Outdistance chamfer

The distances from chamfer are discrete distances using of the masks whose weights are entireties. The distances from chamfer are fast to calculate on a machine. Although they are more precise than the discrete distances d4 and d8, they provide only one approximation of the Euclidean distance.

Definition

A weighting is a couple $(\ mathbf {X}, W)$ where $\ mathbf {X}$ is a point and $w$ an entirety. A mask $\ mathcal {M}$ is a finished whole of pondérations : $\ mathcal {M} = \ left \ {(\ mathbf {X} _i, w_i) \ right \}$ . A mask of chamfer is a mask having a central symmetry, whose weights are strictly positive and nonnull displacements.

It is possible to find a way between two points $p$ and $q$ by using displacements of a mask of chamfer. That is to say $\ mathcal {M} = \ left \ {(\ mathbf {X} _i, w_i) \ right \}$ a mask of chamfer, then

$q = p + \ sum \ lambda_i \ mathbf {X} _i$ ,

where $\ lambda_i$ is an entirety which corresponds to the number of times that displacement $\ mathbf {X} _i$ .

Outdistance chamfer

A distance from chamfer $d_ {\ mathcal {M}}$ between two points $p$ and $q$ of a discrete space is the minimal cost $\ sum \ lambda_i w_i$ of all the ways finite length $\ sum \ lambda_i \ mathbf {X} _i$ between $p$ and $q$ and using displacements $\ mathbf {X} _i$ of the mask $\ mathcal {M}$ :

$d_ {\ mathcal {M}} = \ min \ left \ {\ sum \ lambda_i w_i \ left| Q = p + \ sum \ lambda_i \ mathbf {X} _i \ right. \right\}$ .

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