and consistent manner. At step  616 , the elastic rod model generator  112  determines the coordinates for the ghost point  200 —setting the position to be the strength distance along the current perpendicular bisector. Notably, the current perpendicular bisector is one of the axes of the material frame  250 . The elastic rod model generator  112  computes the remaining two axes as illustrated in the material frame axes  255  of  FIG. 2 . Advantageously, using the ghost point  200  to encode the strength and material frame for the current segment enables realistic modelling of torsion in the PBD simulation framework  110 . 
         [0053]    At step  618 , if the elastic rod model generator  112  determines that the current edge  220  is not the last edge  220  included in the polyline, then the method  600  proceeds to step  620 . At step  620 , the elastic rod model generator  112  sets the current segment to the center line segment that is represented by the next edge  220 , and the method  600  returns to step  610 . The elastic rod model generator  112  continues to execute steps  610 - 618 , encoding the material frame  250  for each edge  220  via the corresponding ghost point  230  until the elastic rod model  200  includes ghost points  230  for each of the edges  220 . 
         [0054]    If, at step  618 , the elastic rod model generator  112  determines that the edge  220  is the last edge  220  included in the polyline, then the method  600  proceeds directly to step  622 . At step  622  the elastic rod model  200  is complete and various components included in the PBD simulation framework  110  operate on the elastic rod model  200 . Advantageously, the PBD simulation framework  110  analyzes the material frames  250  in conjunction with optimized simulation techniques such as applying the modified discrete Darboux vector  350  and enforcing bidirectional interleaving ordering  410  to increase the speed and robustness of the simulation. 
         [0055]    In sum, the disclosed techniques may be used to efficiently model and simulate complex bending and twisting of elastic rods in position-based dynamics (PBD) frameworks. In operation, an elastic rod model generator partitions an elastic rod into segments that are represented by edges and connected at points. This polyline represents the physical location of the elastic rod and characterizes any bending and stretching, but not torsion. To represent torsion, the elastic rod model generator enhances the polyline-adding a “ghost” point per edge. Notably, the elastic rod model generator selects the location of each ghost point to define the orientation of a material frame that follows the tangential direction of the elastic rod. By encoding the orientation of the material frames in the coordinates of the ghost points, the twist about each edge is indirectly, but accurately, represented. 
         [0056]    Advantageously, the position-based elastic modelling techniques disclosed herein enable realistic, real-time animation of objects that bend and twist. In particular, the insertion of ghost points enables plausible real-time simulation and rendering of curly hair. By contrast, conventional PBD frameworks are unable to represent the torsion in elastic rods, and therefore are unable to simulate phenomena such as naturally-behaving curly hair. Further, since the position-based elastic model leverages the efficiency of position-based dynamic, the elastic rod model and associated techniques may be successfully employed by time-critical applications, such as game engines. 
         [0057]    The descriptions of the various embodiments have been presented for purposes of illustration, but are not intended to be exhaustive or limited to the embodiments disclosed. Many modifications and variations will be apparent to those of ordinary skill in the art without departing from the scope and spirit of the described embodiments. 
         [0058]    Aspects of the present embodiments may be embodied as a system, method or computer program product. Accordingly, aspects of the present disclosure may take the form of an entirely hardware embodiment, an entirely software embodiment (including firmware, resident software, micro-code, etc.) or an embodiment combining software and hardware aspects that may all generally be referred to herein as a “circuit,” “module” or “system.” Furthermore, aspects of the present disclosure may take the form of a computer program product embodied in one or more computer readable medium(s) having computer readable program code embodied thereon. 
         [0059]    Any combination of one or more computer readable medium(s) may be utilized. The computer readable medium may be a computer readable signal medium or a computer readable storage medium. A computer readable storage medium may be, for example, but not limited to, an electronic, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, or device, or any suitable combination of the foregoing. More specific examples (a non-exhaustive list) of the computer readable storage medium would include the following: an electrical connection having one or more wires, a portable computer diskette, a hard disk, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or Flash memory), an optical fiber, a portable compact disc read-only memory (CD-ROM), an optical storage device, a magnetic storage device, or any suitable combination of the foregoing. In the context of this document, a computer readable storage medium may be any tangible medium that can contain, or store a program for use by or in connection with an instruction execution system, apparatus, or device. 
         [0060]    Aspects of the present disclosure are described above with reference to flowchart illustrations and/or block diagrams of methods, apparatus (systems) and computer program products according to embodiments of the disclosure. It will be understood that each block of the flowchart illustrations and/or block diagrams, and combinations of blocks in the flowchart illustrations and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general purpose computer, special purpose computer, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, enable the implementation of the functions/acts specified in the flowchart and/or block diagram block or blocks. Such processors may be, without limitation, general purpose processors, special-purpose processors, application-specific processors, or field-programmable 
         [0061]    The flowchart and block diagrams in the Figures illustrate the architecture, functionality, and operation of possible implementations of systems, methods and computer program products according to various embodiments of the present disclosure. In this regard, each block in the flowchart or block diagrams may represent a module, segment, or portion of code, which comprises one or more executable instructions for implementing the specified logical function(s). It should also be noted that, in some alternative implementations, the functions noted in the block may occur out of the order noted in the figures. For example, two blocks shown in succession may, in fact, be executed substantially concurrently, or the blocks may sometimes be executed in the reverse order, depending upon the functionality involved. It will also be noted that each block of the block diagrams and/or flowchart illustration, and combinations of blocks in the block diagrams and/or flowchart illustration, can be implemented by special purpose hardware-based systems that perform the specified functions or acts, or combinations of special purpose hardware and computer instructions. 
         [0062]    While the preceding is directed to embodiments of the present disclosure, other and further embodiments of the disclosure may be devised without departing from the basic scope thereof, and the scope thereof is determined by the claims that follow.