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approximate_hausdorff_distance

approximate_hausdorff_distance(v1, f1, v2, f2, use_cpp=True)

Approximate the Hausdorff distance between two triangle meshes in 3D, i.e. the maximum distance between any point on one mesh and the closest point on the other mesh.

d = max { d(pA,B), d(pB,A) }

where pA is a point on mesh A and pB is a point on mesh B, and d(pA,B) is the distance between pA and the closest point on B. Our approximation will instead compute

d = max { d(vA,B), d(vB,A) }

where vA is a vertex on mesh A and vB is a vertex on mesh B.

Parameters:

Name Type Description Default
v1 (n1,3) array

Vertices of first mesh.

 required
f1 (m1,3) array

Faces of first mesh.

 required
v2 (n2,3) array

Vertices of second mesh.

 required
f2 (m2,3) array

Faces of second mesh.

 required
use_cpp bool, optional (default: True)

Whether to use the C++ implementation of triangle_triangle_distance.

 True

Returns:

Name Type Description
d float

Minimum distance value.
Notes

If you want an exact Hausdorff distance, you can heavily upsample both meshes using gpytoolbox.subdivide(method='upsample'). The approximated Hausdorff distance returned by this function will converge to the exact Hausdorff distance as the number of iterations of subdivision goes to infinity.

Examples:

# meshes in v,f and u,g
# Minimum distance value
d = gpytoolbox.minimum_distance(v,f,u,g)
Source code in src/gpytoolbox/approximate_hausdorff_distance.py 
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def approximate_hausdorff_distance(v1,f1,v2,f2,use_cpp=True):
    """
    Approximate the Hausdorff distance between two triangle meshes in 3D, i.e.
    the maximum distance between any point on one mesh and the closest point
    on the other mesh.

    d = max { d(pA,B), d(pB,A) }

    where pA is a point on mesh A and pB is a point on mesh B, and d(pA,B) is
    the distance between pA and the closest point on B. Our approximation will instead compute

    d = max { d(vA,B), d(vB,A) }

    where vA is a vertex on mesh A and vB is a vertex on mesh B.

    Parameters
    ----------
    v1 : (n1,3) array
        Vertices of first mesh.
    f1 : (m1,3) array
        Faces of first mesh.
    v2 : (n2,3) array
        Vertices of second mesh.
    f2 : (m2,3) array
        Faces of second mesh.
    use_cpp : bool, optional (default: True)
        Whether to use the C++ implementation of triangle_triangle_distance.

    Returns
    -------
    d : float
        Minimum distance value.

    Notes
    -----
    If you want an **exact** Hausdorff distance, you can heavily upsample both meshes using gpytoolbox.subdivide(method='upsample'). The approximated Hausdorff distance returned by this function will converge to the exact Hausdorff distance as the number of iterations of subdivision goes to infinity.

    Examples
    --------
    ```python
    # meshes in v,f and u,g
    # Minimum distance value
    d = gpytoolbox.minimum_distance(v,f,u,g)
    ```
    """

    # cpp implementation
    if use_cpp:
        try:
            from gpytoolbox_bindings import _hausdorff_distance_cpp_impl
        except:
            raise ImportError("Gpytoolbox cannot import its C++ fast winding number binding.") 
        return _hausdorff_distance_cpp_impl(v1,f1,v2,f2)

    # Let's start by computing the one-sided distance, i.e., max(d(vA,B)). We will do this with a loop, but with pre-computed trees for efficient queriying.

    dim = v1.shape[1]
    # Initialize AABB tree for mesh 1
    C1,W1,CH1,PAR1,D1,tri_ind1,_ = initialize_aabbtree(v1,f1)
    # Initialize AABB tree for mesh 2
    C2,W2,CH2,PAR2,D2,tri_ind2,_ = initialize_aabbtree(v2,f2)

    # print("Computing one-sided distance...")

    current_best_guess_hd = 0.0
    for i in range(v1.shape[0]):
        # print("Vertex %d of %d" % (i+1,v1.shape[0]))
        current_best_guess_dviB = np.Inf # current best guess for d(vi,B)

        queue = [0]
        while (len(queue)>0 and current_best_guess_dviB>current_best_guess_hd):
            q2 = queue.pop()
            # print("-----------")
            # print("Queue length : {}".format(len(queue)))
            # print("q1: ",q1)
            # print("q2: ",q2)
            # print("CH1[q1,]: ",CH1[q1,:])
            # print("CH2[q2,]: ",CH2[q2,:])
            # print("current_best_guess: ",current_best_guess)
            is_leaf2 = (CH2[q2,1]==-1)
            if (is_leaf2):
                # Compute distance between vi and triangle q2
                d = np.sqrt(squared_distance_to_element(v1[i,:],v2,f2[tri_ind2[q2],:])[0])
                if d < current_best_guess_dviB:
                    current_best_guess_dviB = d
            else:
                # Compute distance between vi and bounding box of q2
                # Distance from vi to the bounding box of q2
                d = point_to_box_distance(v1[i,:],C2[q2,:],W2[q2,:])
                d_max = point_to_box_max_distance(v1[i,:],C2[q2,:],W2[q2,:])
                # print(d_max)
                # If d_max is smaller than the current best guess for HD, then we don't need to pursue this part of the tree
                if ((d < current_best_guess_dviB) and (d_max > current_best_guess_hd)):
                    # Add children to queue
                    queue.append(CH2[q2,0])
                    queue.append(CH2[q2,1])
        # We have computed d(vi,B). Does it change our guess for hausdorff?
        if current_best_guess_dviB > current_best_guess_hd:
            current_best_guess_hd = current_best_guess_dviB

    # print("One-sided distance: {}".format(current_best_guess_hd))
    # print("Computing two-sided distance...")

    # Now we do the other side, i.e., max(d(vB,A))
    for i in range(v2.shape[0]):
        # print("Vertex %d of %d" % (i+1,v2.shape[0]))
        current_best_guess_dviA = np.Inf
        queue = [0]
        while (len(queue)>0 and current_best_guess_dviA>current_best_guess_hd):
            q2 = queue.pop()
            is_leaf2 = (CH1[q2,1]==-1)
            if (is_leaf2):
                # Compute distance between vi and triangle q2
                d = np.sqrt(squared_distance_to_element(v2[i,:],v1,f1[tri_ind1[q2],:])[0])
                if d < current_best_guess_dviA:
                    current_best_guess_dviA = d
            else:
                # Compute distance between vi and bounding box of q2
                # Distance from vi to the bounding box of q2
                d = point_to_box_distance(v2[i,:],C1[q2,:],W1[q2,:])
                d_max = point_to_box_max_distance(v2[i,:],C1[q2,:],W1[q2,:])
                # If d_max is smaller than the current best guess for HD, then we don't need to pursue this part of the tree
                if (d < current_best_guess_dviA and d_max > current_best_guess_hd):
                    # Add children to queue
                    queue.append(CH1[q2,0])
                    queue.append(CH1[q2,1])
        # We have computed d(vi,A). Does it change our guess for hausdorff?
        if current_best_guess_dviA > current_best_guess_hd:
            current_best_guess_hd = current_best_guess_dviA

    # We are done!
    return current_best_guess_hd

point_to_box_distance(p, C, W)

Compute the distance between a point and a box.

Parameters:

Name Type Description Default
p (3,) array

Point.

 required
C (3,) array

Center of box.

 required
W (3,) array

Width of box.

 required

Returns:

Name Type Description
d float

Distance between point and box.
Source code in src/gpytoolbox/approximate_hausdorff_distance.py 
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def point_to_box_distance(p,C,W):
    """
    Compute the distance between a point and a box.

    Parameters
    ----------
    p : (3,) array
        Point.
    C : (3,) array
        Center of box.
    W : (3,) array
        Width of box.

    Returns
    -------
    d : float
        Distance between point and box.
    """
    d = 0.0
    for i in range(3):
        if p[i] < C[i]-0.5*W[i]:
            d += (p[i]-C[i]+0.5*W[i])**2
        elif p[i] > C[i]+0.5*W[i]:
            d += (p[i]-C[i]-0.5*W[i])**2
    return np.sqrt(d)

point_to_box_max_distance(p, C, W)

Compute the maximum distance between a point and a box.

Parameters:

Name Type Description Default
p (3,) array

Point.

 required
C (3,) array

Center of box.

 required
W (3,) array

Width of box.

 required

Returns:

Name Type Description
d float

Maximum distance between point and box.
Source code in src/gpytoolbox/approximate_hausdorff_distance.py 
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def point_to_box_max_distance(p,C,W):
    """
    Compute the maximum distance between a point and a box.

    Parameters
    ----------
    p : (3,) array
        Point.
    C : (3,) array
        Center of box.
    W : (3,) array
        Width of box.

    Returns
    -------
    d : float
        Maximum distance between point and box.
    """
    d = 0.0
    for i in range(3):
        if p[i] < C[i]:
            # Distance to oppoisite corner
            d += (p[i]-C[i]-0.5*W[i])**2
        elif p[i] > C[i]:
            # Distance to oppoisite corner
            d += (p[i]-C[i]+0.5*W[i])**2
    return np.sqrt(d)