DGtal
0.9.2

#include <DGtal/topology/helpers/Surfaces.h>
Public Types  
typedef TKSpace  KSpace 
typedef KSpace::Integer  Integer 
typedef KSpace::Point  Point 
typedef KSpace::Cell  Cell 
typedef KSpace::SCell  SCell 
typedef KSpace::DirIterator  DirIterator 
Public Member Functions  
~Surfaces ()  
void  selfDisplay (std::ostream &out) const 
bool  isValid () const 
Static Public Member Functions  
template<typename PointPredicate >  
static SCell  findABel (const KSpace &K, const PointPredicate &pp, unsigned int nbtries=1000) throw (DGtal::InputException) 
template<typename PointPredicate >  
static SCell  findABel (const KSpace &K, const PointPredicate &pp, Point x1, Point x2) 
template<typename SCellSet , typename PointPredicate >  
static void  trackBoundary (SCellSet &surface, const KSpace &K, const SurfelAdjacency< KSpace::dimension > &surfel_adj, const PointPredicate &pp, const SCell &start_surfel) 
template<typename SCellSet , typename PointPredicate >  
static void  trackClosedBoundary (SCellSet &surface, const KSpace &K, const SurfelAdjacency< KSpace::dimension > &surfel_adj, const PointPredicate &pp, const SCell &start_surfel) 
template<typename SCellSet , typename SurfelPredicate >  
static void  trackSurface (SCellSet &surface, const KSpace &K, const SurfelAdjacency< KSpace::dimension > &surfel_adj, const SurfelPredicate &pp, const SCell &start_surfel) 
template<typename SCellSet , typename SurfelPredicate >  
static void  trackClosedSurface (SCellSet &surface, const KSpace &K, const SurfelAdjacency< KSpace::dimension > &surfel_adj, const SurfelPredicate &pp, const SCell &start_surfel) 
template<typename PointPredicate >  
static void  track2DBoundary (std::vector< SCell > &aSCellContour2D, const KSpace &K, const SurfelAdjacency< KSpace::dimension > &surfel_adj, const PointPredicate &pp, const SCell &start_surfel) 
template<typename PointPredicate >  
static void  track2DSliceBoundary (std::vector< SCell > &aSCellContour2D, const KSpace &K, const Dimension &trackDir, const SurfelAdjacency< KSpace::dimension > &surfel_adj, const PointPredicate &pp, const SCell &start_surfel) 
template<typename SurfelPredicate >  
static void  track2DSurface (std::vector< SCell > &aSCellContour, const KSpace &K, const SurfelAdjacency< KSpace::dimension > &surfel_adj, const SurfelPredicate &sp, const SCell &start_surfel) 
template<typename SurfelPredicate >  
static void  track2DSliceSurface (std::vector< SCell > &aSCellContour, const KSpace &K, const Dimension &trackDir, const SurfelAdjacency< KSpace::dimension > &surfel_adj, const SurfelPredicate &sp, const SCell &start_surfel) 
template<typename PointPredicate >  
static void  track2DBoundaryPoints (std::vector< Point > &aVectorOfPoints, const KSpace &K, const SurfelAdjacency< KSpace::dimension > &surfel_adj, const PointPredicate &pp, const SCell &start_surfel) 
template<typename PointPredicate >  
static void  extractAllPointContours4C (std::vector< std::vector< Point > > &aVectPointContour2D, const KSpace &aKSpace, const PointPredicate &pp, const SurfelAdjacency< 2 > &aSAdj) 
template<typename PointPredicate >  
static void  extractAll2DSCellContours (std::vector< std::vector< SCell > > &aVectSCellContour2D, const KSpace &aKSpace, const SurfelAdjacency< KSpace::dimension > &aSurfelAdj, const PointPredicate &pp) 
template<typename PointPredicate >  
static void  extractAllConnectedSCell (std::vector< std::vector< SCell > > &aVectConnectedSCell, const KSpace &aKSpace, const SurfelAdjacency< KSpace::dimension > &aSurfelAdj, const PointPredicate &pp, bool forceOrientCellExterior=false) 
template<typename PointPredicate >  
static void  orientSCellExterior (std::vector< SCell > &aVectOfSCell, const KSpace &aKSpace, const PointPredicate &pp) 
template<typename CellSet , typename PointPredicate >  
static void  uMakeBoundary (CellSet &aBoundary, const KSpace &aKSpace, const PointPredicate &pp, const Point &aLowerBound, const Point &aUpperBound) 
template<typename SCellSet , typename PointPredicate >  
static void  sMakeBoundary (SCellSet &aBoundary, const KSpace &aKSpace, const PointPredicate &pp, const Point &aLowerBound, const Point &aUpperBound) 
template<typename OutputIterator , typename PointPredicate >  
static void  uWriteBoundary (OutputIterator &out_it, const KSpace &aKSpace, const PointPredicate &pp, const Point &aLowerBound, const Point &aUpperBound) 
template<typename OutputIterator , typename PointPredicate >  
static void  sWriteBoundary (OutputIterator &out_it, const KSpace &aKSpace, const PointPredicate &pp, const Point &aLowerBound, const Point &aUpperBound) 
template<typename TSurfelPredicate , typename TImageContainer >  
unsigned static int  uFillInterior (const KSpace &aKSpace, const TSurfelPredicate &aSurfPred, TImageContainer &anImage, const typename TImageContainer::Value &aValue, bool empty_is_inside=false, bool incrementMode=true) 
template<typename SurfelPredicate , typename TImageContainer >  
unsigned static int  uFillExterior (const KSpace &aKSpace, const SurfelPredicate &aSurfPred, TImageContainer &anImage, const typename TImageContainer::Value &aValue, bool empty_is_outside=true, bool incrementMode=true) 
Protected Member Functions  
Surfaces ()  
Private Member Functions  
Surfaces (const Surfaces &other)  
Surfaces &  operator= (const Surfaces &other) 
Aim: A utility class for constructing surfaces (i.e. set of (n1)cells).
Description of template class 'Surfaces'
TKSpace  the type of cellular grid space (e.g. a KhalimskySpaceND). 
Note that many methods requires a surfel adjacency so as to know how connected surfels are linked (first by the interior or first by the exterior). Secondly, many methods are parameterized by a PointPredicate which describes the shape as a characteristic function. This is to be more generic than a simple DigitalSet. With this approach, shapes can be defined implicitly.
Essentially a backport from ImaGene.
Definition at line 78 of file Surfaces.h.
typedef KSpace::Cell DGtal::Surfaces< TKSpace >::Cell 
Definition at line 85 of file Surfaces.h.
typedef KSpace::DirIterator DGtal::Surfaces< TKSpace >::DirIterator 
Definition at line 87 of file Surfaces.h.
typedef KSpace::Integer DGtal::Surfaces< TKSpace >::Integer 
Definition at line 83 of file Surfaces.h.
typedef TKSpace DGtal::Surfaces< TKSpace >::KSpace 
Definition at line 82 of file Surfaces.h.
typedef KSpace::Point DGtal::Surfaces< TKSpace >::Point 
Definition at line 84 of file Surfaces.h.
typedef KSpace::SCell DGtal::Surfaces< TKSpace >::SCell 
Definition at line 86 of file Surfaces.h.
DGtal::Surfaces< TKSpace >::~Surfaces  (  ) 
Destructor.

protected 
Constructor. Forbidden by default (protected to avoid g++ warnings).

private 
Copy constructor.
other  the object to clone. Forbidden by default. 

static 
Extract all contours as a vector containing the set of contours composed of SCell. Each contour is represented by a vector of signed surfels whose elements represents a boundary component of a digital shape defined by the predicate [pp]. The algorithms tracks surfels along the boundary of the shape.
PointPredicate  a model of concepts::CPointPredicate describing the inside of a digital shape, meaning a functor taking a Point and returning 'true' whenever the point belongs to the shape. 
aVectSCellContour2D  (modified) a vector of contour represented by a vector of cells (which are all surfels), containing the ordered list of the boundary component of [spelset]. 
aKSpace  any space. 
aSurfelAdj  the surfel adjacency chosen for the tracking. 
pp  an instance of a model of concepts::CPointPredicate, for instance a SetPredicate for a digital set representing a shape. 

static 
Extract all surfel elements associated to each connected components of the given DigitalSet. The connected surfel set are given as result in a vector containing all components. The orientation of the resulting SCell indicates the exterior orientation according the positive axis.
PointPredicate  a model of concepts::CPointPredicate describing the inside of a digital shape, meaning a functor taking a Point and returning 'true' whenever the point belongs to the shape. 
aVectConnectedSCell  (modified) a vector containing for each connected components a vector of the sequence of connected SCells. 
aKSpace  any space. 
aSurfelAdj  the surfel adjacency chosen for the tracking. 
pp  an instance of a model of concepts::CPointPredicate, for instance a SetPredicate for a digital set representing a shape. 
forceOrientCellExterior  if 'true', used to change the default cell orientation in order to get the direction of shape exterior (default =false). This is used only for displaying cells with Viewer3D. This mechanism should evolve shortly. 

static 
Function that extracts all the boundaries of a 2D shape (specified by a predicate on point) in a 2D KSpace. The boundaries are returned as a vector of vector of points.
Extract all 4connected contours as a vector containing the sequence of contour Points. Each contour is represented by a vector of points defined by the sequence of pointels extracted from the boundary surfels. Calls extractAll2DSCellContours.
PointPredicate  a model of concepts::CPointPredicate describing the inside of a digital shape, meaning a functor taking a Point and returning 'true' whenever the point belongs to the shape. 
aVectPointContour2D  (modified) a vector of contour represented by a vector of cells (which are all surfels), containing the ordered list of the boundary component of [pp]. 
aKSpace  any space. 
pp  an instance of a model of concepts::CPointPredicate, for instance a SetPredicate for a digital set representing a shape. 
aSAdj  the surfel adjacency chosen for the tracking. 

static 
Find a bel in some digital set by random tries then dichotomy.
PointPredicate  a model of concepts::CPointPredicate describing the inside of a digital shape, meaning a functor taking a Point and returning 'true' whenever the point belongs to the shape. 
K  any cellular grid space. 
pp  an instance of a model of concepts::CPointPredicate, for instance a SetPredicate for a digital set representing a shape, which should be at least partially included in the bounds of space [K]. 
nbtries  the maximum number of random tries (default 1000). 

static 
Find a bel in some digital set given two hints (one point inside, one point outside).
PointPredicate  a model of concepts::CPointPredicate describing the inside of a digital shape, meaning a functor taking a Point and returning 'true' whenever the point belongs to the shape. 
K  any cellular grid space. 
pp  an instance of a model of concepts::CPointPredicate, for instance a SetPredicate for a digital set representing a shape, which should be at least partially included in the bounds of space [K]. 
x1  a point within the bounds of K and such that pp( x1 ) != pp( x2 ). 
x2  a point within the bounds of K and such that pp( x1 ) != pp( x2 ). 
bool DGtal::Surfaces< TKSpace >::isValid  (  )  const 
Checks the validity/consistency of the object.

private 
Assignment.
other  the object to copy. 

static 
Orient the SCell positively in the direction of the exterior of the DigitalSet shape. It simply check if the direct incident Cell in the first upper dimension (obtain with KhalimskySpaceND::sDirectIncident) belongs to the DigitalSet or not.
This method is used to change the default cell orientation in order to get the direction of shape exterior (default =false). This is used only for displaying cells with Viewer3D. This mechanism should evolve shortly.
PointPredicate  a model of concepts::CPointPredicate describing the inside of a digital shape, meaning a functor taking a Point and returning 'true' whenever the point belongs to the shape. 
aVectOfSCell  (modified) a vector containing the SCell to be oriented positively in the direction of the exterior. 
aKSpace  any space. 
pp  an instance of a model of concepts::CPointPredicate, for instance a SetPredicate for a digital set representing a shape. 
void DGtal::Surfaces< TKSpace >::selfDisplay  (  std::ostream &  out  )  const 
Writes/Displays the object on an output stream.
out  the output stream where the object is written. 

static 
Creates a set of signed surfels whose elements represents all the boundary components of a digital shape described by the predicate [pp].
SCellSet  a model of a set of SCell (e.g., std::set<SCell>). 
PointPredicate  a model of concepts::CPointPredicate describing the inside of a digital shape, meaning a functor taking a Point and returning 'true' whenever the point belongs to the shape. 
aBoundary  (modified) a set of cells (which are all surfels), the boundary component of [aSpelSet]. 
aKSpace  any space. 
pp  an instance of a model of concepts::CPointPredicate, for instance a SetPredicate for a digital set representing a shape. 
aLowerBound  and 
aUpperBound  points giving the bounds of the extracted boundary. 

static 
Writes on the output iterator out_it the signed surfels whose elements represents all the boundary elements of a digital shape described by the predicate [pp].
OutputIterator  any output iterator (like std::back_insert_iterator< std::vector<SCell> >). 
PointPredicate  a model of concepts::CPointPredicate describing the inside of a digital shape, meaning a functor taking a Point and returning 'true' whenever the point belongs to the shape. 
out_it  any output iterator for writing the signed cells. 
aKSpace  any space. 
pp  an instance of a model of concepts::CPointPredicate, for instance a SetPredicate for a digital set representing a shape. 
aLowerBound  and 
aUpperBound  points giving the bounds of the extracted boundary. 

static 
Function that extracts the boundary of a 2D shape (specified by a predicate on point) in a 2D KSpace. The boundary is returned as a vector of surfels. The surfels are guaranteed to be in the direct orientation ordering.
Creates a vector of signed surfels whose elements represents a 2D boundary component of a digital shape described by a PointPredicate. The algorithm tracks surfels along the boundary of the shape by starting from the given [start_surfel]. It only tracks the boundary of a 2D shape.
PointPredicate  a model of concepts::CPointPredicate describing the inside of a digital shape, meaning a functor taking a Point and returning 'true' whenever the point belongs to the shape. 
aSCellContour2D  (modified) a vector of cells (which are all surfels), containing the ordered list of the boundary component of [spelset] which touches [start_surfel]. 
K  any space of dimension 2. 
surfel_adj  the surfel adjacency chosen for the tracking. 
pp  an instance of a model of concepts::CPointPredicate, for instance a SetPredicate for a digital set representing a shape, which should be at least partially included in the bounds of space [K]. 
start_surfel  a signed surfel which should be between an element of [shape] and an element not in [shape]. 

static 
Function that extracts the boundary of a 2D shape (specified by a predicate on point) in a 2D KSpace. The boundary is returned as a vector of points.
This method uses random tries to find a first linel separating an interior pixel from an exterior one. It then follows direct orientations to extract the 4connected set of points.
aVectorOfPoints  (returns) the sequence of points of the boundary component of the digitized shape containing [start_surfel]. 
K  any space of dimension 2. 
surfel_adj  the surfel adjacency chosen for the tracking. 
pp  an instance of a model of concepts::CPointPredicate, for instance a SetPredicate for a digital set representing a shape, which should be at least partially included in the bounds of space [K]. 
start_surfel  a signed surfel which should be between an element of [shape] and an element not in [shape]. 

static 
Function that extracts a 2D slice of the boundary of a nD shape (specified by a predicate on point) in a nD KSpace. The boundary is returned as a vector of surfels. The surfels are guaranteed to be in the direct orientation ordering.
Creates a vector of signed surfels whose elements represents a 2D boundary component of a digital shape described by a PointPredicate. The algorithms tracks surfels along the boundary of the shape by starting from the given [start_surfel], along the direction specified by [trackDir]. More precisely, it is the boundary of the slice of the shape along directions [trackDir] and the orthogonal direction of [start_surfel].
PointPredicate  a model of concepts::CPointPredicate describing the inside of a digital shape, meaning a functor taking a Point and returning 'true' whenever the point belongs to the shape. 
aSCellContour2D  (modified) a vector of cells (which are all surfels), containing the ordered list of the boundary component of [spelset] which touches [start_surfel]. 
K  any space (dimension is arbitrary). 
trackDir  the initial track direction at [start_surfel], should be different from the orthogonal direction of [start_surfel]. 
surfel_adj  the surfel adjacency chosen for the tracking. 
pp  an instance of a model of concepts::CPointPredicate, for instance a SetPredicate for a digital set representing a shape, which should be at least partially included in the bounds of space [K]. 
start_surfel  a signed surfel which should be between an element of [shape] and an element not in [shape]. 

static 
Function that extracts a 2D slice of a n1 digital surface (specified by a predicate on surfel) in a nD KSpace. The boundary is returned as a vector of surfels. The surfels (of dimension n1) are guaranteed to be in the direct orientation ordering.
Creates a vector of signed surfels whose elements represents a component of a slice of digital surface described by a SurfelPredicate. The algorithm tracks surfels along the surface by starting from the given start_surfel and by moving along the directions trackDir and the orthogonal direction to start_surfel. All surfels are returned so as to follow the direct orientation. If the surface is open, the first surfel is at one extremity (the indirect extremity) while the last surfel is at the other extremity. Otherwise, the first surfel is start_surfel.
SurfelPredicate  a model of CSurfelPredicate, meaning a functor taking a Surfel (SCell) and returning 'true' whenever the surfel belongs to the digital surface. Models include FrontierPredicate and BoundaryPredicate. 
aSCellContour  (modified) a vector of cells (which are all surfels), containing the ordered list of surfels that forms the connected component of the digital surface slice containing surfel start_surfel and the direction trackDir. 
K  any space (dimension is arbitrary). 
trackDir  the initial track direction at [start_surfel], should be different from the orthogonal direction of [start_surfel]. 
surfel_adj  the surfel adjacency chosen for the tracking. 
sp  an instance of a model of CSurfelPredicate, for instance a FrontierPredicate or a BoundaryPredicate. 
start_surfel  a signed surfel such that sp(start_surfel) is true. 

static 
Function that extracts a 1dcontour (specified by a predicate on surfel) in a 2D KSpace. The boundary is returned as a vector of surfels. The surfels are guaranteed to be in the direct orientation ordering.
Creates a vector of signed surfels whose elements represents a component of a digital surface described by a SurfelPredicate. The algorithm tracks surfels along the surface by starting from the given start_surfel. All surfels are returned so as to follow the direct orientation. If the surface is open, the first surfel is at one extremity (the indirect extremity) while the last surfel is at the other extremity. Otherwise, the first surfel is start_surfel.
SurfelPredicate  a model of CSurfelPredicate, meaning a functor taking a Surfel (SCell) and returning 'true' whenever the surfel belongs to the digital surface. Models include FrontierPredicate and BoundaryPredicate. 
aSCellContour  (modified) a vector of cells (which are all surfels), containing the ordered list of surfels that forms the connected component containing surfel [start_surfel]. 
K  any space of dimension 2. 
surfel_adj  the surfel adjacency chosen for the tracking. 
sp  an instance of a model of CSurfelPredicate, for instance a FrontierPredicate or a BoundaryPredicate. 
start_surfel  a signed surfel such that sp(start_surfel) is true. 

static 
Function that extracts the boundary of a nD digital shape (specified by a predicate on point), closed or open, in a nD KSpace. The boundary is returned as a set of surfels.
Creates a set of signed surfels whose elements represents a boundary component of a digital shape described by a PointPredicate. The algorithms tracks surfels along the boundary of the shape.
SCellSet  a model of a set of SCell (e.g., std::set<SCell>). 
PointPredicate  a model of concepts::CPointPredicate describing the inside of a digital shape, meaning a functor taking a Point and returning 'true' whenever the point belongs to the shape. 
surface  (modified) a set of cells (which are all surfels), the boundary component of [spelset] which touches [start_surfel]. 
K  any space. 
surfel_adj  the surfel adjacency chosen for the tracking. 
pp  an instance of a model of concepts::CPointPredicate, for instance a SetPredicate for a digital set representing a shape, which should be at least partially included in the bounds of space [K]. 
start_surfel  a signed surfel which should be between an element of [shape] and an element not in [shape]. 

static 
Function that extracts the closed boundary of a nD digital shape (specified by a predicate on point), in a nD KSpace. The boundary is returned as a set of surfels.
Creates a set of signed surfels whose elements represents a boundary component of a digital shape described by a PointPredicate. The algorithms tracks surfels along the boundary of the shape. It follows only direct orientations, so that it is faster than trackBoundary but requires the object to be fully inside the space. Follows the idea of Artzy, Frieder and Herman algorithm [Artzy:1981cgip], but in nD.
SCellSet  a model of a set of SCell (e.g., std::set<SCell>). 
PointPredicate  a model of concepts::CPointPredicate describing the inside of a digital shape, meaning a functor taking a Point and returning 'true' whenever the point belongs to the shape. 
surface  (modified) a set of cells (which are all surfels), the boundary component of [spelset] which touches [start_surfel]. 
K  any space. 
surfel_adj  the surfel adjacency chosen for the tracking. 
pp  an instance of a model of concepts::CPointPredicate, for instance a SetPredicate for a digital set representing a shape, which should be at least partially included in the bounds of space [K]. 
start_surfel  a signed surfel which should be between an element of [shape] and an element not in [shape]. 

static 
Function that extracts a closed n1 digital surface (specified by a predicate on surfel) in a nD KSpace. The surface is returned as a set of surfels.
Creates a set of signed surfels whose elements represents a boundary component of a digital surface described by a SurfelPredicate. The algorithms tracks surfels along the surface. This is an optimized version of trackSurface, which is valid only when the tracked surface is closed.
SCellSet  a model of a set of SCell (e.g., std::set<SCell>). 
SurfelPredicate  a model of CSurfelPredicate describing whether a surfel belongs or not to the surface. 
surface  (modified) a set of cells (which are all surfels), the boundary component of [spelset] which touches [start_surfel]. 
K  any space. 
surfel_adj  the surfel adjacency chosen for the tracking. 
pp  an instance of a model of CSurfelPredicate. 
start_surfel  a signed surfel which should be part of the surface, ie. 'sp(start_surfel)==true'. 

static 
Function that extracts a n1 digital surface (specified by a predicate on surfel), closed or open, in a nD KSpace. The surface is returned as a set of surfels.
Creates a set of signed surfels whose elements represents a boundary component of a digital surface described by a SurfelPredicate. The algorithms tracks surfels along the surface.
SCellSet  a model of a set of SCell (e.g., std::set<SCell>). 
SurfelPredicate  a model of CSurfelPredicate describing whether a surfel belongs or not to the surface. 
surface  (modified) a set of cells (which are all surfels), the boundary component of [spelset] which touches [start_surfel]. 
K  any space. 
surfel_adj  the surfel adjacency chosen for the tracking. 
pp  an instance of a model of CSurfelPredicate. 
start_surfel  a signed surfel which should be part of the surface, ie. 'sp(start_surfel)==true'. 

static 
Given a boundary surface [aSurfPred] in [aKSpace] (given as SurfelPredicate), fills its exterior in a given image [anImage] by a specific value [aValue] (considered as increment if [incrementMode] is set to true). The technique is to fill line by line and tests the intersection with the surface. Note that the set of the surfel of the boundary has to be a closed surface.
aKSpace  the digital space. 
aSurfPred  the digital Jordan surface. 
anImage  the image to be filled. 
aValue  the value to fill the image. 
empty_is_outside  when 'true', an empty line is considered exterior, otherwise interior. 
incrementMode  if set to 'true' the image value is incremented by [aValue] instead to be set to [aValue] (default). 

static 
Given a boundary surface [aSurfPred] in [aKSpace] (given as SurfelPredicate), fills its interior in a given image [anImage] by a specific value [aValue] (considered as increment if [incrementMode] is set to true). The technique is to fill line by line and tests the intersection with the surface. Note that the set of the surfel of the boundary has to be a closed surface.
aKSpace  the digital space. 
aSurfPred  the digital Jordan surface. 
anImage  the image to be filled. 
aValue  the value to fill the image. 
empty_is_inside  when 'true', an empty line is considered interior, otherwise exterior (set by default to false). 
incrementMode  if set to 'true' the image value is incremented by [aValue] instead to be set to [aValue] (default). 

static 
Creates a set of unsigned surfels whose elements represents all the boundary components of a digital shape described by the predicate [pp].
CellSet  a model of a set of Cell (e.g., std::set<Cell>). 
PointPredicate  a model of concepts::CPointPredicate describing the inside of a digital shape, meaning a functor taking a Point and returning 'true' whenever the point belongs to the shape. 
aBoundary  (modified) a set of cells (which are all surfels), the boundary component of [aSpelSet]. 
aKSpace  any space. 
pp  an instance of a model of concepts::CPointPredicate, for instance a SetPredicate for a digital set representing a shape. 
aLowerBound  and 
aUpperBound  points giving the bounds of the extracted boundary. 

static 
Writes on the output iterator out_it the unsigned surfels whose elements represents all the boundary elements of a digital shape described by the predicate [pp].
OutputIterator  any output iterator (like std::back_insert_iterator< std::vector<Cell> >). 
PointPredicate  a model of concepts::CPointPredicate describing the inside of a digital shape, meaning a functor taking a Point and returning 'true' whenever the point belongs to the shape. 
out_it  any output iterator for writing the cells. 
aKSpace  any space. 
pp  an instance of a model of concepts::CPointPredicate, for instance a SetPredicate for a digital set representing a shape. 
aLowerBound  and 
aUpperBound  points giving the bounds of the extracted boundary. 