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Mathematical Functions

Namespaces

namespace  vigra::detail

Classes

class  BSpline< ORDER, T >
class  BSplineBase< ORDER, T >
class  CatmullRomSpline< T >
class  CoscotFunction< T >
class  Gaussian< T >

Functions

template<class Iterator >
Iterator argMax (Iterator first, Iterator last)
template<class Iterator , class UnaryFunctor >
Iterator argMaxIf (Iterator first, Iterator last, UnaryFunctor condition)
template<class Iterator >
Iterator argMin (Iterator first, Iterator last)
template<class Iterator , class UnaryFunctor >
Iterator argMinIf (Iterator first, Iterator last, UnaryFunctor condition)
UInt32 ceilPower2 (UInt32 x)
double chi2 (unsigned int degreesOfFreedom, double arg, double accuracy=1e-7)
double chi2CDF (unsigned int degreesOfFreedom, double arg, double accuracy=1e-7)
template<class T1 , class T2 >
bool closeAtTolerance (T1 l, T2 r, typename PromoteTraits< T1, T2 >::Promote epsilon)
template<class PointArray1 , class PointArray2 >
void convexHull (const PointArray1 &points, PointArray2 &convex_hull)
 Compute convex hull of a 2D polygon.
double ellipticIntegralE (double x, double k)
double ellipticIntegralF (double x, double k)
UInt32 floorPower2 (UInt32 x)
template<typename IntType >
IntType gcd (IntType n, IntType m)
template<typename IntType >
IntType lcm (IntType n, IntType m)
Int32 log2i (UInt32 x)
double noncentralChi2 (unsigned int degreesOfFreedom, double noncentrality, double arg, double accuracy=1e-7)
double noncentralChi2CDF (unsigned int degreesOfFreedom, double noncentrality, double arg, double accuracy=1e-7)
double noncentralChi2CDFApprox (unsigned int degreesOfFreedom, double noncentrality, double arg)
template<class T >
NormTraits< T >::NormType norm (T const &t)
result_type operator() (argument_type x) const
float round (float t)
int roundi (double t)
template<class T >
sign (T t)
template<class T1 , class T2 >
T1 sign (T1 t1, T2 t2)
template<class T >
int signi (T t)
template<class T >
NumericTraits< T >::Promote sq (T t)
Int32 sqrti (Int32 v)
NormTraits< T >::SquaredNormType squaredNorm (T const &t)
template<class T >
void symmetric2x2Eigenvalues (T a00, T a01, T a11, T *r0, T *r1)
template<class T >
void symmetric3x3Eigenvalues (T a00, T a01, T a02, T a11, T a12, T a22, T *r0, T *r1, T *r2)

Detailed Description

Useful mathematical functions and functors.

Function Documentation

float vigra::round ( float  t)

The rounding function.

   Defined for all floating point types. Rounds towards the nearest integer 
   such that <tt>abs(round(t)) == round(abs(t))</tt> for all <tt>t</tt>.

   <b>\#include</b> \<<a href="mathutil_8hxx-source.html">vigra/mathutil.hxx</a>\><br>
   Namespace: vigra
int vigra::roundi ( double  t)

Round and cast to integer.

   Rounds to the nearest integer like round(), but casts the result to 
   <tt>int</tt> (this will be faster and is usually needed anyway).

   <b>\#include</b> \<<a href="mathutil_8hxx-source.html">vigra/mathutil.hxx</a>\><br>
   Namespace: vigra
UInt32 vigra::ceilPower2 ( UInt32  x)

Round up to the nearest power of 2.

   Efficient algorithm for finding the smallest power of 2 which is not smaller than \a x
   (function clp2() from Henry Warren: "Hacker's Delight", Addison-Wesley, 2003,
    see http://www.hackersdelight.org/).
   If \a x > 2^31, the function will return 0 because integer arithmetic is defined modulo 2^32.

   <b>\#include</b> \<<a href="mathutil_8hxx-source.html">vigra/mathutil.hxx</a>\><br>
   Namespace: vigra
UInt32 vigra::floorPower2 ( UInt32  x)

Round down to the nearest power of 2.

   Efficient algorithm for finding the largest power of 2 which is not greater than \a x
   (function flp2() from Henry Warren: "Hacker's Delight", Addison-Wesley, 2003,
    see http://www.hackersdelight.org/).

   <b>\#include</b> \<<a href="mathutil_8hxx-source.html">vigra/mathutil.hxx</a>\><br>
   Namespace: vigra
Int32 vigra::log2i ( UInt32  x)

Compute the base-2 logarithm of an integer.

   Returns the position of the left-most 1-bit in the given number \a x, or
   -1 if \a x == 0. That is,

   @code 
   assert(k >= 0 && k < 32 && log2i(1 << k) == k);
   \endcode

   The function uses Robert Harley's algorithm to determine the number of leading zeros
   in \a x (algorithm nlz10() at http://www.hackersdelight.org/). But note that the functions
   \ref floorPower2() or \ref ceilPower2() are more efficient and should be preferred when possible.

   <b>\#include</b> \<<a href="mathutil_8hxx-source.html">vigra/mathutil.hxx</a>\><br>
   Namespace: vigra
NumericTraits<T>::Promote vigra::sq ( t)

The square function.

   <tt>sq(x) = x*x</tt> is needed so often that it makes sense to define it as a function.

   <b>\#include</b> \<<a href="mathutil_8hxx-source.html">vigra/mathutil.hxx</a>\><br>
   Namespace: vigra
UInt32 sqrti ( Int32  v)

Signed integer square root.

   Useful for fast fixed-point computations.

   <b>\#include</b> \<<a href="mathutil_8hxx-source.html">vigra/mathutil.hxx</a>\><br>
   Namespace: vigra

Unsigned integer square root.

   Useful for fast fixed-point computations.

   <b>\#include</b> \<<a href="mathutil_8hxx-source.html">vigra/mathutil.hxx</a>\><br>
   Namespace: vigra
T vigra::sign ( t)

The sign function.

   Returns 1, 0, or -1 depending on the sign of \a t, but with the same type as \a t.

   <b>\#include</b> \<<a href="mathutil_8hxx-source.html">vigra/mathutil.hxx</a>\><br>
   Namespace: vigra
int vigra::signi ( t)

The integer sign function.

   Returns 1, 0, or -1 depending on the sign of \a t, converted to int.

   <b>\#include</b> \<<a href="mathutil_8hxx-source.html">vigra/mathutil.hxx</a>\><br>
   Namespace: vigra
T1 vigra::sign ( T1  t1,
T2  t2 
)

The binary sign function.

   Transfers the sign of \a t2 to \a t1.

   <b>\#include</b> \<<a href="mathutil_8hxx-source.html">vigra/mathutil.hxx</a>\><br>
   Namespace: vigra
NormTraits<T>::SquaredNormType vigra::squaredNorm ( T const &  t)

The squared norm of a numerical object.

   For scalar types: equals <tt>vigra::sq(t)</tt><br>.
   For vectorial types: equals <tt>vigra::dot(t, t)</tt><br>.
   For complex types: equals <tt>vigra::sq(t.real()) + vigra::sq(t.imag())</tt><br>.
   For matrix types: results in the squared Frobenius norm (sum of squares of the matrix elements).
NormTraits<T>::NormType vigra::norm ( T const &  t)

The norm of a numerical object.

   For scalar types: implemented as <tt>abs(t)</tt><br>
   otherwise: implemented as <tt>sqrt(squaredNorm(t))</tt>.

   <b>\#include</b> \<<a href="mathutil_8hxx-source.html">vigra/mathutil.hxx</a>\><br>
   Namespace: vigra
Iterator vigra::argMin ( Iterator  first,
Iterator  last 
)

Find the minimum element in a sequence.

   The function returns the iterator refering to the minimum element.

   <b>Required Interface:</b>

   @code 
   Iterator is a standard forward iterator.

   bool f = *first < NumericTraits<typename std::iterator_traits<Iterator>::value_type>::max();
   \endcode

   <b>\#include</b> \<<a href="mathutil_8hxx-source.html">vigra/mathutil.hxx</a>\><br>
   Namespace: vigra
Iterator vigra::argMax ( Iterator  first,
Iterator  last 
)

Find the maximum element in a sequence.

   The function returns the iterator refering to the maximum element.

   <b>Required Interface:</b>

   @code 
   Iterator is a standard forward iterator.

   bool f = NumericTraits<typename std::iterator_traits<Iterator>::value_type>::min() < *first;
   \endcode

   <b>\#include</b> \<<a href="mathutil_8hxx-source.html">vigra/mathutil.hxx</a>\><br>
   Namespace: vigra
Iterator vigra::argMinIf ( Iterator  first,
Iterator  last,
UnaryFunctor  condition 
)

Find the minimum element in a sequence conforming to a condition.

   The function returns the iterator refering to the minimum element,
   where only elements conforming to the condition (i.e. where 
   <tt>condition(*iterator)</tt> evaluates to <tt>true</tt>) are considered.
   If no element conforms to the condition, or the sequence is empty,
   the end iterator \a last is returned.

   <b>Required Interface:</b>

   @code 
   Iterator is a standard forward iterator.

   bool c = condition(*first);

   bool f = *first < NumericTraits<typename std::iterator_traits<Iterator>::value_type>::max();
   \endcode

   <b>\#include</b> \<<a href="mathutil_8hxx-source.html">vigra/mathutil.hxx</a>\><br>
   Namespace: vigra
Iterator vigra::argMaxIf ( Iterator  first,
Iterator  last,
UnaryFunctor  condition 
)

Find the maximum element in a sequence conforming to a condition.

   The function returns the iterator refering to the maximum element,
   where only elements conforming to the condition (i.e. where 
   <tt>condition(*iterator)</tt> evaluates to <tt>true</tt>) are considered.
   If no element conforms to the condition, or the sequence is empty,
   the end iterator \a last is returned.

   <b>Required Interface:</b>

   @code 
   Iterator is a standard forward iterator.

   bool c = condition(*first);

   bool f = NumericTraits<typename std::iterator_traits<Iterator>::value_type>::min() < *first;
   \endcode

   <b>\#include</b> \<<a href="mathutil_8hxx-source.html">vigra/mathutil.hxx</a>\><br>
   Namespace: vigra
void vigra::symmetric2x2Eigenvalues ( a00,
a01,
a11,
T *  r0,
T *  r1 
)

Compute the eigenvalues of a 2x2 real symmetric matrix.

   This uses the analytical eigenvalue formula 

\[ \lambda_{1,2} = \frac{1}{2}\left(a_{00} + a_{11} \pm \sqrt{(a_{00} - a_{11})^2 + 4 a_{01}^2}\right) \]

  <b>\#include</b> \<<a href="mathutil_8hxx-source.html">vigra/mathutil.hxx</a>\><br>
  Namespace: vigra
void vigra::symmetric3x3Eigenvalues ( a00,
a01,
a02,
a11,
a12,
a22,
T *  r0,
T *  r1,
T *  r2 
)

Compute the eigenvalues of a 3x3 real symmetric matrix.

   This uses a numerically stable version of the analytical eigenvalue formula according to
   <p>
   David Eberly: <a href="http://www.geometrictools.com/Documentation/EigenSymmetric3x3.pdf">
   <em>"Eigensystems for 3 × 3 Symmetric Matrices (Revisited)"</em></a>, Geometric Tools Documentation, 2006

   <b>\#include</b> \<<a href="mathutil_8hxx-source.html">vigra/mathutil.hxx</a>\><br>
   Namespace: vigra
double vigra::ellipticIntegralF ( double  x,
double  k 
)

The incomplete elliptic integral of the first kind.

   Computes

\[ \mbox{F}(x, k) = \int_0^x \frac{1}{\sqrt{1 - k^2 \sin(t)^2}} dt \]

  according to the algorithm given in Press et al. "Numerical Recipes". 

  Note: In some libraries (e.g. Mathematica), the second parameter of the elliptic integral
  functions must be k^2 rather than k. Check the documentation when results disagree!

  <b>\#include</b> \<<a href="mathutil_8hxx-source.html">vigra/mathutil.hxx</a>\><br>
  Namespace: vigra
double vigra::ellipticIntegralE ( double  x,
double  k 
)

The incomplete elliptic integral of the second kind.

   Computes

\[ \mbox{E}(x, k) = \int_0^x \sqrt{1 - k^2 \sin(t)^2} dt \]

  according to the algorithm given in Press et al. "Numerical Recipes". The
  complete elliptic integral of the second kind is simply <tt>ellipticIntegralE(M_PI/2, k)</TT>.

  Note: In some libraries (e.g. Mathematica), the second parameter of the elliptic integral
  functions must be k^2 rather than k. Check the documentation when results disagree!

  <b>\#include</b> \<<a href="mathutil_8hxx-source.html">vigra/mathutil.hxx</a>\><br>
  Namespace: vigra
double vigra::chi2 ( unsigned int  degreesOfFreedom,
double  arg,
double  accuracy = 1e-7 
)

Chi square distribution.

   Computes the density of a chi square distribution with \a degreesOfFreedom 
   and tolerance \a accuracy at the given argument \a arg
   by calling <tt>noncentralChi2(degreesOfFreedom, 0.0, arg, accuracy)</tt>.

   <b>\#include</b> \<<a href="mathutil_8hxx-source.html">vigra/mathutil.hxx</a>\><br>
   Namespace: vigra
double vigra::chi2CDF ( unsigned int  degreesOfFreedom,
double  arg,
double  accuracy = 1e-7 
)

Cumulative chi square distribution.

   Computes the cumulative density of a chi square distribution with \a degreesOfFreedom 
   and tolerance \a accuracy at the given argument \a arg, i.e. the probability that
   a random number drawn from the distribution is below \a arg
   by calling <tt>noncentralChi2CDF(degreesOfFreedom, 0.0, arg, accuracy)</tt>.

   <b>\#include</b> \<<a href="mathutil_8hxx-source.html">vigra/mathutil.hxx</a>\><br>
   Namespace: vigra
double vigra::noncentralChi2 ( unsigned int  degreesOfFreedom,
double  noncentrality,
double  arg,
double  accuracy = 1e-7 
)

Non-central chi square distribution.

   Computes the density of a non-central chi square distribution with \a degreesOfFreedom, 
   noncentrality parameter \a noncentrality and tolerance \a accuracy at the given argument 
   \a arg. It uses Algorithm AS 231 from Appl. Statist. (1987) Vol.36, No.3 (code ported from 
   http://lib.stat.cmu.edu/apstat/231). The algorithm has linear complexity in the number of
   degrees of freedom.

   <b>\#include</b> \<<a href="mathutil_8hxx-source.html">vigra/mathutil.hxx</a>\><br>
   Namespace: vigra
double vigra::noncentralChi2CDF ( unsigned int  degreesOfFreedom,
double  noncentrality,
double  arg,
double  accuracy = 1e-7 
)

Cumulative non-central chi square distribution.

   Computes the cumulative density of a chi square distribution with \a degreesOfFreedom, 
   noncentrality parameter \a noncentrality and tolerance \a accuracy at the given argument 
   \a arg, i.e. the probability that a random number drawn from the distribution is below \a arg
   It uses Algorithm AS 231 from Appl. Statist. (1987) Vol.36, No.3 (code ported from 
   http://lib.stat.cmu.edu/apstat/231). The algorithm has linear complexity in the number of
   degrees of freedom (see noncentralChi2CDFApprox() for a constant-time algorithm).

   <b>\#include</b> \<<a href="mathutil_8hxx-source.html">vigra/mathutil.hxx</a>\><br>
   Namespace: vigra
double vigra::noncentralChi2CDFApprox ( unsigned int  degreesOfFreedom,
double  noncentrality,
double  arg 
)

Cumulative non-central chi square distribution (approximate).

   Computes approximate values of the cumulative density of a chi square distribution with \a degreesOfFreedom, 
   and noncentrality parameter \a noncentrality at the given argument 
   \a arg, i.e. the probability that a random number drawn from the distribution is below \a arg
   It uses the approximate transform into a normal distribution due to Wilson and Hilferty 
   (see Abramovitz, Stegun: "Handbook of Mathematical Functions", formula 26.3.32). 
   The algorithm's running time is independent of the inputs, i.e. is should be used
   when noncentralChi2CDF() is too slow, and approximate values are sufficient. The accuracy is only 
   about 0.1 for few degrees of freedom, but reaches about 0.001 above dof = 5.

   <b>\#include</b> \<<a href="mathutil_8hxx-source.html">vigra/mathutil.hxx</a>\><br>
   Namespace: vigra
bool vigra::closeAtTolerance ( T1  l,
T2  r,
typename PromoteTraits< T1, T2 >::Promote  epsilon 
)

Tolerance based floating-point comparison.

   Check whether two floating point numbers are equal within the given tolerance.
   This is useful because floating point numbers that should be equal in theory are
   rarely exactly equal in practice. If the tolerance \a epsilon is not given,
   twice the machine epsilon is used.

   <b>\#include</b> \<<a href="mathutil_8hxx-source.html">vigra/mathutil.hxx</a>\><br>
   Namespace: vigra
void vigra::convexHull ( const PointArray1 &  points,
PointArray2 &  convex_hull 
)

Compute convex hull of a 2D polygon.

The input array points contains a (not necessarily ordered) set of 2D points whose convex hull is to be computed. The array's value_type (i.e. the point type) must be compatible with std::vector (in particular, it must support indexing, copying, and have size() == 2). The points of the convex hull will be appended to the output array convex_hull (which must support std::back_inserter(convex_hull)). Since the convex hull is a closed polygon, the first and last point of the output will be the same (i.e. the first point will simply be inserted at the end again). The points of the convex hull will be ordered counter-clockwise, starting with the leftmost point of the imput.

IntType vigra::gcd ( IntType  n,
IntType  m 
)

Calculate the greatest common divisor.

This function works for arbitrary integer types, including user-defined (e.g. infinite precision) ones.

#include <vigra/rational.hxx>
Namespace: vigra

IntType vigra::lcm ( IntType  n,
IntType  m 
)

Calculate the lowest common multiple.

This function works for arbitrary integer types, including user-defined (e.g. infinite precision) ones.

#include <vigra/rational.hxx>
Namespace: vigra

CatmullRomSpline< T >::result_type operator() ( argument_type  x) const

function (functor) call

© Ullrich Köthe (ullrich.koethe@iwr.uni-heidelberg.de)
Heidelberg Collaboratory for Image Processing, University of Heidelberg, Germany

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vigra 1.7.1 (Wed Mar 12 2014)