GRAPH CUTS AND APPROXIMATION OF THE EUCLIDEAN

METRIC ON ANISOTROPIC GRIDS

Ondˇrej Danˇek and Pavel Matula

Centre for Biomedical Image Analysis, Faculty of Informatics, Masaryk University, Brno, Czech Republic

Keywords:

Graph cuts, Euclidean metric approximation, Anisotropic grids, Voronoi diagrams, Image segmentation.

Abstract:

Graph cuts can be used to ﬁnd globally minimal contours and surfaces in 2D and 3D space, respectively.

To achieve this, weights of the edges in the graph are set so that the capacity of the cut approximates the

contour length or surface area under chosen metric. Formulas giving good approximation in the case of the

Euclidean metric are known, however, they assume isotropic resolution of the underlying grid of pixels or

voxels. Anisotropy has to be simulated using more general Riemannian metrics. In this paper we show how

to circumvent this and obtain a good approximation of the Euclidean metric on anisotropic grids directly by

exploiting the well-known Cauchy-Crofton formulas and Voronoi diagrams theory. Furthermore, we show

that our approach yields much smaller metrication errors and most interestingly, it is in particular situations

better even in the isotropic case due to its invariance to mirroring. Finally, we demonstrate an application of

the derived formulas to biomedical image segmentation.

1 INTRODUCTION

Graph cuts were originally developed as an elegant

tool for interactive image segmentation (Boykov and

Funka-Lea, 2006) with applicability to N-D problems

and allowing integration of various types of regional

or geometric constraints. Nevertheless, they quickly

emerged as a general technique to solve diverse com-

puter vision and image processing problems (Boykov

and Veksler, 2006). Particularly, graph cuts are suit-

able to ﬁnd global minima of certain classes of en-

ergy functionals (Kolmogorov and Zabih, 2004) fre-

quently used in computer vision in polynomial time.

Among others, these may include energy terms de-

pendent on contour length or surface area. This is due

to (Boykov and Kolmogorov, 2003) who proved that

despite their discrete nature graph cuts can approxi-

mate any Euclidean or Riemannian metric with arbi-

trarily small error and derived the required formulas

for edge weights.

In the following text we focus on the Euclidean

metric as it is essential for graph cut based mini-

mization of many popular energy functionals such as

the Chan-Vese model for image segmentation (Chan

and Vese, 2001) (Zeng et al., 2006). The formu-

las derived in (Boykov and Kolmogorov, 2003) as-

sume isotropic resolution of the underlying grid of

pixels/voxels which is a limitation in some ﬁelds. For

instance, volumetric images produced by optical mi-

croscopes often have notably lower resolution in the

z axis than in the xy plane. Hence, before process-

ing it is necessary to either upsample the z direction

which substantially increases computational demands

or downsample the xy plane which causes loss of in-

formation. Last option is to simulate the anisotropy

using the more general Riemannian metrics. Unfor-

tunately, it turns out that the corresponding formu-

las have signiﬁcantly larger approximation error that

once again can be reduced only for the price of slower

and more memory intensive computation taking into

account larger neighbourhood.

In this paper we show how to solve the above

mentioned problem and derive the weights required

for the approximation of the Euclidean metric on

anisotropic grids directly. For this purpose we fol-

low (Boykov and Kolmogorov, 2003) and exploit the

well-known Cauchy-Crofton formulas from integral

geometry. However, several amendments allow us to

obtain a better approximation. Namely, we employ

Voronoi diagrams theory to calculate the partitioning

of angular orientations of lines which is required dur-

ing the discretization of the Cauchy-Crofton formu-

las. This among other things makes our approxima-

tion invariant to image mirroring. Moreover,we show

that our approach has much smaller metrication error,

especially in the case of small neighbourhood or large

Dan

ek O. and Matula P. (2010).

GRAPH CUTS AND APPROXIMATION OF THE EUCLIDEAN METRIC ON ANISOTROPIC GRIDS.

In Proceedings of the International Conference on Computer Vision Theory and Applications, pages 68-73

DOI: 10.5220/0002833000680073

 SciTePress

Figure 1: (a) 8-neighbourhood 2D grid graph. (b) 16-

neighbourhood system on a grid with isotropic resolution.

anisotropy and that under speciﬁc conditions it is bet-

ter even in the isotropic case.

The paper is structured as follows. The notation

and known results are brieﬂy reviewed in Section 2.

In Section 3 we present our contribution and derive

the formulas approximating the Euclidean metric on

both 2D and 3D grids with anisotropic resolution.

Section 4 contains detailed discussion of the approxi-

mation error and gives example of an application of

our results to biomedical image segmentation. We

conclude the paper in Section 5.

2 CUT METRICS

Consider an undirected graph G embedded in a reg-

ular orthogonal 2D grid with all nodes having topo-

logically identical neighbourhood system and with

isotropic spacing δ between the nodes. Example of

such graph with 8-neighbourhood system is depicted

in Fig.1a. Further, let the neighbourhood system N

be described by a set of vectors N = {e

,...,e

}. We

assume that the vectors are listed in the increasing or-

der of their angular orientation 0 ≤ φ

< π. We also

assume that vectors e

are undirected (we do not dif-

ferentiate between e

and −e

) and shortest possible

in given direction, e.g. 16-neighbourhood would be

represented by a set of 8 vectors N

= {e

,...,e

}

as depicted in Fig.1b. Finally, we deﬁne the distance

between the nearest lines generated by vector e

in the

grid as ρ

(for 8-neighbourhood these are depicted in

Fig.1a).

Lets assume each edge e

is assigned particular

weight w

and imagine we are given a contour as

shown in Fig.1a. This contour divides the nodes of

the graph into two groups based on whether they lie

inside or outside the contour. A cut C is deﬁned as the

set of all edges joining the inner nodes with the outer

ones. The cut capacity |C |

is the sum of the weights

of the cut edges. The question stands whether it is

possible to set weights w

so that the capacity of the

cut approximatesthe Euclidean length |C |

of the con-

Figure 2: (a) 8-neighbourhood system on a grid with

anisotropic resolution. (b) Computation of ∆φ

tour. Since algorithms for ﬁnding minimal cuts con-

stitute well studied part of combinatorial optimiza-

tion (Boykov and Kolmogorov, 2004) this would al-

low us to effectively ﬁnd globally minimal contours

or surfaces satisfying certain criterion.

The technical result of (Boykov and Kolmogorov,

2003) answers the question positively. Based on the

Cauchy-Crofton formula from integral geometry the

weights for a 2D grid should be set to:

∆φ

2|e

(1)

The whole derivation of the formula is omitted here,

so is the extension to 3D grids. Both are being ex-

plained in more detail in the remaining text. Never-

theless, as already suggested in the introduction Eu-

clidean metric is not the only one that can be approxi-

mated using graph cuts. The complete discussion can

be found in (Kolmogorov and Boykov, 2005).

3 EUCLIDEAN METRIC ON

ANISOTROPIC GRIDS

Up until now we assumed that the grid of graph nodes

has isotropic resolution δ. In this section we investi-

gate the anisotropic case and adjust the edge weight

formulas appropriately.

3.1 2D Grids

Consider an undirected graph G embedded in a reg-

ular orthogonal 2D grid with all nodes having topo-

logically identical neighbourhood system. However,

let the spacing of the grid nodes be δ

and δ

in hori-

zontal and vertical directions, respectively. Otherwise

the whole notation remains unchanged. Example of

an 8-neighbourhood system over an anisotropic grid

is depicted in Fig. 2a.

GRAPH CUTS AND APPROXIMATION OF THE EUCLIDEAN METRIC ON ANISOTROPIC GRIDS

Now, consider the Cauchy-Crofton formula that

links Euclidean length |C |

of contour C with a mea-

sure of a set of lines intersecting it:

|C |

dL (2)

where L is the space of all lines and n

(l) is the num-

ber of intersections of line l with contour C . Every

line in a plane is uniquely identiﬁed by its angular

orientation φ and distance ρ from the origin. Thus,

the formula can be rewritten in the form:

|C |

+∞

−∞

(φ,ρ)

dρdφ (3)

and discretized by partitioning the space of all lines

according to the neighbourhood N = {e

,...,e

|C |

≈

∑

k=1

∑

(k,i)

∆ρ

∆φ

(4)

where i enumerates lines generated by vector e

. Fur-

ther, let n

(k) =

∑

(k,i) be the total number of in-

tersections of contour C with all lines generated by

vector e

. We obtain:

|C |

≈

∑

k=1

(k)

∆ρ

∆φ

(5)

From the last equation it can be seen, that if we set

∆ρ

∆φ

(6)

then (proof omitted):

|C |

,δ

→0

−−−−−−−−−−−−→

sup∆φ

→0,sup|e

|→0

|C |

(7)

Finally, the distance between the closest lines gener-

ated by vector e

in the grid equals to:

∆ρ

(8)

and if we substitute Eq.8 into Eq.6 we obtain the

above mentioned Eq.1.

So far we have followed the method of (Boykov

and Kolmogorov, 2003). However, when δ

6= δ

this

approach has a serious ﬂaw. One may notice that in

the example depicted in Fig. 2a edges e

and e

will

be assigned different weights because ∆φ

6= ∆φ

and

∆ρ

= ∆ρ

. But this means that if we mirror the con-

tour horizontally we will obtain different cut capacity.

Hence, edge weights derived this way are not invari-

ant to mirroring, which is rather inconvenient prop-

erty causing additional bias of the approximation. In

fact, this bias is present in the isotropic case as well,

but not for all neighbourhoods. For instance, in the

16-neighbourhood depicted in Fig.1b edges e

and e

will be assigned different weights and it indeed has a

negative effect on the approximation as we will show

in the following section

The solution lies in different partitioning of the

unit circle of angular orientations. We do not uti-

lize ∆φ

in the way it has been used so far. Instead

we introduce new symbol ∆φ

which from a proba-

bilistic point of view can be interpreted as a measure

of lines closest to e

in terms of their angular ori-

entation. The computation is done as follows. Let

S = {

,...,

} be a set of points lying on a unit

circle. We calculate the Voronoi diagram of S on the

1D circle manifold and deﬁne ∆φ

to be the size of the

Voronoi cell corresponding to point

. The whole

process is depicted in Fig. 2b. It reduces to the fol-

lowing formula:

∆φ

+ ∆φ

k−1

(9)

Putting this together with Eq.6 and Eq.8 the ﬁnal edge

weights for a 2D grid with anisotropic resolution are

calculated as:

′

∆ρ

∆φ

(∆φ

+ ∆φ

k−1

)

4|e

(10)

Such edge weights still follow the distribution of the

angular orientations of lines generated by vectors in

N but in a smarter way causing the approximation to

be invariant to contour mirroring while not breaking

the convergence of the original approach at the same

time.

3.2 3D Grids

In three dimensions the contour C is replaced by a

surface C

and the graph G is embedded in a reg-

ular orthogonal 3D grid with δ

, δ

and δ

spacing

between the nodes in x, y and z directions, respec-

tively, with all nodes having topologically identical

3D neighbourhood system N = {e

,...,e

} (e.g. 6-,

18- or 26-neighbourhood).

This time ∆ρ

expresses the ”density” of lines

generated by vector e

. It is calculated by intersect-

ing these lines with a plane perpendicular to them and

computing the area of cells in the obtained 2D grid of

points. It can be easily computed using this formula:

∆ρ

(11)

Each vector e

is now determined by two angular

orientations ϕ

and ψ

with ∆φ

corresponding to

the partitioning of the unit sphere among the angu-

lar orientations of vectors in N . In fact, this formu-

lation is rather vague and it is unclear how to cal-

culate ∆φ

the way it is being described in (Boykov

VISAPP 2010 - International Conference on Computer Vision Theory and Applications

0 2 4 6 8

−30

−20

−10

angular orientation

multiplicative error

Our

BK ’03

0 2 4 6 8

−6

−4

−2

angular orientation

multiplicative error

Our

BK ’03

0 2 4 6 8

−3

−2

−1

angular orientation

multiplicative error

Our

BK ’03

0 2 4 6 8

−3

−2

−1

angular orientation

multiplicative error

Our

BK ’03

0 2 4 6 8

−100

−50

100

150

angular orientation

multiplicative error

Our

BK ’03

0 2 4 6 8

−60

−40

−20

angular orientation

multiplicative error

Our

BK ’03

0 2 4 6 8

−25

−20

−15

−10

−5

angular orientation

multiplicative error

Our

BK ’03

0 2 4 6 8

−10

−5

angular orientation

multiplicative error

Our

BK ’03

0 2 4 6 8

−100

100

200

300

400

angular orientation

multiplicative error

Our

BK ’03

0 2 4 6 8

−100

−50

100

150

angular orientation

multiplicative error

Our

BK ’03

0 2 4 6 8

−60

−40

−20

angular orientation

multiplicative error

Our

BK ’03

0 2 4 6 8

−40

−30

−20

−10

angular orientation

multiplicative error

Our

BK ’03

N16

N32

1:1

3:1

6:1

Figure 3: Metrication error (in percents) in 2D for several combinations of neighbourhood system and anisotropy ratio.

and Kolmogorov, 2003). Particularly because for al-

most all common 3D neighbourhoods (e.g. 18- or 26-

neighbourhood) the distribution of the angular orien-

tations is not uniform (this stems from the fact, that

it is not possible to create a Platonic solid for such

number of points).

The capacity of a cut is analogously deﬁned as the

sum of the weights of the edges joining grid nodes en-

closed by the surface C

with those lying outside and

the goal is to set the weights w

so that the capacity

of the cut approximates the area of the surface under

Euclidean metric. The Cauchy-Crofton formula for

surface area in 3D is:

dL (12)

and using the same derivation steps as in the case of

2D grids yields the following edge weights:

∆ρ

∆φ

π|e

(13)

The problem with the clarity of ∆φ

is addressed

easily by extending our concept of Voronoi diagram

based weights ∆φ

. Let S = {

,...,

} be a set of

points this time lying on a unit sphere. We calculate

the Voronoi diagram of S on the 2D sphere surface

manifold and deﬁne ∆φ

to be the area of the Voronoi

cell corresponding to point

. This is a general

and explicit approach that can be used for any type

of neighbourhood. Unfortunately, the spherical case

can not be reduced to a simple formula. To compute

the spherical Voronoi diagram we recommend to use

the convex hull based method described in (Brown,

1979). Putting this all together we end up with the

ﬁnal formula for 3D anisotropic grids:

′

∆ρ

∆φ

π|e

(14)

To conclude this section, this approach can be the-

oretically extended to any number of dimensions. In

the general N-D case one would have to calculate

Voronoi diagram of points on a hypersphereto get ∆φ

weights. The adjustment of ∆ρ

is straightforward.

4 EXPERIMENTAL RESULTS

4.1 Approximation Error

To benchmark the approximations we chose to mea-

sure the multiplicative error they give under partic-

ular angular orientations in 2D. Graphs of the er-

ror are available in Fig. 3. The ﬁgure contains 12

graphs where each column corresponds to a partic-

ular 2D neighbourhood and each row to a particu-

lar anisotropy ratio. We compared our approxima-

tion with the method described in (Boykov and Kol-

mogorov,2003). To simulate the anisotropy we had to

embed it into a Riemannian metric in the latter case.

According to the referenced paper the weights for a

GRAPH CUTS AND APPROXIMATION OF THE EUCLIDEAN METRIC ON ANISOTROPIC GRIDS

Figure 4: Two examples of biomedical image segmentation using the Chan-Vese model. (a) yz cross-section of the segmented

image. (b) Level-set based method. (c) Graph cuts with edge weights for isotropic resolution. (d) Graph cuts with anisotropy

embedded into the Riemannian metric. (e) Graph cuts with our edge weights.

Riemannian metric with a constant metric tensor D

over an isotropic 2D grid should be set to:

∆φ

2|e

detD

· D· u

)

3/2

(15)

where u

and to:

∆φ

π|e

detD

· D· u

)

(16)

in case of a 3D grid. Resolution change corresponds

to a constant metric tensor with eigenvectors aligned

with the coordinate system and eigenvalues δ

and δ

Hence, the metric tensor simulating the anisotropic

grid has the following form:

D =



0 δ



(17)

Notice that if D is the identity matrix the second term

in Eq.15 and Eq.16 vanishes and the formulas reduce

to the isotropic case.

The multiplicative error measures in percents the

difference between the approximated value and the

factual length, i.e. zero is the ideal meaning no er-

ror. As can be seen from Fig. 3 both approaches per-

form equivalently in the isotropic case for 4- and 8-

neighbourhood. For larger neighbourhoods our ap-

proach is almost two times better and its invariance

to mirroring is also apparent as the graph is sym-

metrical around values

, π and

3π

. With increasing

anisotropy the gap widens and especially for smaller

neighbourhoods the difference is really huge. How-

ever, note that the maximal error depends primarily

on sup ∆φ

and that this value increases with increas-

ing anisotropy. Thus, for high anisotropy ratios using

larger neighbourhood is inevitable.

4.2 Applications to Image Segmentation

In this subsection we show the practical impact of our

results and evaluate the beneﬁts of the improved ap-

proximation in biomedical image segmentation. We

chose the Chan-Vese segmentation model (Chan and

Vese, 2001) that is being very popular in this ﬁeld

for its robust segmentation of highly degraded data.

The Chan-Vese model is a binary segmentation model

VISAPP 2010 - International Conference on Computer Vision Theory and Applications

which corresponds to piecewise-constant specializa-

tion of the well-known Mumford-Shah energy func-

tional. In simple terms, it segments the image into

two regions trying to minimize the length of the fron-

tier between them and their intensity variance. This

functional can be minimized using graph cuts (Zeng

et al., 2006) and as it tries to minimize the boundary

length it obviously depends on the Euclidean metric

approximation.

To test the improvement of our approximation

over the previous approach also in 3D we plugged

the derived formulas into the algorithm and used it to

segment low-quality volumetric images of cell clus-

ters acquired by an optical microscope. The yz cross-

sections of the segmented images are depicted in

Fig.4a. The dimensions of the images are 280×360×

50, with resolution in the xy plane being about 4.5

times the resolution in the z direction. We used 26-

neighbourhood to segment the images.

In Fig. 4b is the Chan-Vese segmentation com-

puted using level-sets. This technique was much

slower than the graph cuts, however, it does not suf-

fer from the metrication errors so we used its results

as the ground truth. Figure 4c shows the graph cut

based segmentation when the anisotropy is ignored.

The results obtained using the Riemannian metric and

our weights are depicted in Fig. 4d and Fig. 4e, re-

spectively. Clearly, our method gives a result closest

to the level-sets. On the other hand, the segmenta-

tion based on the Riemannian metric seems too ﬂat

or chopped. Based on the results from the previous

subsection it could be probably greatly improved us-

ing a larger neighbourhood, but at the cost of higher

computational demands.

5 CONCLUSIONS AND FUTURE

WORK

In this paper we addressed the problem of approxima-

tion of the Euclidean metric on 2D and 3D anisotropic

grids via graph cuts. We derived the required formu-

las and showed that our approach has a signiﬁcantly

smaller metrication error than the previous one and

that it is invariant to image mirroring. Using the pre-

sented results it is possible to exploit graph cut based

energy minimization dependent on contour length or

surface area over images with anisotropic resolution

directly without the need to resample them or to use

large neighbourhoods for better precision. A possi-

ble application of the results was demonstrated on a

biomedical image segmentation.

As explained in Section 4.1 anisotropic grids cor-

respond to a special case of the Riemannian met-

ric with a constant metric tensor with eigenvectors

aligned with the coordinate system. However, the

general case of this metric is also being widely used

in several ﬁelds including image segmentation. Tak-

ing into account the relatively high error of the current

formulas we would like to make use of the presented

results and focus on better approximations of the gen-

eral case of the Riemannian metric in our future work.

ACKNOWLEDGEMENTS

This work has been supported by the Ministry of Ed-

ucation of the Czech Republic (Projects No. MSM-

0021622419, No. LC535 and No. 2B06052).

REFERENCES

Boykov, Y. and Funka-Lea, G. (2006). Graph cuts and efﬁ-

cient n-d image segmentation (review). International

Journal of Computer Vision, 70(2):109–131.

Boykov, Y. and Kolmogorov, V. (2003). Computing geo-

desics and minimal surfaces via graph cuts. In ICCV

’03: Proceedings of the Ninth IEEE International

Conference on Computer Vision, page 26.

Boykov, Y. and Kolmogorov, V. (2004). An experi-

mental comparison of min-cut/max-ﬂow algorithms

for energy minimization in vision. IEEE Transac-

tions on Pattern Analysis and Machine Intelligence,

26(9):1124–1137.

Boykov, Y. and Veksler, O. (2006). Graph cuts in vision and

graphics: Theories and applications. In Handbook of

Mathematical Models in Computer Vision, pages 79–

96. Springer-Verlag.

Brown, K. Q. (1979). Geometric transforms for fast geo-

metric algorithms. PhD thesis, Pittsburgh, PA, USA.

Chan, T. F. and Vese, L. A. (2001). Active contours with-

out edges. IEEE Transactions on Image Processing,

10(2):266–277.

Kolmogorov, V. and Boykov, Y. (2005). What metrics can

be approximated by geo-cuts, or global optimization

of length/area and ﬂux. In ICCV ’05: Proceedings

of the Tenth IEEE International Conference on Com-

puter Vision, pages 564–571.

Kolmogorov, V. and Zabih, R. (2004). What energy func-

tions can be minimized via graph cuts? IEEE Trans-

actions on Pattern Analysis and Machine Intelligence,

26(2):147–159.

Zeng, Y., Chen, W., and Peng, Q. (2006). Efﬁciently solv-

ing the piecewise constant mumford-shah model using

graph cuts. Technical report, Dept. of Computer Sci-

ence, Zhejiang University, P.R. China.

GRAPH CUTS AND APPROXIMATION OF THE EUCLIDEAN METRIC ON ANISOTROPIC GRIDS