A metric space of geometric shapes

My research involves geometric shapes in $R^2$, and I need a metric with several properties such as:

• Families of similar shapes, such as squares, are closed in this metric. Also more general families, such as the family of 2-fat objects, are closed in this metric.
• Converging sequences of interior-disjoint shapes, converge to interior-disjoint limits.
• Every continuous measure on a converging sequence, converges to the measure of the limit.

I tried the Hausdorff distance, which is a metric on the space of closed sets, but found out that it doesn't say much about measures.

I tried the Symmetric distance (defined as the area of the symmetric difference), but found out that it is only a pseudo-metric. I tried to make it a metric by restricting the underlying space of shapes, but found out that it is tricky even when only polygons are considered. I thought of converting the pseudo-metric to a metric on equivalence classes and then selecting a representative shape from each equivalence class, but found no simple way to do this selection.

I thought of defining a new metric which is the maximum of the Hausdorff distance and the Symmetric distance and enjoy the best of the two worlds, but at that point, it began to feel like reinventing the wheel. Surely I am not the first who needs a metric between plain geometric shapes.

So my question is:

Is there a paper or a book that explicitly studies the topic of metrics between shapes in the plane, not in the context of general topology but with attention to the specific geometric properties?

• As Wlodzimierz notes in their answer, the 'converging sequences of shapes converge to non-overlapping limits' seems misguided - what non-overlapping convergent limit would you expect to get out of $\{[0,1]^2\cup [0,1]\times[1+\frac1n, 2+\frac1n]\}$ as $n\to\infty$? – Steven Stadnicki Aug 27 '14 at 18:30
• By "non-overlapping" I meant "interior-disjoint"; fixed. – Erel Segal-Halevi Aug 28 '14 at 4:59
• Erel, your 2nd condition would mean, I'd say, that the limit set (shape) is a boundary set (i.e. it would have empty interior)--in which case this would be a clearer formulation. – Włodzimierz Holsztyński Aug 28 '14 at 6:23

The following paper gives an overview on Riemannian geometries on shape spaces and diffeomorphism group.

• Martin Bauer, Martins Bruveris, Peter W. Michor: Overview of the Geometries of Shape Spaces and Diffeomorphism Groups. Journal of Mathematical Imaging and Vision, 50, 1-2, 60-97, 2014. (pdf)

Edit:

A metric on the space of plane shapes that it somewhat more easy to use (since it allows for explicit solutions of the PDE which is the geodesic equation, but it does not see translations) is in the following paper:

• Martin Bauer, Martins Bruveris, Stephen Marsland, Peter W. Michor: Constructing reparametrization invariant metrics on spaces of plane curves. Differential Geometry and its Applications 34 (2014), 139–165. (pdf)
• Thanks, the second paper looks like what I need, but I still find it slightly difficult to understand. What background material would you recommend? – Erel Segal-Halevi Sep 3 '14 at 10:17
• MR2723182 Mumford, David; Desolneux, Agnès: Pattern theory. The stochastic analysis of real-world signals. Applying Mathematics. A K Peters, Ltd., Natick, MA, 2010. xii+407 pp. ISBN: 978-1-56881-579-4 (Reviewer: Laurent Younes) AND MR2656312 Younes, Laurent: Shapes and diffeomorphisms. Applied Mathematical Sciences, 171. Springer-Verlag, Berlin, 2010. xviii+434 pp. ISBN: 978-3-642-12054-1 – Peter Michor Sep 3 '14 at 10:27

I can't say for certain if this satisfies all of your requirements, but this paper of Sharon and Mumford studies 2D shapes by comparing the (suitably normalized) Riemann maps from its interior and exterior to the unit circle, by comparing these maps on the circle one represents the shape by an equivalence class of diffeomorphisms of the circle, and these can be made into a metric space. (They require smooth shapes; I don't know if this can be weakened.)

Your symmetric distance is fine, as long as we're particularly careful in the definition of a 'shape'. I believe the following will suffice:

A shape is an equivalence class of Borel sets, where two Borel sets are equivalent if their symmetric difference has zero Lebesgue measure.

Then this is a true metric, by definition, and it's possible to take limits of Cauchy sequences: given a Cauchy sequence $A_1, A_2, \dots$ of Borel sets, we define $A_{\omega}$ to consist of all points which occur in cofinitely many $A_i$, no points which occur in finitely many $A_i$, and we don't care about any other points (since their total Lebesgue measure is zero). And it's easy to show that such a limit can be constructed by careful application of countable unions and intersections, so is indeed Borel.

• I thought of converting the pseudo-metric to a metric using equivalence classes, but in my application the output should be a single well-defined shape. In a previous question, I asked about a natural way to define representative shapes for the equivalence classes, but currently got no reply: math.stackexchange.com/questions/845610/… – Erel Segal-Halevi Aug 27 '14 at 10:29
• I suppose it's subjective what you mean by 'shape'. Whilst the open and closed unit discs are distinct as sets, I would say that they are the same 'shape'. – Adam P. Goucher Aug 27 '14 at 19:18
• I agree that they are the same shape, but what about the union of the open unit disc with an additional segment e.g. ${0}\times[1,2]$? Most people would say that it has the shape of a baloon or a lollipop, not of a disc... I want to make sure that only the disc is selected, without any additions of zero area. – Erel Segal-Halevi Aug 28 '14 at 4:57

You may check papers by Stefan Banach on measure theory. He would consider metric invariant finitely additive measures but for arbitrary sets--in $\mathbb R^2$ too.

Also, certain ideas by Karol Borsuk, in a paper presented at the Mathematical Congress of 1954 (I am quite sure it was 1954) may be of interest to you. Karol Borsuk considered several metrics for compact spaces which were sensitive to topological properties.

BTW, in my opinion the condition on limit being disjoint for the disjoint shapes seems unnatural to me.