ITK  5.4.0 Insight Toolkit
itk::Math Namespace Reference

Detail

## Functions

bool abs (bool x)

unsigned char abs (char x)

unsigned short abs (short x)

unsigned char abs (signed char x)

unsigned char abs (unsigned char x)

unsigned int abs (unsigned int x)

unsigned long long abs (unsigned long long x)

unsigned long abs (unsigned long x)

unsigned short abs (unsigned short x)

template<typename T1 , typename T2 >
bool AlmostEquals (T1 x1, T2 x2)

template<typename TReturn , typename TInput >
TReturn CastWithRangeCheck (TInput x)

template<TReturn , typename TInput >
Ceil (TInput x)

template<typename TInput1 , typename TInput2 >
bool ExactlyEquals (const TInput1 &x1, const TInput2 &x2)

template<typename T >
FloatAddULP (T x, typename Detail::FloatIEEE< T >::IntType ulps)

template<typename T >
bool FloatAlmostEqual (T x1, T x2, typename Detail::FloatIEEE< T >::IntType maxUlps=4, typename Detail::FloatIEEE< T >::FloatType maxAbsoluteDifference=0.1 *itk::NumericTraits< T >::epsilon())

template<typename T >
Detail::FloatIEEE< T >::IntType FloatDifferenceULP (T x1, T x2)

template<TReturn , typename TInput >
Floor (TInput x)

template<typename T1 , typename T2 >
bool NotAlmostEquals (T1 x1, T2 x2)

template<typename TInput1 , typename TInput2 >
bool NotExactlyEquals (const TInput1 &x1, const TInput2 &x2)

template<typename TReturn , typename TInput >
TReturn Round (TInput x)

template<TReturn , typename TInput >
RoundHalfIntegerToEven (TInput x)

template<TReturn , typename TInput >
RoundHalfIntegerUp (TInput x)

template<typename TReturnType = uintmax_t>
constexpr TReturnType UnsignedPower (const uintmax_t base, const uintmax_t exponent) noexcept

template<typename TReturnType = uintmax_t>
constexpr TReturnType UnsignedProduct (const uintmax_t a, const uintmax_t b) noexcept

ITKCommon_EXPORT bool IsPrime (unsigned short n)

ITKCommon_EXPORT bool IsPrime (unsigned int n)

ITKCommon_EXPORT bool IsPrime (unsigned long n)

ITKCommon_EXPORT bool IsPrime (unsigned long long n)

ITKCommon_EXPORT unsigned short GreatestPrimeFactor (unsigned short n)

ITKCommon_EXPORT unsigned int GreatestPrimeFactor (unsigned int n)

ITKCommon_EXPORT unsigned long GreatestPrimeFactor (unsigned long n)

ITKCommon_EXPORT unsigned long long GreatestPrimeFactor (unsigned long long n)

## Variables

static constexpr double e = vnl_math::e

static constexpr double eps = vnl_math::eps

static constexpr double euler = vnl_math::euler

static constexpr float float_eps = vnl_math::float_eps

static constexpr float float_sqrteps = vnl_math::float_sqrteps

static constexpr double ln10 = vnl_math::ln10

static constexpr double ln2 = vnl_math::ln2

static constexpr double log10e = vnl_math::log10e

static constexpr double log2e = vnl_math::log2e

static constexpr double one_over_pi = vnl_math::one_over_pi

static constexpr double one_over_sqrt2pi = vnl_math::one_over_sqrt2pi

static constexpr double pi = vnl_math::pi

static constexpr double pi_over_180 = vnl_math::pi_over_180

static constexpr double pi_over_2 = vnl_math::pi_over_2

static constexpr double pi_over_4 = vnl_math::pi_over_4

static constexpr double sqrt1_2 = vnl_math::sqrt1_2

static constexpr double sqrt1_3 = vnl_math::sqrt1_3

static constexpr double sqrt2 = vnl_math::sqrt2

static constexpr double sqrt2pi = vnl_math::sqrt2pi

static constexpr double sqrteps = vnl_math::sqrteps

static constexpr double two_over_pi = vnl_math::two_over_pi

static constexpr double two_over_sqrtpi = vnl_math::two_over_sqrtpi

static constexpr double twopi = vnl_math::twopi

## ◆ abs() [2/9]

 unsigned char itk::Math::abs ( char x )
inline

Definition at line 871 of file itkMath.h.

## ◆ abs() [3/9]

 unsigned short itk::Math::abs ( short x )
inline

Definition at line 876 of file itkMath.h.

## ◆ abs() [4/9]

 unsigned char itk::Math::abs ( signed char x )
inline

Definition at line 866 of file itkMath.h.

## ◆ abs() [5/9]

 unsigned char itk::Math::abs ( unsigned char x )
inline

Definition at line 861 of file itkMath.h.

## ◆ abs() [6/9]

 unsigned int itk::Math::abs ( unsigned int x )
inline

Definition at line 886 of file itkMath.h.

## ◆ abs() [7/9]

 unsigned long long itk::Math::abs ( unsigned long long x )
inline

Definition at line 897 of file itkMath.h.

Referenced by FloatAlmostEqual().

## ◆ abs() [8/9]

 unsigned long itk::Math::abs ( unsigned long x )
inline

Definition at line 891 of file itkMath.h.

## ◆ abs() [9/9]

 unsigned short itk::Math::abs ( unsigned short x )
inline

Definition at line 881 of file itkMath.h.

## ◆ AlmostEquals()

template<typename T1 , typename T2 >
 bool itk::Math::AlmostEquals ( T1 x1, T2 x2 )
inline

Provide consistent equality checks between values of potentially different scalar or complex types.

template< typename T1, typename T2 > AlmostEquals( T1 x1, T2 x2 )

template< typename T1, typename T2 > NotAlmostEquals( T1 x1, T2 x2 )

This function compares two scalar or complex values of potentially different types. For maximum extensibility the function is implemented through a series of templated structs which direct the AlmostEquals() call to the correct function by evaluating the parameter's types.

Overall algorithm: If both values are complex... separate values into real and imaginary components and compare them separately

If one of the values is complex.. see if the imaginary part of the complex value is approximately 0 ... compare real part of complex value with scalar value

If both values are scalars...

To compare two floating point types... use FloatAlmostEqual.

To compare a floating point and an integer type... Use static_cast<FloatingPointType>(integerValue) and call FloatAlmostEqual

To compare signed and unsigned integers... Check for negative value or overflow, then cast and use ==

To compare two signed or two unsigned integers ... Use ==

To compare anything else ... Use ==

Parameters
 x1 first scalar value to compare x2 second scalar value to compare

Definition at line 690 of file itkMath.h.

## ◆ CastWithRangeCheck()

template<typename TReturn , typename TInput >
 TReturn itk::Math::CastWithRangeCheck ( TInput x )
inline

Definition at line 214 of file itkMath.h.

References itkConceptMacro.

## ◆ Ceil()

template<TReturn , typename TInput >
 itk::Math::Ceil ( TInput x )

Round towards plus infinity.

The behavior of overflow is undefined due to numerous implementations.

Warning
argument absolute value must be less than INT_MAX/2 for vnl_math_ceil to be guaranteed to work.
We also assume that the rounding mode is not changed from the default one (or at least that it is always restored to the default one).

## ◆ ExactlyEquals()

template<typename TInput1 , typename TInput2 >
 bool itk::Math::ExactlyEquals ( const TInput1 & x1, const TInput2 & x2 )
inline

Return the result of an exact comparison between two scalar values of potentially different types.

template <typename TInput1, typename TInput2> inline bool ExactlyEquals( const TInput & x1, const TInput & x2 )

template <typename TInput1, typename TInput2> inline bool NotExactlyEquals( const TInput & x1, const TInput & x2 )

These functions complement the EqualsComparison functions and determine if two scalar values are exactly equal using the compilers casting rules when comparing two different types. While this is also easily accomplished by using the equality operators, use of this function demonstrates the intent of an exact equality check and thus improves readability and clarity of code. In addition, this function suppresses float-equal warnings produced when using Clang.

Parameters
 x1 first floating point value to compare x2 second floating point value to compare

Definition at line 728 of file itkMath.h.

template<typename T >
 T itk::Math::FloatAddULP ( T x, typename Detail::FloatIEEE< T >::IntType ulps )
inline

Add the given ULPs (units in the last place) to a float.

If the given ULPs can are negative, they are subtracted.

FloatAlmostEqual
FloatDifferenceULP

Definition at line 271 of file itkMath.h.

## ◆ FloatAlmostEqual()

template<typename T >
 bool itk::Math::FloatAlmostEqual ( T x1, T x2, typename Detail::FloatIEEE< T >::IntType maxUlps = 4, typename Detail::FloatIEEE< T >::FloatType maxAbsoluteDifference = 0.1 * itk::NumericTraits::epsilon() )
inline

Compare two floats and return if they are effectively equal.

Determining when floats are almost equal is difficult because of their IEEE bit representation. This function uses the integer representation of the float to determine if they are almost equal.

The implementation is based off the explanation in the white papers:

This function is not a cure-all, and reading those articles is important to understand its appropriate use in the context of ULPs, zeros, subnormals, infinities, and NANs. For example, it is preferable to use this function on two floats directly instead of subtracting them and comparing them to zero.

The tolerance is specified in ULPs (units in the last place), i.e. how many floats there are in between the numbers. Therefore, the tolerance depends on the magnitude of the values that are being compared. A second tolerance is a maximum difference allowed, which is important when comparing numbers close to zero.

A NAN compares as not equal to a number, but two NAN's may compare as equal to each other.

Parameters
 x1 first floating value to compare x2 second floating values to compare maxUlps maximum units in the last place to be considered equal maxAbsoluteDifference maximum absolute difference to be considered equal

Definition at line 310 of file itkMath.h.

References abs(), and FloatDifferenceULP().

## ◆ FloatDifferenceULP()

template<typename T >
 Detail::FloatIEEE::IntType itk::Math::FloatDifferenceULP ( T x1, T x2 )
inline

Return the signed distance in ULPs (units in the last place) between two floats.

This is the signed distance, i.e., if x1 > x2, then the result is positive.

FloatAlmostEqual

Definition at line 255 of file itkMath.h.

References itk::Math::Detail::FloatIEEE< T >::AsULP().

Referenced by FloatAlmostEqual().

## ◆ Floor()

template<TReturn , typename TInput >
 itk::Math::Floor ( TInput x )

Round towards minus infinity.

The behavior of overflow is undefined due to numerous implementations.

Warning
argument absolute value must be less than NumbericTraits<TReturn>::max()/2 for vnl_math_floor to be guaranteed to work.
We also assume that the rounding mode is not changed from the default one (or at least that it is always restored to the default one).

## ◆ GreatestPrimeFactor() [1/4]

 ITKCommon_EXPORT unsigned int itk::Math::GreatestPrimeFactor ( unsigned int n )

Return the greatest factor of the decomposition in prime numbers.

## ◆ GreatestPrimeFactor() [2/4]

 ITKCommon_EXPORT unsigned long long itk::Math::GreatestPrimeFactor ( unsigned long long n )

Return the greatest factor of the decomposition in prime numbers.

## ◆ GreatestPrimeFactor() [3/4]

 ITKCommon_EXPORT unsigned long itk::Math::GreatestPrimeFactor ( unsigned long n )

Return the greatest factor of the decomposition in prime numbers.

## ◆ GreatestPrimeFactor() [4/4]

 ITKCommon_EXPORT unsigned short itk::Math::GreatestPrimeFactor ( unsigned short n )

Return the greatest factor of the decomposition in prime numbers.

## ◆ IsPrime() [1/4]

 ITKCommon_EXPORT bool itk::Math::IsPrime ( unsigned int n )

Return whether the number is a prime number or not.

Note
Negative numbers cannot be prime.

## ◆ IsPrime() [2/4]

 ITKCommon_EXPORT bool itk::Math::IsPrime ( unsigned long long n )

Return whether the number is a prime number or not.

Note
Negative numbers cannot be prime.

## ◆ IsPrime() [3/4]

 ITKCommon_EXPORT bool itk::Math::IsPrime ( unsigned long n )

Return whether the number is a prime number or not.

Note
Negative numbers cannot be prime.

## ◆ IsPrime() [4/4]

 ITKCommon_EXPORT bool itk::Math::IsPrime ( unsigned short n )

Return whether the number is a prime number or not.

Note
Negative numbers cannot be prime.

## ◆ NotAlmostEquals()

template<typename T1 , typename T2 >
 bool itk::Math::NotAlmostEquals ( T1 x1, T2 x2 )
inline
Examples
Examples/Statistics/KdTree.cxx.

Definition at line 698 of file itkMath.h.

References AlmostEquals().

## ◆ Round()

template<typename TReturn , typename TInput >
 TReturn itk::Math::Round ( TInput x )
inline

Round towards nearest integer (This is a synonym for RoundHalfIntegerUp)

Template Parameters
 TReturn must be an integer type TInput must be float or double
RoundHalfIntegerUp<TReturn, TInput>()

Definition at line 179 of file itkMath.h.

## ◆ RoundHalfIntegerToEven()

template<TReturn , typename TInput >
 itk::Math::RoundHalfIntegerToEven ( TInput x )

Round towards nearest integer.

Template Parameters
 TReturn must be an integer type TInput must be float or double  halfway cases are rounded towards the nearest even integer, e.g.  RoundHalfIntegerToEven( 1.5) == 2 RoundHalfIntegerToEven(-1.5) == -2 RoundHalfIntegerToEven( 2.5) == 2 RoundHalfIntegerToEven( 3.5) == 4

The behavior of overflow is undefined due to numerous implementations.

Warning
We assume that the rounding mode is not changed from the default one (or at least that it is always restored to the default one).

## ◆ RoundHalfIntegerUp()

template<TReturn , typename TInput >
 itk::Math::RoundHalfIntegerUp ( TInput x )

Round towards nearest integer.

Template Parameters
 TReturn must be an integer type TInput must be float or double  halfway cases are rounded upward, e.g.  RoundHalfIntegerUp( 1.5) == 2 RoundHalfIntegerUp(-1.5) == -1 RoundHalfIntegerUp( 2.5) == 3

The behavior of overflow is undefined due to numerous implementations.

Warning
The argument absolute value must be less than NumbericTraits<TReturn>::max()/2 for RoundHalfIntegerUp to be guaranteed to work.
We also assume that the rounding mode is not changed from the default one (or at least that it is always restored to the default one).

## ◆ UnsignedPower()

template<typename TReturnType = uintmax_t>
 constexpr TReturnType itk::Math::UnsignedPower ( const uintmax_t base, const uintmax_t exponent )
constexprnoexcept

Calculates base ^ exponent. Numeric overflow triggers a compilation error in "constexpr context" and a debug assert failure at run-time. Otherwise equivalent to C++11 static_cast<uintmax_t>(std::pow(base, exponent))

Note
UnsignedPower(0, 0) is not supported, as zero to the power of zero has no agreed-upon value: https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero

Definition at line 807 of file itkMath.h.

References ITK_X_ASSERT.

## ◆ UnsignedProduct()

template<typename TReturnType = uintmax_t>
 constexpr TReturnType itk::Math::UnsignedProduct ( const uintmax_t a, const uintmax_t b )
constexprnoexcept

Returns a * b. Numeric overflow triggers a compilation error in "constexpr context" and a debug assert failure at run-time.

Definition at line 785 of file itkMath.h.

References ITK_X_ASSERT.

## Variable Documentation

staticconstexpr

$\frac{180}{\pi}$

Definition at line 79 of file itkMath.h.

## ◆ eps

 constexpr double itk::Math::eps = vnl_math::eps
staticconstexpr

## ◆ euler

 constexpr double itk::Math::euler = vnl_math::euler
staticconstexpr

euler constant

Definition at line 93 of file itkMath.h.

## ◆ float_eps

 constexpr float itk::Math::float_eps = vnl_math::float_eps
staticconstexpr

Definition at line 99 of file itkMath.h.

## ◆ float_sqrteps

 constexpr float itk::Math::float_sqrteps = vnl_math::float_sqrteps
staticconstexpr

Definition at line 100 of file itkMath.h.

## ◆ ln10

 constexpr double itk::Math::ln10 = vnl_math::ln10
staticconstexpr

$\log_e 10$

Definition at line 63 of file itkMath.h.

## ◆ ln2

 constexpr double itk::Math::ln2 = vnl_math::ln2
staticconstexpr

$\log_e 2$

Definition at line 61 of file itkMath.h.

## ◆ log10e

 constexpr double itk::Math::log10e = vnl_math::log10e
staticconstexpr

$\log_{10} e$

Definition at line 59 of file itkMath.h.

## ◆ log2e

 constexpr double itk::Math::log2e = vnl_math::log2e
staticconstexpr

$\log_2 e$

Definition at line 57 of file itkMath.h.

## ◆ one_over_pi

 constexpr double itk::Math::one_over_pi = vnl_math::one_over_pi
staticconstexpr

$\frac{1}{\pi}$

Definition at line 75 of file itkMath.h.

## ◆ one_over_sqrt2pi

 constexpr double itk::Math::one_over_sqrt2pi = vnl_math::one_over_sqrt2pi
staticconstexpr

$\frac{1}{\sqrt{2\pi}}$

Definition at line 85 of file itkMath.h.

## ◆ pi_over_180

 constexpr double itk::Math::pi_over_180 = vnl_math::pi_over_180
staticconstexpr

$\frac{\pi}{180}$

Definition at line 73 of file itkMath.h.

## ◆ pi_over_2

 constexpr double itk::Math::pi_over_2 = vnl_math::pi_over_2
staticconstexpr

$\frac{\pi}{2}$

Definition at line 69 of file itkMath.h.

## ◆ pi_over_4

 constexpr double itk::Math::pi_over_4 = vnl_math::pi_over_4
staticconstexpr

$\frac{\pi}{4}$

Definition at line 71 of file itkMath.h.

## ◆ sqrt1_2

 constexpr double itk::Math::sqrt1_2 = vnl_math::sqrt1_2
staticconstexpr

$\sqrt{ \frac{1}{2}}$

Definition at line 89 of file itkMath.h.

## ◆ sqrt1_3

 constexpr double itk::Math::sqrt1_3 = vnl_math::sqrt1_3
staticconstexpr

$\sqrt{ \frac{1}{3}}$

Definition at line 91 of file itkMath.h.

## ◆ sqrt2

 constexpr double itk::Math::sqrt2 = vnl_math::sqrt2
staticconstexpr

$\sqrt{2}$

Definition at line 87 of file itkMath.h.

## ◆ sqrt2pi

 constexpr double itk::Math::sqrt2pi = vnl_math::sqrt2pi
staticconstexpr

$\sqrt{2\pi}$

Definition at line 81 of file itkMath.h.

## ◆ sqrteps

 constexpr double itk::Math::sqrteps = vnl_math::sqrteps
staticconstexpr

Definition at line 97 of file itkMath.h.

## ◆ two_over_pi

 constexpr double itk::Math::two_over_pi = vnl_math::two_over_pi
staticconstexpr

$\frac{2}{\pi}$

Definition at line 77 of file itkMath.h.

## ◆ two_over_sqrtpi

 constexpr double itk::Math::two_over_sqrtpi = vnl_math::two_over_sqrtpi
staticconstexpr

$\frac{2}{\sqrt{\pi}}$

Definition at line 83 of file itkMath.h.

## ◆ twopi

 constexpr double itk::Math::twopi = vnl_math::twopi
staticconstexpr

$2\pi$

Examples
Examples/Segmentation/HoughTransform2DCirclesImageFilter.cxx.

Definition at line 67 of file itkMath.h.

itk::Math::RoundHalfIntegerUp
RoundHalfIntegerUp(TInput x)
Round towards nearest integer.
itk::Math::RoundHalfIntegerToEven
RoundHalfIntegerToEven(TInput x)
Round towards nearest integer.