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not applicable in this context because of the complex
appearance of the cell, including the surrounding zona
pellucida, clutter, and artifacts; this also hinders the
application of straightforward edge-based segmenta-
tion algorithms, as many spurious contours are de-
tected.
Iterative energy minimization methods such as ac-
tive contours (Xu, 1998) and level sets are frequently
employed in biomedical imaging: in this context,
their application is not straightforward because de-
bris are likely to generate several local minima in
the energy function, which makes quick and robust
convergence problematic; for example, in (Morales
et al., 2008) active contours are used for measuring
the thickness of the zona pellucida in embryo images,
but only after a preprocessing step aimed at removing
debris and other artifacts.
In (Beuchat et al., 2008) a semisupervised tech-
nique for measuring various zygote features is used,
where the cell shape is approximated by an ellipse:
in our case, instead, we recover the actual shape of
the cell, which is often not well approximated by an
ellipse.
The technique we are presenting includes a global
energy minimization step, and may be classified as a
specialized graph-cut (Zabih and Kolmogorov, 2004)
approach, where: a) priors on the cell shape are ac-
counted for by operating on a spatially-transformed
image and searching for a minimum-cost path on a
directed acyclic graph; b) priors on the contour ap-
pearance due to HMC lighting are directly integrated
in the energy terms; c) information at different fo-
cal planes is simultaneously represented in a single
large graph. We are therefore extending the approach
in (Giusti et al., 2009) by operating on information
from many focal planes at the same time.
Interestingly, several previous works handled the
peculiar lighting in HMC and DIC images as an ob-
stacle to segmentation (Kuijper and Heise, 2008),
and adopted preprocessing techniques for removing
it, whereas we actually exploit such appearance for
improving robustness.
3 PRELIMINARIES, MODEL AND
NOTATION
Our algorithm is designed to operate on a Z-stack
of N images taken with Hoffman Modulation Con-
trast (HMC) microscopy
1
. We denote the input im-
1
a technique delivering visually similar results is Dif-
ferential Interference Contrast (DIC), which is also a likely
application scenario for our technique.
ages as I
1
,I
2
,...,I
N
, and their respective focal planes
z = z
1
,z
2
..z
N
. Such focal planes can be considered
horizontal slices at different depths of a 3D space
whose cartesian axes are (x,y, z).
HMC is an imaging technique converting optical
slopes to variations of the light intensity: it is rou-
tinely used in IVF labs for observing zygotes, as it
provides a large amount of contrast for transparent
specimens and eases human observation as the objects
appear three-dimensional and side-lit, as if a light
source was illuminating them from a side (apparent
lighting direction).
The underlying imaging model is considerably
complex, especially if the effect of out-of-focus fea-
tures is taken into account. Still, several intuitive prin-
ciples hold, on which we base our approach:
• structures which lie on or near the current focal
plane z
i
appear sharp and exhibit strong localized
gradients in the image intensity I
i
;
• as the focal plane depth moves farther from the
structure’s depth, the structure image becomes
blurred. Consequently, its gradients of the struc-
ture’s image lose locality and strength, although
the global contrast and visibility of the feature
may not be affected, or may even be emphatised
in some situations
2
.
In this work, the main feature of interest is the cell
contour; as the cell is a 3D object, in order to explain
the appearance of its contour at different focus levels,
we provide the following formalization: Let S be the
surface of the cell, which we assume to be smooth,
in the 3D space (x,y,z). The contour generator curve
Γ is a curve in 3D space, identified by the locus of
points P on S such that the tangent plane to S in P
contains the z direction
3
. Although this definition al-
lows Γ to be composed by several disjoint curve parts,
the regularity of the cell shape, which is convex and
ellipsoid-like, allows us to assume that Γ is a single,
closed curve in the following.
We are interested in detecting the image of the
contour generator curve Γ in our input images I
i
. In
particular, let γ be the 2D apparent contour, i.e. the or-
thogonal projection of Γ on the (x, y) plane. Follow-
ing the principles introduced previously in this sec-
tion, a part of γ is visible and well-focused in an image
I
i
if the corresponding part of Γ is on or near the z = z
i
2
in fact, a slightly defocused feature imaged through a
phase contrast technique may appear more evident to an hu-
man operator than the same feature in perfect focus; this
makes manual focusing inherently operator-dependent and
hardly repeatable.
3
Note that this is similar to the concept of contour gen-
erator curve in projective geometry where an ortographic
camera is considered.
SIMULTANEOUS FOCUSING AND CONTOURING OF HUMAN ZYGOTES FOR IN VITRO FERTILIZATION
153