computational geometry - Find the volume of Phobos and Deimos

How can we calculate the volume of a 3D object using the new-in-10 computational geometry functions?

For simple objects this works:

shuttle = ExampleData[{"Geometry3D", "SpaceShuttle"}]

Mathematica graphics

Volume@BoundaryDiscretizeGraphics[shuttle]
(* 55.5217 *)

But BoundaryDiscretizeRegion tends to fail on more complex geometries:

{deimos = ExampleData[{"Geometry3D", "Deimos"}], 
 phobos = ExampleData[{"Geometry3D", "Phobos"}]}

Mathematica graphics

BoundaryDiscretizeGraphics[deimos]

BoundaryMeshRegion::binsect: The boundary curves self-intersect or cross each other in BoundaryMeshRegion[{{-0.473472,5.76096,0.000999842},{-0.32066,5.70981,0.233476},{-0.0747657,5.65315,0.231889},<<46>>,{-2.88191,5.74069,0.489521},<<9360>>},{{},{},{Polygon[{{1,2,3},<<49>>,<<18766>>}]}}]. >>

In general, what is a good way to proceed when we need to treat an object as a solid 3D volume (not surface!) for various purposes, such as triangulation, solving a PDE inside it, calculating a volume, or using it as an integration region?

Here are two more practical test cases:

bunny = ExampleData[{"Geometry3D", "StanfordBunny"}]
horse = ExampleData[{"Geometry3D", "Horse"}]

The Stanford bunny is especially relevant because originally it was created from voxel data (a common application).

Comments

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mathematical optimization - Minimizing using indices, error: Part::pkspec1: The expression cannot be used as a part specification

I want to use Minimize where the variables to minimize are indices pointing into an array. Here a MWE that hopefully shows what my problem is. vars = u@# & /@ Range[3]; cons = Flatten@ { Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; Minimize[{Total@((vec1[[#]] - vec2[[u[#]]])^2 & /@ Range[1, 3]), cons}, vars, Integers] The error I get: Part::pkspec1: The expression u[1] cannot be used as a part specification. >> Answer Ok, it seems that one can get around Mathematica trying to evaluate vec2[[u[1]]] too early by using the function Indexed[vec2,u[1]] . The working MWE would then look like the following: vars = u@# & /@ Range[3]; cons = Flatten@{ Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; NMinimize[ {Total@((vec1[[#]] - Indexed[vec2, u[#]])^2 & /@ R...

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