Dual mesh resampling

An apparatus comprising:

• means for obtaining a polygon mesh having at least one primal mesh signal and a polygon mesh connectivity;
• means for resampling said primal mesh signal to produce a dual mesh signal of said polygon mesh;
• means for using said polygon mesh connectivity to define an energy function relating said dual mesh signal with said primal mesh signal;
• and
• means for minimizing said energy function with respect to said dual mesh signal while keeping said primal mesh signal fixed to provide values for said dual mesh signal.

A method comprising:

• obtaining a first polygon mesh;
• modifying said first polygon mesh employing a resampling mesh operator to form a second polygon mesh;
• replacing said first mesh with said second polygon mesh;
• and
• repeating said steps of modifying and replacing until a stopping criterion is met.

A method comprising:

• obtaining a polygon mesh, wherein:
• said polygon mesh including a primal mesh signal and a polygon mesh connectivity,
• said polygon mesh connectivity including a plurality of polygon mesh vertices, a plurality of polygon mesh faces, and at least one regular edge;
• each polygon mesh face having at least three corners, each corner being a particular polygon mesh vertex from among said plurality of polygon mesh vertices, each regular edge being incident to two polygon mesh vertices and two polygon mesh faces;
• and
• said primal mesh signal including a plurality of primal points, each primal point being a point in an N-dimensional space, each primal point corresponding to a particular polygon mesh vertex from said plurality of polygon mesh vertices, and
• resampling the polygon mesh to produce a dual mesh signal, wherein said dual mesh signal including a plurality of dual points, each dual point being a point in said N-dimensional space, each dual point corresponding to a particular polygon mesh face from said plurality of polygon mesh faces;
• reading said polygon mesh from an input source;
• minimizing an energy function of the primal points and the dual points with respect to the dual points while the primal points are kept fixed, wherein:
• said energy function being a sum of a plurality of squares of terms, each square term being associated with a particular regular edge of said polygon mesh connectivity,
• said square term being the square of a difference term, said difference term being computed by subtracting the sum of two dual points from the sum of two primal points,
• said two dual points corresponding to the two mesh faces to which said regular edge is incident, and
• said two primal point corresponding to the two vertices to which said regular edge is incident;
• and
• writing said dual mesh signal to an output source.

A method comprising:

• creating a dual polygon mesh from a polygon mesh, wherein:
• said polygon mesh including a primal mesh signal and a polygon mesh connectivity,
• said polygon mesh connectivity including a plurality of polygon mesh vertices, a plurality of polygon mesh faces, and at least one regular edge,
• each polygon mesh face having at least three corners, each corner being a particular polygon mesh vertex from among said plurality of polygon mesh vertices, each regular edge being incident to two polygon mesh vertices and two polygon mesh faces,
• said primal mesh signal including a plurality of primal points, each primal point being a point in an N-dimensional space, each primal point corresponding to a particular polygon mesh vertex from said plurality of polygon mesh vertices,
• said dual polygon mesh including a dual mesh signal and a dual mesh connectivity, said dual mesh signal including a plurality of dual points, each dual point being a point in said N-dimensional space, each dual point corresponding to a particular polygon mesh face from said plurality of polygon mesh faces, said dual mesh connectivity having a plurality of dual mesh vertices, and a plurality of dual mesh faces,
• each dual mesh vertex corresponding to one said polygon mesh face of said polygon mesh connectivity, each dual mesh face corresponding to one polygon mesh vertex of said polygon mesh connectivity;
• reading said polygon mesh from an input source;
• minimizing an energy function of the primal points and the dual points with respect to the dual points while the primal points are kept fixed, wherein:
• said energy function is a sum of a plurality of squares of terms, each square term being associated with a particular regular edge of said polygon mesh connectivity, said square term being the square of a difference term,
• said difference term being computed by subtracting the sum of two dual points from the sum of two primal points,
• said two dual points corresponding to the two mesh faces to which said regular edge is incident, and
• said two primal points corresponding to the two vertices to which said regular edge is incident;
• constructing said dual mesh connectivity from said polygon mesh connectivity;
• combining said dual mesh signal and said dual mesh connectivity as said dual polygon mesh;
• and
• writing said dual polygon mesh to an output source.

A method comprising:

• smoothing a polygon mesh, wherein:
• said polygon mesh includes a primal mesh signal and a polygon mesh connectivity,
• said polygon mesh connectivity including a plurality of polygon mesh vertices, a plurality of polygon mesh faces, and at least one regular edge,
• each polygon mesh face having at least three corners, each corner being a particular polygon mesh vertex from among said plurality of polygon mesh vertices, each regular edge being incident to two polygon mesh vertices and two polygon mesh faces, and
• said primal mesh signal including a plurality of primal points, each primal point being a point in an N-dimensional space, each primal point corresponding to a particular polygon mesh vertex from said plurality of polygon mesh vertices;
• reading said polygon mesh from an input source;
• creating a temporary dual mesh signal, wherein said temporary dual mesh signal including a plurality of dual points, each dual point being a point in said N-dimensional space, each dual point corresponding to a particular polygon mesh face from said plurality of polygon mesh faces;
• minimizing an energy function of the primal points and the dual points with respect to the dual points while the primal points are kept fixed, wherein:
• said energy function is a sum of a plurality of squares of terms, each square term being associated with a particular regular edge of said polygon mesh connectivity, said square term being the square of a difference term,
• said difference term being computed by subtracting the sum of two dual points from the sum of two primal points, and
• said two dual points corresponding to the two mesh faces to which said regular edge is incident, said two primal points corresponding to the two vertices to which said regular edge is incident;
• recomputing said primal mesh signal by minimizing said energy function with respect to the primal points while the dual points are kept fixed;
• and
• writing said primal mesh to an output source.

A method comprising:

• refining a polygon mesh producing a primal-dual mesh, wherein:
• said polygon mesh including a primal mesh signal and a polygon mesh connectivity, said polygon mesh connectivity including a plurality of polygon mesh vertices, a plurality of polygon mesh faces, and at least one regular edge,
• each polygon mesh face having at least three corners, each corner being a particular polygon mesh vertex from among said plurality of polygon mesh vertices, each regular edge being incident to two polygon mesh vertices and two polygon mesh faces,
• said primal mesh signal including a plurality of primal points, each primal point being a point in an N-dimensional space, each primal point corresponding to a particular polygon mesh vertex from said plurality of polygon mesh vertices;
• and
• reading said polygon mesh from an input source;
• creating a temporary dual mesh signal, wherein said temporary dual mesh signal including a plurality of dual points, each dual point being a point in said N-dimensional space, each dual point corresponding to a particular polygon mesh face from said plurality of polygon mesh faces;
• minimizing an energy function of the primal points and the dual points with respect to the dual points while the primal points are kept fixed, wherein:
• said energy function being a sum of a plurality of squares of terms, each square term being associated with a particular regular edge of said polygon mesh connectivity,
• said square term being the square of a difference term, said difference term being computed by subtracting the sum of two dual points from the sum of two primal points, and
• said two dual points corresponding to the two mesh faces to which said regular edge is incident, said two primal points corresponding to the two vertices to which said regular edge is incident;
• constructing a primal-dual mesh signal by concatenating said primal mesh signal with said dual mesh signal;
• constructing a primal-dual mesh connectivity from said polygon mesh connectivity, wherein said primal-dual mesh connectivity has a plurality of primal-dual mesh vertices, and a plurality of primal-dual mesh faces, each primal-dual mesh face corresponding to one particular regular edge of said polygon mesh connectivity, each prima-dual mesh vertex corresponds to one of a particular polygon mesh vertex and a particular polygon mesh face of the polygon mesh connectivity;
• constructing said primal-dual mesh by combining said primal-dual mesh signal and said primal-dual mesh connectivity;
• and
• writing said primal-dual mesh to an output source.

A system for resampling a primal mesh signal to produce a dual mesh signal comprising:

• a computer with operating system, computer memory, input/output interface, and a central processing unit;
• and
• an application program executed by the central processing unit for resampling a primal mesh to produce a dual mesh signal, wherein:
• said polygon mesh including a primal mesh signal and a polygon mesh connectivity, said polygon mesh connectivity including a plurality of polygon mesh vertices, a plurality of polygon mesh faces, and at least one regular edge,
• each polygon mesh face having at least three corners, each corner being a particular polygon mesh vertex from among said plurality of polygon mesh vertices, each regular edge being incident to two polygon mesh vertices and two polygon mesh faces,
• said primal mesh signal including a plurality of primal points, each primal point being a point in an N-dimensional space, each primal point corresponding to a particular polygon mesh vertex from said plurality of polygon mesh vertices,
• said dual mesh signal including a plurality of dual points, each dual point being a point in said N-dimensional space, each dual point corresponding to a particular polygon mesh face from said plurality of polygon mesh faces,
• said application program comprising the computer implemented steps of:
• reading said polygon mesh from one of the computer memory and the peripheral device through the input/output interface;
• and
• minimizing an energy function of the primal points and the dual points with respect to the dual points while the primal points are kept fixed, wherein:
• said energy function is a sum of a plurality of squares of terms, each square term being associated with a particular regular edge of said polygon mesh connectivity,
• said square term being the square of a difference term, said difference term being computed by subtracting the sum of two dual points from the sum of two primal points,
• said two dual points corresponding to the two mesh faces to which said regular edge is incident, and
• said two primal points corresponding to the two vertices to which
• said regular edge is incident;
• and
• writing said dual mesh signal to computer memory or peripheral device through the input/output interface.

A system for resampling a polygon mesh to produce a dual polygon mesh comprising:

• a computer with operating system, computer memory, input/output interface, and a central processing unit;
• and an application program executed by the central processing unit for resampling a polygon mesh to produce a dual mesh, wherein:
• said polygon mesh connectivity includes a plurality of polygon mesh vertices, a plurality of polygon mesh faces, and at least one regular edge,
• each polygon mesh face having three or more corners, each corner being a particular polygon mesh vertex from among said plurality of polygon mesh vertices, each regular edge being incident to two polygon mesh vertices and two polygon mesh faces,
• said primal mesh signal including a plurality of primal points, each primal point being a point in an N-dimensional space, each primal point corresponding to a particular polygon mesh vertex from said plurality of polygon mesh vertices,
• said dual polygon mesh including a dual mesh signal and a dual mesh connectivity, said dual mesh signal including a plurality of dual points, each dual point being a point in said N-dimensional space, each dual point corresponding to a particular polygon mesh face from said plurality of polygon mesh faces, said dual mesh connectivity having a plurality of dual mesh vertices, and a plurality of dual mesh faces, each dual mesh vertex corresponding to one said polygon mesh face of said polygon mesh connectivity, each dual mesh face corresponding to one polygon mesh vertex of said polygon mesh connectivity,
• said application program comprising the computer implemented steps of:
• reading said polygon mesh from the computer memory or peripheral device through the input/output interface;
• minimizing an energy function of the primal points and the dual points with respect to the dual points while the primal points are kept fixed, said energy function being a sum of a plurality of squares of terms, each square term being associated with a particular regular edge of said polygon mesh connectivity, said square term being the square of a difference term, said difference term being computed by subtracting the sum of two dual points from the sum of two primal points, said two dual points corresponding to the two mesh faces to which said regular edge is incident, said two primal points corresponding to the two vertices to which said regular edge is incident;
• constructing said dual mesh connectivity from said polygon mesh connectivity;
• combining said dual mesh signal and said dual mesh connectivity as said dual polygon mesh;
• and
• writing said dual polygon mesh to the computer memory or peripheral device through the input/output interface.

A system for smoothing a polygon mesh to produce a smoothed polygon mesh comprising:

• a computer with operating system, computer memory, input/output interface, and a central processing unit;
• and
• an application program executed by the central processing unit for smoothing a polygon mesh to produce a smoothed polygon mesh signal, wherein:
• said polygon mesh including a primal mesh signal and a polygon mesh connectivity, said polygon mesh connectivity including a plurality of polygon mesh vertices, a plurality of polygon mesh faces, and at least one regular edge,
• each polygon mesh face having three or more corners, each corner being a particular polygon mesh vertex from among said plurality of polygon mesh vertices, each regular edge being incident to two polygon mesh vertices and two polygon mesh faces,
• said primal mesh signal including a plurality of primal points, each primal point being a point in an N-dimensional space, each primal point corresponding to a particular polygon mesh vertex from said plurality of polygon mesh vertices, comprising the computer implemented steps of:
• reading said polygon mesh from the computer memory or peripheral device through the input/output interface;
• creating a temporary dual mesh signal, said temporary dual mesh signal including a plurality of dual points, each dual point being a point in said N-dimensional space, each dual point corresponding to a particular polygon mesh face from said plurality of polygon mesh faces;
• minimizing an energy function of the primal points and the dual points with respect to the dual points while the primal points are kept fixed, said energy function being a sum of a plurality of squares of terms, each square term being associated with a particular regular edge of said polygon mesh connectivity, said square term being the square of a difference term, said difference term being computed by subtracting the sum of two dual points from the sum of two primal points, said two dual points corresponding to the two mesh faces to which said regular edge is incident, said two primal points corresponding to the two vertices to which said regular edge is incident;
• and
• recomputing said primal mesh signal by minimizing said energy function with respect to the primal points while the dual points are kept fixed;
• and
• writing said smoothed polygon mesh to the computer memory or peripheral device through the input/output interface.

A system for refining a polygon mesh to produce a primal-dual mesh comprising:

• a computer with operating system, computer memory, input/output interface, and a central processing unit;
• and
• an application program executed by the central processing unit for refining a polygon mesh to produce a primal-dual mesh comprising, wherein:
• said polygon mesh including a primal mesh signal and a polygon mesh connectivity, said polygon mesh connectivity including a plurality of polygon mesh vertices, a plurality of polygon mesh faces, and at least one regular edge,
• each polygon mesh face having three or more corners, each corner being a particular polygon mesh vertex from among said plurality of polygon mesh vertices, each regular edge being incident to two polygon mesh vertices and two polygon mesh faces,
• said primal mesh signal including a plurality of primal points, each primal point being a point in an N-dimensional space, each primal point corresponding to a particular polygon mesh vertex from said plurality of polygon mesh vertices, comprising the computer implemented steps of:
• reading said polygon mesh from the computer memory or peripheral device through the input/output interface;
• creating a temporary dual mesh signal, said temporary dual mesh signal including a plurality of dual points, each dual point being a point in said N-dimensional space, each dual point corresponding to a particular polygon mesh face from said plurality of polygon mesh faces;
• minimizing an energy function of the primal points and the dual points with respect to the dual points while the primal points are kept fixed, said energy function being a sum of a plurality of squares of terms, each square term being associated with a particular regular edge of said polygon mesh connectivity, said square term being the square of a difference term, said difference term being computed by subtracting the sum of two dual points from the sum of two primal points, said two dual points corresponding to the two mesh faces to which said regular edge is incident, said two primal points corresponding to the two vertices to which said regular edge is incident;
• and
• constructing a primal-dual mesh signal by concatenating said primal mesh signal with said dual mesh signal;
• constructing a primal-dual mesh connectivity from said polygon mesh connectivity, said primal-dual mesh connectivity having a plurality of primal-dual mesh vertices, and a plurality of primal-dual mesh faces, each primal-dual mesh face corresponding to one particular regular edge of said polygon mesh connectivity, each prima-dual mesh vertex corresponding to either one particular polygon mesh vertex or one particular polygon mesh face of the polygon mesh connectivity;
• constructing said primal-dual mesh by combining said primal-dual mesh signal and said primal-dual mesh connectivity;
• and
• writing said primal-dual mesh to the computer memory or peripheral device through the input/output interface.