Tangent Estimation
of 3D Digital Curves
Kacper Pluta, Technion – Israel Institute of Technology, Center for Graphics and Geometric Computing
Technion 04/11/2018
Y. Kenmochi
CNRS, France
a joint work with
J.-O. Lachaud
University of Savoy, France
Digital Geometry
Digital Geometry
Digital Geometry
A classical problem of digital geometry is to estimate geometric properties of digitized shapes without any knowledge about the underlying continuous shape.
Digital Geometry
Digital Geometry
Definition (Multigrid Discrete Space)
Let
h > 0
Figure comes from R. Klette & A. Rosenfeld: Digital Geometry: Geometrical Mehods For Digital Picture Analysis, Morgan Kaufmann 2004
be a grid resolution and
\mathbb{Z}_h = \left\{\frac{i}{h}| i \in \mathbb{Z}\right\}
, then
\mathbb{Z}^n_h
is the
set of n-D discrete points in a grid of the resolution h.
Digital Geometry in Image Processing Pipeline
Digital Geometry in Image Processing Pipeline
Digital Geometry in Image Processing Pipeline
Digital Geometry in Image Processing Pipeline
Digital Geometry in Image Processing Pipeline
The information about the tangent orientation at the integer points of the skeleton allows for local measurements.
Methods for Tangent Estimating
Approximation techniques in the Euclidean space
Methods which work directly in discrete spaces
+ very good accuracy
- can be costly
- poor behavior on sharp corners
+ good accuracy
+ simple and fast
- often poor behavior on corrupted/noisy curves
Adaptivity to Local Geometry
Some methods fix size of the computational window globally. Therefore, they cannot adapt to local geometry of the shape.
\mathbf{x}_0
\mathbf{x}_1
Adaptivity to Local Geometry
Some methods fix size of the computational window globally. Change of the resolution involves change of the window size.
\mathbf{x}_0
\mathbf{x}_1
Adaptivity to Local Geometry
The size of the computational window can be adopted to the local curve's geometry thanks to the notion of Digital Straight Segments.
What is a curve in ?
\mathbb{Z}^3
Digital Curves
3D Digital Curve
Any set
\mathbf{C}
such that
|\mathbf{C}| \geq 2
is called a 3D digital k-curve iff there
\mathbf{p},\mathbf{q} \in \mathbf{C}
such that
|\Gamma_k(\mathbf{p}) \cap \mathbf{C}| = |\Gamma_k(\mathbf{q}) \cap \mathbf{C}| = 2
and for any
\mathbf{x} \in \mathbf{C} \setminus \{\mathbf{p}, \mathbf{q}\},
we have
|\Gamma_k(\mathbf{x}) \cap \mathbf{C}| = 3 .
are possibly two points
Adjacency Set
Given a point
\mathbf{x} \in \mathbb{Z}^3,
the adjacency set of
\mathbf{x}
is defined as:
\Gamma_k(\mathbf{x}) = \{\mathbf{y} | d^2(\mathbf{x},\mathbf{y}) \leq n \} \setminus \{\mathbf{x}\},
where
d
stands for the Euclidean distance function and k for an
adjacency relation i.e., 6 (face), 18 (edge), 26 (vertex).
Digital Curves
A digital 26-curve i.e.,
n \leq 3
The Tangent Estimation Algorithm, In Short
- Decompose the digital k-curve into digital k-line segments. Then, such line segments approximate the tangent directions at their points
- For each point of the curve estimate a discrete tangent as a weighted sum of the directions given by the the k-line segments passing through the point
The Building Blocks
Arithmetic Digital Line
An arithmetic line of parameters
2D Arithmetic Line
(a, b, c)
and of arithmetic thickness
\omega
is defined as
D(a, b, c, \omega) = \{ (p, q) \in \mathbb{Z}^2 | c \leq a p - b q < c + \omega \}
where
a, b, c, \omega \in \mathbb{Z}
and
\gcd(a, b) = 1.
\omega
Theorem [Reveillès 1991]
D(a, b, c, \omega)
Let
be an arithmetic line, then:
1. If
\omega < \max(|a|, |b|)
then
D(a, b, c, \omega)
is not connected,
2. If
\omega = \max(|a|, |b|)
then
D(a, b, c, \omega)
is 8-connected,
3. If
\omega = |a| + |b|
then
D(a, b, c, \omega)
is 4-connected.
1.
2.
3.
Arithmetic Digital Line
Definition (Remainder)
Remainder and Leaning Point
The remainder associated to a point
\mathbf{p} = (p_1, p_2)
of
D(a, b, c, \omega)
is an
integer value defined as
r(\mathbf{p}) = a p_1 - b p_2.
- When
r(\mathbf{p}) = c
then
\mathbf{p}
is called a
lower leaning point
- When
r(\mathbf{p}) = c + \omega - 1
then
\mathbf{p}
called an upper leaning point
is
Arithmetic Digital Line Recognition
Let
- M(i, j) be a segment of a digital 8-line
D(a, b, c, \omega)
with
0 \leq a < b,
- be the point of the greatest abscissa of M,
\mathbf{q} = (q_1, q_2)
- l and l' be the lower leaning points of minimum and maximum abscissae of M,
- u and u' be the upper leaning points of minimum and maximum abscissae of M,
By adding a point
\mathbf{p} = (p_1, p_2)
connected to M such that
p_1 = q_1 + 1
we verify if
M' = M \cup \{\mathbf{p}\}
is an 8-conneted line segment.
Arithmetic Digital Line Recognition
Theorem [Debled-Rennesson and Reveillès, 1995]
We have
1. if
c \leq r(\mathbf{p}) < b
then M' is an 8-connected line segment
2. if
r(\mathbf{p}) < c - 1
D(a, b, c, \omega)
or
r(\mathbf{p}) > b
then M' is not 8-connected
3. if
r(\mathbf{p}) = c - 1
then M' is an 8-connected line segment
D(p_2 - u_2, p_1 - u_1, -a p_1 + b p_2, \omega)
4. if
r(\mathbf{p}) = b
then M' is an 8-connected line
D(p_2 - l_2, p_1 - l_1, -a p_1 + b p_2 + b - 1, \omega)
1.
4.
Arithmetic Digital Line Recognition
It is then possible to decompose a sequence of k-connected points into digital k-connected line segments.
\mathit{M}_1(0, 18)
\mathit{M}_2(18, 29)
\mathit{M}_3(29, 42)
\mathit{M}_4(42, 60)
\mathbf{C}(0,60)
a 8-connected digital curve i.e., a sequence of points from 0 to 60.
Figure comes from I. Debled-Rennesson: Reconnaissance des droites et plans discrets, PhD Thesis 1995
Arithmetic Digital Line Recognition
\mathit{M}_1(0, 18)
\mathit{M}_2(18, 29)
\mathit{M}_3(29, 42)
\mathit{M}_4(42, 60)
Figure comes from I. Debled-Rennesson: Reconnaissance des droites et plans discrets, PhD Thesis 1995
Over the years several discrete tangent estimators based on a digital line recognition has been proposed.
They all try to make the right balance between the longest and the most centered DSS around a point of interest.
How about 3D?
Figure comes from I. Debled-Rennesson: Reconnaissance des droites et plans discrets, PhD Thesis 1995
Curve Functionality
A set C of 3D integer points is functional along an axis (say x) if the points of C can be sorted such that the x-component increase by one between two consecutive points.
The projections and have the same cardinal as C i.e.,
these projections are bijective.
\pi_{xy}(\mathbf{C})
\pi_{zx}(\mathbf{C})
Definition
Proposition
Curve Functionality
\pi_{xy}(\mathbf{C})
\pi_{zx}(\mathbf{C})
Curve Functionality
From now on we consider a 3D 26-connected curve
C, which is functional.
Any 26-connected 3D digital curve can be decomposed into functional parts of size at least two with an overlap of at least one point.
3D k-digital Line Segment
In the 3D case, M(i, j) is verified iff two of the three projections of C
on the planes
O_{xy}, O_{xz}
and
O_{yz}
are 2D digital line segments.
M is 26-connected iff the two valid projections are 8-connected.
\pi_{xy}(\mathbf{C})
\pi_{zx}(\mathbf{C})
3D Maximal Line Segments
A digital line segment M is maximal if it cannot be extended anymore from any end.
Definition
Let M be a 3D maximal segment along C. Then there exists one 2D maximal DSS along and a 2D maximal DSS along such that M is the intersection of their back projection onto C.
\pi_{xy}(\mathbf{C})
\pi_{zx}(\mathbf{C})
Proposition
Tangent Estimation Algorithm
The Tangent Estimation Algorithm, In Short
- Decompose the digital k-curve into maximal k-line segments. Then, such line segments approximate the tangent directions at their points
- For each point of the curve estimate a discrete tangent as a weighted sum of the directions given by the the k-line segments passing through the point
Tangential Cover
For any digital curve C, there is a unique set
\mathcal{M}
segments called the tangential cover [Feschet and Tougne 1999].
of maximal
Tangential Cover
Tangential Cover
The set of maximal segments going through a point
\mathbf{x} \in \mathbf{C}
is called
the pencil of maximal segments at
\mathbf{x}
and defined as
P(\mathbf{x}) = \{ M_i \in \mathcal{M} | \mathbf{x} \in M_i \}.
Eccentricity
The eccentricity
\mathbf{x}
M_i
L_i
m_i
e_i(\mathbf{x})
of a point
\mathbf{x}
with respect to a maximal
segment
M_i
is the relative position between the extremities of
M_i
such that
e_i(\mathbf{x}) =
\begin{cases}
\| \mathbf{x} - m_i \|_1 & \quad \text{if } M_i \in P(\mathbf{x}),\\
0 & \quad \text{otherwise.}
\end{cases}
\lambda
-Maximal-segment Tangent
Estimator
The
\lambda
direction
\mathbf{t}(\mathbf{x})
at the point
\mathbf{x}
of a digital k-curve is defined
as a weighted combination of the vectors
\mathbf{t}_i
of the covering
maximal segments
M_i
such that
\mathbf{t} (\mathbf{x}) = \frac{\sum_{M_i \in P(\mathbf{x})} \lambda(e_{i}(\mathbf{x}))\frac{\mathbf{t}_i}{|\mathbf{t}_i|}}{\sum_{M_i
\in P(\mathbf{x})} \lambda(e_{i}(\mathbf{x}))}.
\lambda
-Functions
\lambda: x \mapsto \sin(\pi x)
\lambda: x \mapsto 64 (-x^6 + 3 x^5 - 3 x^4 + x^3)
\lambda: x \mapsto \frac{2}{e^{15 (x - 1/2)} + e^{-15 (x - 1/2)}}
A function
\lambda
maps from
[0,1]
to
\mathbb{R}^+
with
\lambda(0) = \lambda(1) = 0
and
\lambda > 0
elsewhere.
Multigrid Convergence
What does it mean multigrid convergent?
D(X,h)
X
Continuous shape
and its digitization
with respect to h.
Definition [Coeurjolly et al. 12]
The estimator
\hat{Q}
is multigrid-convergent for a family of shapes
\mathbb{X}
iff
for any
X \in \mathbb{X},
there exists a grid step
h_X > 0
such that the
estimate
\hat{Q}(D(X,h), \mathbf{y} , h)
is defined for all
\mathbf{y} \in \partial D(X,h)
with
0 < h < h_X,
and for any
\mathbf{x} \in \partial X,
\forall \mathbf{y} \in \partial D(X,h)
with
\| \mathbf{y} - \mathbf{x} \|_1 \leq h, | \hat{Q} (D(X,h), \mathbf{y}, h ) - Q(X,\mathbf{x}) | \leq \tau_{X,\mathbf{x}}(h).
\tau_{X,\mathbf{x}}
Where
is a function defining the speed of convergence of
Any digital point sufficiently close to the point of the interest has its estimated geometric quantity, which tends toward the expected local value of the geometric function as h tends toward 0.
\hat{Q}
towards
Q
at a point
\mathbf{x} \in \partial X.
Multigrid Convergence
Theorem [Lachaud et al. 07]
The 2D MST estimator is multigrid convergent toward the tangent direction along the boundary of any convex shape with three time differentiable boundary and continuous curvature. An upper bound for its average rate of convergence is , as the grid step h tends toward 0.
\lambda-
O(h^{\frac{1}{3}})
Steps of Proving Convergence in 2D
- Maximal segments have unbounded digital length with a finer and finer resolution
- The slope of a maximal segment is then proved to tend toward the tangent direction of any point it covers
For the details see [Lachaud et al. 07]
Convergence in 2D
h = 1
h = 0.1
h
h
Average Absolute Error
Average Absolute Error
For the details see [Coeurjolly et al. 12]
The plots comes from [Coeurjolly et al. 12]
Convergence in 2D
For the details see [Coeurjolly et al. 12]
h = 1
h = 0.1
h
h
Average Absolute Error
Average Absolute Error
The plots comes from [Coeurjolly et al. 12]
How About 3D?
Properties of 3D Maximal Segments
In contrast to their 2D counterparts, 3D maximal segments can be short, even for very small h. This poses a problem for proving the convergence of 3D MST.
\lambda-
Bad News
Good News
We can derive two other methods for which we can prove the bound of average convergence, and then experimentally compare the convergence rate of the three methods.
Proposition
Let us denote by the Euclidean length of the curve .
Then the asymptotic average digital length of 3D maximal segments along h-digitizations of with gridstep h follows
\mathcal{L}(\gamma)
\gamma
\bar{\Lambda}(h)
\gamma
\bar{\Lambda}(h) = \mathcal{L}(\gamma) \Omega\left(h^{-\frac{1}{3}}\right).
Properties of 3D Maximal Segments
Properties of 3D Maximal Segments
Proposition
For any point of a functional curve C, there exists some 3D maximal segment such that:
\mathbf{x}
M_k
- it is in the vicinity of
\mathbf{x}: d(\mathbf{x}, M_k) \leq \frac{\ell}{2},
- it is long enough:
L(M_k) \geq \frac{\ell}{2},
where is the digital length of the shortest 2D maximal segments along the two valid projections.
\ell
Asymptotic Properties and Multigrid Convergence
Filtered 3D MST
\lambda-
Using the previous proposition we discard all the 3D maximal
L \leq \frac{\ell}{2}.
segments with length
3D DSS is considered as being at the beginning (resp. end) of a 3D DSS that is not further away than
\frac{\ell}{2}.
Then, each point not covered by a
Asymptotic Properties and Multigrid Convergence
3D MST by 2D
We consider two valid projection with respect to the main axis. We use 2D MST on the 2D projective curves, and finally we combine the two 2D estimations of the tangent directions into a 3D tangent direction at each 3D point.
\lambda-
\lambda-
Asymptotic Properties and Multigrid Convergence
Theorem
The 3D tangent estimator by combination of 2D maximal segments, and the 3D tangent estimator by filtered 3D maximal segments are both multigrid convergent toward the 3D tangent. An upper bound for their average rate of convergence is , as the grid step h tends toward 0.
O(h^{\frac{1}{3}})
Experimental Evaluation of the Tangent Estimators
Experimental Evaluation of the Tangent Estimators
Conclusion and Perspectives
- Understand behavior of the estimator on non-functional curves
- Future improvements for noisy data e.g., skeletons obtained from segmented CT scans
- Comparison with other methods
Perspectives
We studied a 3D tangent estimator that is fast, simple and has good asymptotic behaviors
Conclusion
The source code has been submitted to DGtal C++ library.
References
- J.-P. Reveillès: Géométrie Discrète, Calcul en Nombres Entiers et Algorithmique, Habilitation à diriger des recherches, Université Louis-Pasteur, 1991, https://hal.archives-ouvertes.fr/tel-01279525
-
I. Debled-Rennesson and J.-P. Reveillès: A linear algorithm for segmentation of digital curves, International Journal of Pattern Recognition and Artificial Intelligence, vol. 9(6), 1995.
-
F. Feschet, L. Tougne: Optimal time computation of the tangent of a discrete curve: application to the curvature, in: Proceedings of DGCI’99, pp. 31–40, Springer 1999
-
J.-O. Lachaud, A. Vialard, F. de Vieilleville: Fast, accurate and convergent tangent estimation on digital contours, Image and Vision Computing, pp. 1572-1587, vol. 25(10), 2007
-
D. Coeurjolly, J.-O. Lachaud, T. Roussillon: Multigrid Convergence of Discrete Geometric Estimators, Lecture Notes in Computational Vision and Biomechanics, pp. 395-424, vol. 2, Springer 2012
Thank you for tour Attention!
Tangent Estimation of 3D Digital Curves
By Kacper Pluta
Tangent Estimation of 3D Digital Curves
A seminary talk about 3D Lambda-maximal segment tangent estimator.
