Patent Publication Number: US-11030773-B2

Title: Hand tracking based on articulated distance field

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     The present application is a continuation of U.S. patent application Ser. No. 15/994,563, entitled “Hand Tracking Based on Articulated Distance Field,” filed on May 31, 2017, and which is related to and claims priority to U.S. Provisional Patent Application Serial No. 62/513,199, entitled “Articulated Distance Fields for Ultra-Fast Tracking of Hands Interacting,” filed May 31, 2017, the entirety of which are incorporated by reference herein. The present application claims priority to and benefit of applications 15/994,563 and 62/513,199 and incorporates all such applications herein by reference. 
    
    
     BACKGROUND 
     Field of the Disclosure 
     The present disclosure relates generally to imagery capture and processing and more particularly to hand tracking using captured imagery. 
     Description of the Related Art 
     Hand tracking allows articulated hand gestures to be used as an input mechanism for virtual reality and augmented reality systems, thereby supporting a more immersive user experience. A generative hand tracking system captures images and depth data of the user&#39;s hand and fits a generative model to the captured image or depth data. To fit the model to the captured data, the hand tracking system defines and optimizes an energy function to find a minimum that corresponds to the correct hand pose. However, conventional hand tracking systems typically have accuracy and latency issues that can result in an unsatisfying user experience. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The present disclosure may be better understood, and its numerous features and advantages made apparent to those skilled in the art by referencing the accompanying drawings. The use of the same reference symbols in different drawings indicates similar or identical items. 
         FIG. 1  is a diagram illustrating a hand tracking system estimating a current pose of a hand based on a depth image in accordance with at least one embodiment of the present disclosure. 
         FIG. 2  is a diagram illustrating a hand tracking module of the hand tracking system of  FIG. 1  configured to estimate a current pose of a hand based on a depth image in accordance with at least one embodiment of the present disclosure. 
         FIG. 3  is a diagram illustrating interpolation of a grid of precomputed signed distances to generate a smooth signed distance field for estimating a distance from a point to a model in accordance with at least one embodiment of the present disclosure. 
         FIG. 4  is a diagram illustrating a base pose of a skinned tetrahedral volumetric mesh in accordance with at least one embodiment of the present disclosure. 
         FIG. 5  is a diagram illustrating a deformed pose of the tetrahedral volumetric mesh in accordance with at least one embodiment of the present disclosure. 
         FIG. 6  is a diagram illustrating a two-dimensional cross-section of the end of a finger in a base pose contained inside a triangular mesh in accordance with at least one embodiment of the present disclosure. 
         FIG. 7  is a diagram illustrating a two-dimensional cross-section of the end of a finger in a query pose contained inside a deformed triangular mesh in accordance with at least one embodiment of the present disclosure. 
         FIG. 8  is a diagram of an energy function based on a distance between each point of a three-dimensional (3D) point cloud based on a depth image and a candidate pose in accordance with at least one embodiment of the present disclosure. 
         FIG. 9  is a flow diagram illustrating a method of estimating a current pose of a hand based on a captured depth image in accordance with at least one embodiment of the present disclosure. 
         FIG. 10  is a flow diagram illustrating a method of minimizing an energy function by initializing using the pose from the previous frame and one or more poses derived from a coarse global predicted pose in accordance with at least one embodiment of the present disclosure. 
         FIG. 11  is a flow diagram illustrating a method of predicting a coarse global predicted pose of a hand in accordance with at least one embodiment of the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     The following description is intended to convey a thorough understanding of the present disclosure by providing a number of specific embodiments and details involving estimating a pose of a hand by volumetrically deforming a signed distance field based on a skinned tetrahedral mesh. It is understood, however, that the present disclosure is not limited to these specific embodiments and details, which are examples only, and the scope of the disclosure is accordingly intended to be limited only by the following claims and equivalents thereof. It is further understood that one possessing ordinary skill in the art, in light of known systems and methods, would appreciate the use of the disclosure for its intended purposes and benefits in any number of alternative embodiments, depending upon specific design and other needs. 
       FIGS. 1-11  illustrate techniques for estimating a pose of at least one hand by volumetrically deforming a signed distance field using a skinned tetrahedral mesh to locate a local minimum of an energy function, wherein the local minimum corresponds to the hand pose. A hand tracking module receives depth images of a hand from a depth camera and identifies a pose of the hand by fitting an implicit surface model of a hand, defined as the zero crossings of an articulated signed distance function, to the pixels of a depth image that correspond to the hand. The hand tracking module fits the model to the pixels by first volumetrically warping the pixels into a base pose and then interpolating 3D grid of precomputed signed distance values to estimate the distance to the implicit surface model. The volumetric warp is performed using a skinned tetrahedral mesh. The hand tracking module uses the skinned tetrahedral mesh to warp space from a base pose to a deformed pose to define an articulated signed distance field from which the hand tracking module derives candidate poses of the hand. Explicitly generating the articulated signed distance function is, however, avoided, by instead warping the pixels from the deformed pose to the base pose where the distance to the surface can be estimated by interpolating the precomputed 3D grid of signed distance values. The hand tracking module then minimizes the energy function based on the distance of each corresponding pixel as to identify the candidate pose that most closely approximates the pose of the hand. 
     In some embodiments, the hand tracking module initializes the candidate poses using the pose from the previous frame, that is, the depth image immediately preceding the current depth image. The hand tracking system leverages a depth camera with an extremely high frame rate to minimize the difference between the true pose from the previous frame and the true pose in the current frame. In some embodiments, the hand tracking module further initializes the candidate poses by a predicted pose. To predict a pose, the hand tracking module segments the pixels of the depth images based on a probability for each pixel representing a left hand, a right hand, or a background. The hand tracking module generates a three-dimensional (3D) point cloud of at least one of the left hand and the right hand based on the corresponding pixels and predicts a global orientation of the hand based a comparison of the 3D point cloud to a plurality of known poses to generate the predicted current pose. 
       FIG. 1  illustrates a hand tracking system  100  configured to support hand tracking functionality for AR/VR applications, using depth sensor data in accordance with at least one embodiment of the present disclosure. The hand tracking system  100  can include a user-portable mobile device, such as a tablet computer, computing-enabled cellular phone (e.g., a “smartphone”), a head-mounted display (HMD), a notebook computer, a personal digital assistant (PDA), a gaming system remote, a television remote, camera attachments with or without a screen, and the like. In other embodiments, the hand tracking system  100  can include another type of mobile device, such as an automobile, robot, remote-controlled drone or other airborne device, and the like. For ease of illustration, the hand tracking system  100  is generally described herein in the example context of a mobile device, such as a tablet computer or a smartphone; however, the hand tracking system  100  is not limited to these example implementations. The hand tracking system  100  includes a hand tracking module  110  estimating a current pose  140  of a hand  120  based on a depth image  115  captured by a depth camera  105  in accordance with at least one embodiment of the present disclosure. In this example, the hand  120  is a right hand making a pointing gesture, with the thumb and index finger extended and the remaining fingers curled down to the palm. 
     The depth camera  105 , in one embodiment, uses a modulated light projector (not shown) to project modulated light patterns into the local environment, and uses one or more imaging sensors  106  to capture reflections of the modulated light patterns as they reflect back from objects in the local environment  112 . These modulated light patterns can be either spatially-modulated light patterns or temporally-modulated light patterns. The captured reflections of the modulated light patterns are referred to herein as “depth images”  115 . In some embodiments, the depth camera  105  calculates the depths of the objects, that is, the distances of the objects from the depth camera  105 , based on the analysis of the depth images  115 . 
     The hand tracking module  110  receives a depth image  115  from the depth camera  105  and identifies a pose of the hand  120  by fitting a hand model to the pixels of the depth image  115  that correspond to the hand  120 . In some embodiments, the model is parameterized by 28 values (e.g., four joint articulations of each of the five fingers, two degrees of freedom at the wrist, and six degrees of freedom for global orientation). In some embodiments, the hand tracking module  110  parameterizes the global rotation of the model using a quaternion so that the pose vector θ is 29-dimensional. The hand tracking module  110  segments out of and back projects from the depth image  115  a set of 3D data points corresponding to the hand  120 . The hand tracking module  110  then fits a parameterized implicit surface model S(θ) ⊆   3 , formulated as the zero crossings of an articulated signed distance function, to the set of 3D data points {x n } n=1   N  ⊆   3 . The hand tracking module  110  minimizes the distance from each 3D data point to the surface by minimizing the energy 
                       E   data     ⁡     (   θ   )       =         ∑     n   =   1     N     ⁢       D   ⁡     (       x   n     ,   θ     )       2       =       ∑     n   =   1     N     ⁢       min     y   ∈     s   ⁡     (   θ   )           ⁢              x   n     -   y          2                   (   1   )               
where E data (θ) is the energy of the pose θ, D(x n ,θ) is the distance from each 3D data point x n  to the nearest pointy of the surface model in the pose θ, and N is the number of 3D data points in the set.
 
     To facilitate increased accuracy and efficiency of minimizing the energy, the hand tracking module  110  defines the distance D (x, θ) to an implicit surface of the hand model in a way that is relatively easy and fast to compute. The hand tracking module  110  builds a tetrahedral mesh (not shown) and skins the vertices to a skeleton (not shown). By defining x in relation to its barycentric coordinates in a tetrahedron of the mesh, the hand tracking module  110  defines a function that warps the space from a base pose to a deformed pose, as is described in more detail below. Based on the deformed pose, the hand tracking module  110  defines an articulated signed distance field. A point in the space of the current pose can be warped back to the base pose where the distance to the surface can be estimated efficiently by interpolating a precomputed 3D grid of signed distances. The hand tracking module  110  leverages this as part of its process to rapidly estimate a current pose  140  of the hand  120 . 
     In some embodiments, the hand tracking module  110  uses the current pose estimate  140  to update graphical data  135  on a display  130 . In some embodiments, the display  130  is a physical surface, such as a tablet, mobile phone, smart device, display monitor, array(s) of display monitors, laptop, signage and the like or a projection onto a physical surface. In some embodiments, the display  130  is planar. In some embodiments, the display  130  is curved. In some embodiments, the display  130  is a virtual surface, such as a three-dimensional or holographic projection of objects in space including virtual reality and augmented reality. In some embodiments in which the display  130  is a virtual surface, the virtual surface is displayed within an HMD of a user. The location of the virtual surface may be relative to stationary objects (such as walls or furniture) within the local environment  112  of the user. 
       FIG. 2  is a diagram illustrating the hand tracking module  110  of the hand tracking system  100  of  FIG. 1  in accordance with at least one embodiment of the present disclosure. The hand tracking module  110  includes a memory  205 , a pixel segmenter  210 , a reinitializer  215 , an interpolator  220 , and a volumetric deformer  225 . Each of these modules represents hardware, software, or a combination thereof, configured to execute the operations as described herein. The hand tracking module  110  is configured to receive a depth image  115  from the depth camera (not shown) and to generate a current pose estimate  140  based on the depth image  115 . 
     The memory  205  is a memory device generally configured to store data, and therefore may be a random access memory (RAM) memory module, non-volatile memory device (e.g., flash memory), and the like. The memory  205  may form part of a memory hierarchy of the hand tracking system  100  and may include other memory modules, such as additional caches not illustrated at  FIG. 1 . The memory  205  is configured to receive and store the depth image  115  from the depth camera (not shown). 
     The pixel segmenter  210  is a module configured to segment the pixels of the depth image  115  into pixels corresponding to a left hand, a right hand, and a background. In some embodiments, the pixel segmenter  210  assigns a probability for each pixel of the depth image  115  as corresponding to a left hand p left , a right hand p right , and a background p bg  ∈ [0,1] to produce a probability map P. In some embodiments, the pixel segmenter  210  thresholds P with a high value η high  ∈ [0,1], convolves the output with a large bandwidth Gaussian filter, and then finds the location of the maximum value, which the hand segmenter  210  assigns as a hand position. The hand segmenter  210  then thresholds P with a smaller value η low  and intersects P with a sphere of radius r sphere  ∈   to segment the hand pixels. 
     In some embodiments, the pixel segmenter  210  also trains a Randomized Decision Forest (RDF) classifier to produce P. The RDF classifier (not shown) employs depth and translation invariant features which threshold the depth difference of two pixels at depth-normalized offsets around the central pixel. For each pixel p at coordinate (u, v), on a depth image I, each split node in the tree evaluates the function: 
                       I   ⁡     (       u   +       Δ   ⁢           ⁢     u   i       Γ       ,     v   +       Δ   ⁢           ⁢     v   1       Γ         )       -     I   ⁡     (       u   +       Δ   ⁢           ⁢     u   2       Γ       ,     v   +       Δ   ⁢           ⁢     v   2       Γ         )         &gt;   τ           (   2   )               
where is Γ is I(u,v), Δu i  and Δv i  are the two offsets and τ is the threshold for that split node. In some embodiments, to enhance the feature pool for subtasks that are invariant to rotations, such as a single extended hand, the pixel segmenter  210  introduces a new rotationally invariant family of features, which threshold the average depth of two co-centric rings:
 
                         R   ⁡     (     u   ,   v   ,     r   1     ,   I     )       K     -       R   ⁡     (     u   ,   v   ,     r   2     ,   I     )       K       &gt;   τ           (   3   )               
where R(u,v,r,I) is the sum over K depth pixels found on a ring of depth-scaled radius r around the central pixel. In some embodiments, the pixel segmenter  210  approximates the ring with a fixed number of points k:
 
     
       
         
           
             
               
                 
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     In some embodiments, the pixel segmenter  210  additionally defines a unary version of this feature as follows: 
     
       
         
           
             
               
                 
                   
                     
                       
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     At training time, the pixel segmenter  210  samples from a pool of binary and unary rotationally dependent and invariant features based on a learned prior pose. In some embodiments, for each considered feature, the pixel segmenter  210  uniformly samples multiple τ values from a fixed range and selects the value that maximizes the information gain. The pixel segmenter  210  outputs a segmented depth image R per hand. 
     In some embodiments, the pixel segmenter  210  uses a convolutional neural network (CNN) or a randomized decision forest (RDF) or both to produce a probability map that encodes for each pixel, the probability of the pixel belonging to the left hand, the right hand, and the background, respectively. To detect the right hand, the pixel segmenter  210  temporarily sets all values of the probability map p right to zero that are below a high value η high  ∈ [0,1]. The pixel segmenter  210  convolves the output with a large bandwidth Gaussian filter, and then uses the location of the maximum value. The pixel segmenter  210  then removes outliers from the original segmentation p right  by setting to zero the value of any pixels whose probability is less than η low  ∈ [0,η high ] or whose 3D location is not contained in a sphere of radius r sphere  ε   around the hand detection. The pixel segmenter  210  thus ensures that pixels far from the most prominent hand (e.g., pixels on other people&#39;s hands in the background) do not contaminate the segmentation while allowing the machine learning method to discard nearby pixels that are recognized as not belonging to the hand (e.g., pixels on the user&#39;s chest). The hand segmenter  210  back projects the pixels that pass the test into 3D space using the depth camera  105  parameters to form a point cloud {x n } n=1   N  ⊆    3  as to define the energy 
     
       
         
           
             
               
                 
                   
                     
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     The reinitializer  215  receives the segmented depth image R from the pixel segmenter  210 . The reinitializer  215  resets the hand tracking module  110  by generating a coarse global predicted pose when the hand tracking module  110  loses track of the hand  120  of  FIG. 1 . In some embodiments, the hand tracking module  110  uses coarse global predicted pose as a candidate pose of the hand. In some embodiments, the reinitializer  215  uses an RDF to estimate the six degrees of freedom (6 DOF) hand pose by locating three joints on the palm, which is assumed to be planar. The three joints are the wrist joint q w , the base of the metacarpophalangeal (MCP) joint q i , and the base of the pinky MCP q p . The reinitializer  215  locates the three joints by evaluating each pixel p in R to produce a single vote for the three-dimensional (3D) offset of each joint relative top. The trees of the RDF are trained with a regression objective to minimize the vote variance in the leaves. Each pixel votes for all the joints, which are aggregated separately to form a vote distribution per joint. The reinitializer  215  selects the modes of the distributions as final estimates for the three joints. In some embodiments, the reinitializer  215  converts the three joints into a reinitialization pose by setting the global translation to q w  and deriving the global orientation by finding the orientation of the three-dimensional triangle defined by the three joints. The reinitializer  215  then samples a set of finger poses randomly from the prior pose to generate the coarse global predicted pose. 
     The interpolator  220  precomputes a 3D grid of signed distance values in a base pose θ 0  and uses tricubic interpolation to define a signed distance D(x,θ 0 )={tilde over (D)}(x) ∈   to the surface for any point x ε    3 . Tricubic interpolation gives access to smooth first and second order derivatives with respect to x. Thus, the signed distance field smoothly captures details of the model using tricubic interpolation. 
     The volumetric deformer  225  uses a linear skinned tetrahedral mesh to define a signed distance field into an arbitrary pose θ as a volumetric warp of the signed distance field of the interpolator  220 . Instead of explicitly generating the deformed signed distance function, the volumetric deformer  225  can efficiently warp a point in the current pose back into the base pose so the distance to the implicit surface, and its derivatives, can be rapidly estimated by the interpolator. The volumetric deformer  225  defines the deformation of the vertices of the tetrahedral mesh via linear blend skinning. 
     Strictly speaking, the tetrahedral mesh actually defines a warp y=W(x,θ) from the base pose to the deformed pose. The function is largely invertible, such that the set of points in the base pose that deform to a point in the current pose is typically 1, unless the deformation causes tetrahedra to self-intersect. In the latter case, the ambiguity is resolved by simply picking the point in the base pose with a smaller absolute distance to the implicit surface as defined by the interpolator  220 . This thus defines a function W −1 (x,θ) that warps the space from the deformed pose to the base pose. The distance to the surface D(x,θ) for an arbitrary pose θ is thus defined as D(x,θ)={tilde over (D)}(W −1 (x,θ)), which can be easily evaluated without explicitly generating a dense signed distance field in the deformed pose. Thus, the tetrahedral mesh transforms the detail of the signed distance field into different poses. The tetrahedral mesh warp introduces artifacts only at articulation points, which can be addressed by densifying the tetrahedral mesh only at the articulation points. 
     The hand tracking module  110  composes the precomputed signed distance field {tilde over (D)}(x) ∈ R from the interpolator  220  and the volumetric deformation W(x,θ) from the skinned volumetric tetrahedral mesh to define an articulated signed distance field D (x,θ)={tilde over (D)}(W −1 (x,θ)) that yields the estimated distance to the surface of the point x in the estimated pose. The hand tracking module  110  uses the articulated signed distance field D(x,θ) to define an energy function E(θ)=Σ n=1   N D(x n ,θ) 2 , although other terms encoding prior knowledge could be incorporated. 
     In some embodiments, the hand tracking module  110  initializes the candidate poses θ first using the pose θ prev  output from the system in the previous frame. In some embodiments, the hand tracking module  110  initializes further candidate poses θ by using a coarse global predicted pose θ pred  generated by the reinitializer  215 . In some embodiments, the depth camera (not shown) employs a high frame rate, such that the difference between the pose θ prev  in the previous frame and the true pose in the current frame is minimized. By minimizing the energy function, the hand tracking module  110  generates a current pose estimate  140 . 
     In some embodiments, the hand tracking module  110  tracks two hands by jointly) optimizing over poses Θ={θ left ,θ right } and a set of right handed assignments Y={η n } n=1   N  ⊆{0,1} N  which implicitly define a set of left handed assignments Γ( )={1−η n } n=1   N . The hand tracking module  110  then formulates the full energy to be optimized as
 
 {tilde over (E)} (Θ)= E (θ left ;Γ( ))+ E (θ right ;( )+λ assign Σ n=1   N (η n γ n   right +(1−η n )γ n   left )  (7)
 
where γ n   right  and γ n   left  are penalties output from the segmentation forest for assigning data point n to the right and the left hand pose, respectively. To optimize this function, the hand tracking module  110  performs alternation between Θ and γ, updating Θ with Levenberg updates and updating γ by discretely considering whether assigning the data point to the left or right hand will lower the energy.
 
       FIG. 3  illustrates interpolation of a pixel  320  of a depth image based on a precomputed distance function to generate a smooth signed distance field (SDF)  330  for estimating a distance  325  from the pixel  320  to a model  305  in a base pose θ 0  in accordance with at least one embodiment of the present disclosure. The interpolator  220  of  FIG. 2  precomputes a dense grid  310  of signed distances  315  in the base pose θ 0 . The interpolator  220  then uses tricubic interpolation to define the signed distance function  325  D(x,θ)={tilde over (D)}(x) ε   to the surface for any point x ε    3  in the neutral, or base, pose. Precomputing and interpolating the grid of signed distances  315  eases the computational burden of evaluating distances D(x,θ) and smoothly captures the high frequency details of the model  305 . 
       FIG. 4  illustrates a base pose  400  of a tetrahedral volumetric mesh  410  of the volumetric deformer  225  of  FIG. 2  with vertices skinned to the dense SDF  330  of  FIG. 3  in accordance with at least one embodiment of the present disclosure. The skinned tetrahedral mesh  410  transforms the detail of the dense SDF  330  into different poses. The skinned tetrahedral mesh  410  introduces artifacts only at articulation points. In some embodiments, the skinned tetrahedral mesh  410  is densified at the articulation points, e.g.,  415 ,  420 ,  425 , while the dense SDF  330  represents the geometry of the pose in other areas. In some embodiments, the volumetric deformer (not shown) applies arbitrary mesh skinning techniques to deform a single SDF  330 . Thus, the deformation function and detail representation are decoupled, allowing a coarse tetrahedral mesh to be used to transfer detailed static geometry represented by the SDF  330 . This may also allow the possibility of modifying the static geometry in the SDF  330  online without having to modify the deformation function. 
       FIG. 5  illustrates a deformed pose  500  of the tetrahedral volumetric mesh  410  of  FIG. 4  in accordance with at least one embodiment of the present disclosure. The volumetric deformer  225  of  FIG. 2  uses the tetrahedral volumetric mesh  410  to warp a point x to W(x,θ). Thus, the volumetric deformer  225  uses the tetrahedral volumetric mesh  410  to provide a function y=W(x,θ) that warps space from the base pose to a deformed pose. The function is largely invertible, such that it is also possible to define a function x=W −1 (y,θ) that warps the space from the deformed pose to the base pose. This allows the hand tracking module  110  to avoid explicitly warping and densely generating a signed distance function in a new pose which would be prohibitively expensive to perform continually while searching for a correct pose. Instead, the hand tracking module  110  can estimate the distance D(x,θ) of a point x to the implicit surface in any pose θ by instead warping x back into the base pose where the distance to the surface can be rapidly evaluated by interpolating a precomputed 3D grid of signed distance values. Further, as the warp and the signed distance field are differentiable almost everywhere, the hand tracking module  110  can also rapidly query derivatives to enable rapid local search of energy functions defined in terms of distances to the surface. 
       FIG. 6  illustrates a two-dimensional (2D) cross-section of the end of a finger  605  in a base pose contained inside a triangular mesh  610  in accordance with at least one embodiment of the present disclosure. The tetrahedral volumetric mesh  410  of  FIGS. 4 and 5  is depicted as a 2D equivalent triangular mesh  610  for ease of reference. The triangular mesh  610  includes triangles  614 ,  616 ,  618 ,  620 ,  622 ,  624 ,  626 , and  628 . 
       FIG. 7  illustrates a 2D cross-section of the end of the finger  605  of  FIG. 6  in a query pose θ contained inside a deformed triangular mesh  710  in accordance with at least one embodiment of the present disclosure. A triangular mesh in 2D is the analogue of a tetrahedral mesh in 3D and is thus used to more simply illustrate the technique. The tetrahedral mesh (illustrated as triangular mesh  710 ) includes tetrahedra (illustrated as triangles  714 ,  716 ,  718 ,  720 ,  722 ,  724 ,  726 , and  728 ), which correspond to tetrahedra (or triangles)  614 ,  616 ,  618 ,  620 ,  622 ,  624 ,  626 , and  628 , respectively, of  FIG. 6 . When the mesh  710  is deformed, each tetrahedra (or triangle)  714 ,  716 ,  718 ,  720 ,  722 ,  724 ,  726 , and  728  defines an affine transform between the base pose of  FIG. 6  and the query pose θ. This defines a volumetric warp W(x,θ) from the base pose to the query pose. Using the inverse affine transforms of each tetrahedra (or triangle), one can try to define an inverse warp W −1 (x,θ). Using this, the volumetric deformer  225  of  FIG. 2  implicitly defines a signed distance field D(x,θ) as described further herein. For a query point x (e.g., point  730 ) that falls inside the deformed mesh  710 , a tetrahedra (or triangle) τ that contains the point can use its inverse affine transform sends the query point to B τ (x,θ) where the distance to the implicitly encoded surface can be queried as {tilde over (D)}(B τ (x,θ)). For a pointy (e.g., point  732 ) that falls outside the deformed mesh  710 , the volumetric deformer  225  first measures the distance to the closest point contained in the tetrahedral mesh. To this distance, the volumetric deformer  225  then adds the distance obtained by evaluating the distance of this closest point to the surface using the aforementioned technique. 
     In more detail, for any point x, the volumetric deformer  225  uses the closest point q τ (x,θ)=V τ (θ){circumflex over (β)} τ (x,θ) where τ is the tetrahedron (or triangle) containing the closest point and V τ (θ) ∈    3×4  (or    2×3 ) is a matrix with the positions of the tetrahedron τ&#39;s four vertices (or triangle τ&#39;s three vertices) in pose θ stored in its columns and {circumflex over (β)} τ  (x,θ) ∈    4  (or {circumflex over (β)} τ (x,θ) ∈    3 ) is the barycentric coordinate of the closest point in the tetrahedron (or triangle) T under pose θ. In some embodiments, the volumetric deformer  225  warps the closest point back to the base pose as B τ (x,θ)=V τ (θ 0 ){circumflex over (β)} τ (x,θ) to query its distance to the implicitly encoded surface. When the query point x lies in the tetrahedral mesh, q τ (x,θ)=x, whereas when x lies outside the tetrahedral mesh (e.g., point  732 ), the volumetric deformer accounts for the additional distance between q τ (x,θ) and x. In some cases, the deformation of the tetrahedral mesh causes the query point x to fall in multiple overlapping tetrahedra, causing the volumetric warp to not be strictly invertible. The volumetric deformer  225  therefore resolves this issue by defining the set of tetrahedra (or triangles) that contain x as
 
 ( x,θ )={τ: q   τ   9   x, θ)= x}   (8)
 
The volumetric deformer  225  then chooses the tetrahedron (or triangle) τ* (x,θ) that will be used to warp the point back into the base pose as
 
                       τ   *     ⁡     (     x   ,   θ     )       =     {           arg   ⁢           ⁢     min     τ   ∈     T   ⁡     (     x   ,   θ     )                          D   ~     (       B   τ     ⁡     (     x   ,   θ     )                      when   ⁢           ⁢     T   (     x   ,   θ     )       ≠   ∅               arg   ⁢           ⁢     min   τ                  x   -       q   τ     ⁡     (     x   ,   θ     )                      when   ⁢           ⁢     T   ⁡     (     x   ,   θ     )         =   ∅                     (   9   )               
The first case selects the containing tetrahedron (or triangle) which warps the point back of minimum absolute distance to the surface in the base pose. The second case selects the tetrahedron (or triangle) that the point is closest to in the current pose. The volumetric deformer  225  then defines the articulated signed distance function to the surface to be
 
 D ( x,θ )=∥ x−q   τ*(x,θ) ( x, θ)∥+ {tilde over (D)} ( B   τ*(x,θ) ( x, θ))  (10)
 
where the first term measures the distance to the closest point in the selected tetrahedron (or triangle) and the second term warps that closest point back to the base pose to evaluate the signed distance to evaluate its distance to the surface.
 
     Thus, the volumetric deformer  225  divides the space into a discrete set of cells as τ*(x,θ) jumps from one tetrahedron (or triangle) to another. When x lands in at least one tetrahedron (or triangle), the volumetric deformer  225  uses an affine transform defined by the selected tetrahedron (or triangle) to map the space in the current pose back into the base pose for SDF evaluation. When x lands outside the tetrahedral mesh  510  (or triangular mesh  710 ), the volumetric deformer  225  selects the closest tetrahedron (triangle) and similarly uses the affine transform to warp the closest point on the closest tetrahedron&#39;s boundary into the base pose for SDF evaluation. The volumetric deformer  225  adds to this value the distance from x to the closest point on the tetrahedron boundary to compensate for the query point being outside the tetrahedral mesh. In some embodiments, the volumetric deformer  225  adds more tetrahedra (or triangles) to smooth out bumps around joints. 
       FIG. 8  is a diagram of an energy function  810  of a distance between each point of a three-dimensional (3D) point cloud based on the depth image  115  of  FIG. 1  and a candidate pose based on the articulated signed distance function in accordance with at least one embodiment of the present disclosure. The hand tracking module  110  of  FIGS. 1 and 2  generates the energy function  810  to evaluate how well the points of the 3D point cloud are explained by the candidate hand pose θ. The hand tracking module  110  defines the energy function as 
                     E   ⁡     (   θ   )       =         ∑     n   =   1     N     ⁢       min     y   ∈     S   ⁡     (   θ   )           ⁢              x   n     -   y          2         =       ∑     n   =   1     N     ⁢       D   ⁡     (       x   n     ,   θ     )       2                 (   11   )               
The articulated signed distance field defined allows D (x,θ) to be rapidly queried for distances and derivatives. As a result, the energy function above can be rapidly queried for both its value and descent directions so that rapid local search can be performed from initialization poses.
 
     In some embodiments, the hand tracking module  110  performs a local search to minimize the energy by bounding the candidate pose by the pose from the previous frame  820  of the depth camera  105  of  FIG. 1 . In some embodiments, the depth camera  105  is a high frame rate depth camera, such that the pose in the previous frame  825  is extremely likely to be close to the true pose in the current frame due to the short time interval between the frames. Rapidly minimizing the aforementioned energy function facilitates processing of depth frames at a high frame rate. In some embodiments, the hand tracking module  110  further initializes the candidate pose by the coarse global predicted pose  830  generated by the reinitializer  215 . By initializing the candidate pose by one or both of the pose of the previous frame and the coarse global predicted pose  830 , the hand tracking module  110  avoids local minima of the energy function  810 . 
       FIG. 9  is a flow diagram illustrating a method  900  of estimating a current pose of a hand based on a captured depth image in accordance with at least one embodiment of the present disclosure. At block  902 , the depth camera  105  of  FIG. 1  captures a depth image  115  of the hand  120 . At block  904 , the interpolator  220  of the hand tracking module  110  defines a dense signed distance field  330  based on the depth image  115 . At block  906 , the volumetric deformer  225  volumetrically defines the dense signed distance field  330  based on the tetrahedral mesh  510 . At block  908 , the volumetric deformer  225  defines the articulated signed distance function based on the volumetric deformation of the dense signed distance field  330 . At block  910 , the hand tracking module  110  minimizes the energy function  810  to estimate the current pose  140  by exploiting the deformer and interpolator that allows extremely rapid querying of distances to the implicit surface, and corresponding derivatives, in arbitrary poses. 
       FIG. 10  is a flow diagram illustrating a method  1000  of minimizing the energy function  810  for a candidate pose that is initialized by the pose in the previous frame  825  and a coarse global predicted pose  830  in accordance with at least one embodiment of the present disclosure. At block  1002 , the hand tracking module  110  sets the pose from the previous frame  825  as a first initialization of the candidate pose. At block  1004 , the hand tracking module  110  sets the coarse global predicted pose  830  as a second initialization of the candidate pose. At block  1006 , the hand tracking module  110  leverages an articulated signed distance function to provide rapid local search from each initialization. At block  1008 , the hand tracking module  110  estimates the current pose  140  as the candidate pose with the minimum energy function  810 . 
       FIG. 11  is a flow diagram illustrating a method  1100  of generating a coarse global predicted pose  830  of a hand  120  in accordance with at least one embodiment of the present disclosure. At block  1102 , the memory  205  receives a depth image  115 . At block  1104 , the pixel segmenter  210  segments the pixels of the depth image  115  into pixels corresponding to the left hand, the right hand, and the background. At block  1106 , each segmented pixel votes for locations on the palm of the hand  120  to generate point clouds. At block  1108 , the reinitializer  215  finds the center of each point cloud to generate the coarse global predicted pose  830  of the hand  120 . 
     In some embodiments, certain aspects of the techniques described above may implemented by one or more processors of a processing system executing software. The software comprises one or more sets of executable instructions stored or otherwise tangibly embodied on a non-transitory computer readable storage medium. The software can include the instructions and certain data that, when executed by the one or more processors, manipulate the one or more processors to perform one or more aspects of the techniques described above. The non-transitory computer readable storage medium can include, for example, a magnetic or optical disk storage device, solid state storage devices such as Flash memory, a cache, random access memory (RAM) or other non-volatile memory device or devices, and the like. The executable instructions stored on the non-transitory computer readable storage medium may be in source code, assembly language code, object code, or other instruction format that is interpreted or otherwise executable by one or more processors. 
     A computer readable storage medium may include any storage medium, or combination of storage media, accessible by a computer system during use to provide instructions and/or data to the computer system. Such storage media can include, but is not limited to, optical media (e.g., compact disc (CD), digital versatile disc (DVD), Blu-Ray disc), magnetic media (e.g., floppy disc, magnetic tape, or magnetic hard drive), volatile memory (e.g., random access memory (RAM) or cache), non-volatile memory (e.g., read-only memory (ROM) or Flash memory), or microelectromechanical systems (MEMS)-based storage media. The computer readable storage medium may be embedded in the computing system (e.g., system RAM or ROM), fixedly attached to the computing system (e.g., a magnetic hard drive), removably attached to the computing system (e.g., an optical disc or Universal Serial Bus (USB)-based Flash memory), or coupled to the computer system via a wired or wireless network (e.g., network accessible storage (NAS)). 
     Note that not all of the activities or elements described above in the general description are required, that a portion of a specific activity or device may not be required, and that one or more further activities may be performed, or elements included, in addition to those described. Still further, the order in which activities are listed are not necessarily the order in which they are performed. Also, the concepts have been described with reference to specific embodiments. However, one of ordinary skill in the art appreciates that various modifications and changes can be made without departing from the scope of the present disclosure as set forth in the claims below. Accordingly, the specification and figures are to be regarded in an illustrative rather than a restrictive sense, and all such modifications are intended to be included within the scope of the present disclosure. 
     Benefits, other advantages, and solutions to problems have been described above with regard to specific embodiments. However, the benefits, advantages, solutions to problems, and any feature(s) that may cause any benefit, advantage, or solution to occur or become more pronounced are not to be construed as a critical, required, or essential feature of any or all the claims. Moreover, the particular embodiments disclosed above are illustrative only, as the disclosed subject matter may be modified and practiced in different but equivalent manners apparent to those skilled in the art having the benefit of the teachings herein. No limitations are intended to the details of construction or design herein shown, other than as described in the claims below. It is therefore evident that the particular embodiments disclosed above may be altered or modified and all such variations are considered within the scope of the disclosed subject matter. Accordingly, the protection sought herein is as set forth in the claims below.