DGtal  1.2.0
Number and Integer Concepts

# Overview

When implementing digital geometry algorithms, the choice of integer type to implement or approximate $$Z$$ is crucial in many situations. In DGtal, objects, datastructure and algorithms are orthogonal to this choice in the sense that the user can specify the best Integer type for his own pruposes.

First of all, approximation with built-in integer types with bounded ranges can be considered. For example, DGtal::int32_t defines OS independent 32 bits signed integers.

If more precision is required and if GMP (Gnu multiprecision library) is available, DGtal::BigInteger defines arbitrary precision integers. In this case, performances can be impacted.

As detailed in the Digital Spaces, Points, Vectors and Domains documentation, the integer type choice is specified as template parameter of templated classes (e.g. SpaceND). The main constraint on the type is that it induces a commutative ring with identity for the classical addition, and multiplication operators.

# Number and Integer Concepts

In many DGtal classes, we control user specified template type thanks to concept checking mechanisms. In many situations, when using a template type associating with numbers, following basic concepts are involved:

Concept name Short description Model examples
concepts::CSignedNumber Signed numbers DGtal::int32_t, double, ....
concepts::CUnsignedNumber Unsigned numbers DGtal::uint32_t,...
concepts::CBoundedNumber Numbers with bounded capacity DGtal::int64_t, ... (but not BigInteger)
concepts::CIntegralNumber Integral numbers DGtal::int64_t, BigInteger,...

When we want to perform some computations on these numbers, we may use the following concepts:

Concept name Short description Model examples
concepts::CCommutativeRing the model can be associated to a commutative ring (operators +,-,*, ....) DGtal::int32_t, double, ....
concepts::CEuclideanRing Refinement of concepts::CCommutativeRing with a "/" operator DGtal::int32_t, double, .....
concepts::CInteger Refinement of both concepts::CEuclideanRing and concepts::CIntegralNumber DGtal::int32_t, .....

# Basic integer types and NumberTraits

First of all, BasicTypes refines in DGtal classical built-in integer types such as DGtal::uint64_t or DGtal::int32_t. However, not all these types are models of the concept concepts::CCommutativeRing. Indeed, DGtal::uint64_t has no inverse for the addition operator.

Hence, if we have a templated class SpaceND with two arguments: a static dimension and a type of integer. And if we add a constraint that the integer type must be a model of concepts::CCommutativeRing, the following code:

will produce a compiler error.

In DGtal algorithms, we sometimes need information or associated types attached to a integer type. NumberTraits structure is used to decorate integer types with tags (e.g. NumberTraits<T>::IsSigned), associated types (e.g. NumberTraits<T>::UnsignedVersion), constant values (e.g. NumberTraits<T>::ONE and NumberTraits<T>::ZERO), or conversion functions (e.g. NumberTraits<T>::castToDouble()).