hausdorff is not symmetric - that should be added to the documentation

[barendgehrels]

The Hausdorff algorithm is not symmetrical and can give unexpected results.

SEE COMMENTS BELOW - we should fix the documentation.

Minimal case:

#include <boost/geometry.hpp>

namespace bg = boost::geometry;

template <typename Geometry>
double fixed_hausdorff(const Geometry& geometry1, const Geometry& geometry2)
{
    return std::max(bg::discrete_hausdorff_distance(geometry1, geometry2),
        bg::discrete_hausdorff_distance(geometry2, geometry1));
}

template <typename Geometry>
void test_case(Geometry const& p, Geometry const& q)
{
    std::cout 
        << "Frechet: " << boost::geometry::discrete_frechet_distance(p, q)
        << " Hausdorff: " << boost::geometry::discrete_hausdorff_distance(p, q)
        << " Hausdorff (q/p): " << boost::geometry::discrete_hausdorff_distance(q, p)
        << " Fixed Hausdorff: " << fixed_hausdorff(p, q)
        << std::endl;
}

int main()
{
    using coordinate_type = double;
    using point = boost::geometry::model::d2::point_xy<coordinate_type>;
    using linestring = boost::geometry::model::linestring<point>;

    // Two lines consistently 0.1 apart
    std::string const simplex1 = "LINESTRING(1.0 1.0, 2.0 1.0)";
    std::string const simplex2 = "LINESTRING(1.0 1.1, 2.0 1.1)";

    // With extra point in between
    std::string const simplex3 = "LINESTRING(1.0 1.1, 1.5 1.1,2.0 1.1)";

    // With a "spike"
    std::string const simplex4 = "LINESTRING(1.0 1.1, 1.49 1.1, 1.5 2.5, 1.51 1.1, 2.0 1.1)";

    test_case(bg::from_wkt<linestring>(simplex1), bg::from_wkt<linestring>(simplex2));
    test_case(bg::from_wkt<linestring>(simplex1), bg::from_wkt<linestring>(simplex3));
    test_case(bg::from_wkt<linestring>(simplex1), bg::from_wkt<linestring>(simplex4));

   return 0;
}

It reports:

Frechet: 0.1 Hausdorff: 0.1 Hausdorff (q/p): 0.1 Fixed Hausdorff: 0.1
Frechet: 0.509902 Hausdorff: 0.1 Hausdorff (q/p): 0.509902 Fixed Hausdorff: 0.509902
Frechet: 1.58114 Hausdorff: 0.1 Hausdorff (q/p): 1.58114 Fixed Hausdorff: 1.58114

and you can see it is wrong. I verified it with another (non open source) version of Hausdorff and that gives the same result as here labeled as Fixed (also for other testcases).

The fixed_hausdorff presented here is probably not optimized. I only tested for linestrings.

The correct version is equal to Frechet, for these testcases (for others, obviously, it's not).

[vissarion]

Indeed, there are places in the bibliography where Hausdorff distance is defined oriented or assymetric $h(A,B) = max{a\in A}(min{b\in B}(d(a,b))$ where $d(a,b)$ is the distance between $a$ and $b$. But maybe the most common is the symmetric one i.e. $H(A,B)=max( h(A,B),h(B,A ))$ (the one you call "fixed").

Thus, it is not exactly wrong, it is a matter of definition. However, it is problematic the definition is not given in the documentation.

I am OK to implement the symmetric Hausdorff and also describe it in the documentation. I am also OK if we keep the assymetric explicitly say this in the documentation and let the user decide whether to use the symmetric one by writing a function as your fixed_hausdorff().

[barendgehrels]

hi @vissarion ,

All right, it makes sense so we don't need to change the implementation. But we should add it to the documentation, so I can leave this ticket open.

I will rename fixed_hausdorff later in my own PR to, for example, symmetric_hausdorff.

Thanks.