![](bg3.png)
are the coordinates given by the rotation angle. Hence
they are defined as:
˜x = xcosθ+ ysinθ
˜y = xsinθ − ycosθ
(2)
We use anisotropic Gaussians with σ = σ
x
= 2σ
y
.
Therefore, the Gaussian function results in:
G
σ,θ
=
1
(2π)2σ
2
e
−
˜x
2
2(2σ)
2
+
˜y
2
2σ
2
(3)
Hence the kernel will be defined as:
∂
2
˜y
G
σ,θ
=
˜y
2
− 1
σ
4
G
σ,θ
(4)
We apply a normalization so that the geometry of
the valleys is prioritized:
G
N
σ,θ
:=
k∂
2
˜y
G
σ,θ
∗ Ik
k∂
2
˜y
G
σ,θ
kkIk
(5)
where k · k stands for the L
2
integral norm and ∗ de-
noting the convolution operator.
The kernels are applied to different scales and in
different equally distributed orientations in a range
centered in the joint orientation. The final outcome
are 24 output images, each of them corresponding to
a determined orientation and scale. Hence, the output
I
ridges
must be a combination of all of them, defined
as the maximum of the outputs from each filter:
I
ridges
= max
i, j
G
N
σ
i
,θ
j
(6)
On the other hand, the edge space image is ob-
tained as the gradient of the input image after apply-
ing structure preserving diffusion (Gil et al., 2009).
Diffusion filtering proved its success in improving the
quality of the edge detection by smoothing the image
irregularities while keeping the main image structure.
The second stage performs the feature extraction
using Non-maximum Suppresion algorithm (Canny,
1986), which only keeps pixels that are local maxima
along the gradient direction. Gradients are computed
using the structure tensor of the space image.
In the third stage hysteresis thresholding algo-
rithm (Canny, 1986) is used to binarize the non-
maximum suppressed images while preserving fea-
ture connectivity and removing weak responses.
Finally, the two images forwarded by the third
stage must go through the fourth processing step in or-
der to provide the final sclerosis and lower bone seg-
mentations. This stage is different for the two thresh-
olded images:
• The final sclerosis segmentation is the ridge in the
binarized image that is closer to the center of the
image following the finger orientation .
• As far as lower bone is concerned, the correspond-
ing binarized image is processed to remove the
edges in the upper part and the margins. After-
wards, the endpoints in the remaining processed
edges are linked. The final lower bone segmenta-
tion is obtained by computing the convex hull of
the linked-edge image.
4 EXPERIMENTAL SETUP
Two subsets were created from our dataset of 320 an-
notated images to perform our experiments: 1) Tune
Dataset, with 40 randomly selected healthy joints (20
MCP and 20 PIP); 2) Test Dataset, with the remaining
280 joints (140 MCP and 140 PIP). The Tune Dataset
was used to tune the parameters of the system for scle-
rosis and lower bone contour segmentation, and the
Test Dataset was used to compute the output of our
system and compare the proposed distance measure
with the SvdH score. We added 20 DIP joints from
the non-annotated dataset to enrich the variability of
the Tune Dataset -these joints can be safely included
because, although not having a SvdH score, they fit
our model-. Both sclerosis and lower bone were man-
ually segmented by an expert using OsiriX (Rosset
et al., 2004) software exclusively for the 60 images of
the Tune Dataset.
Performance metrics were developed to evaluate
the performance of the system, i.e. the quality of our
detections, and tune the parameters. We based our
metrics on the Average Surface Distance (ASD), de-
fined as follows:
ASD(U, V) =
1
|S(U)|
∑
s
U
∈S(U)
d(s
U
, S(V))
!
(7)
where, given a pixel p and a region R conformed by a
set of pixels S(R), d(p, S(R)) is defined as:
d(p, S(R)) = min
s
R
∈S(R)
kp − S
R
k (8)
where k.k stands for the Euclidean distance.
Thus, if A denotes our automatic segmentation and
M denotes the manual delineation, we define:
Caught = ASD(A, M) (9)
Missed = ASD(M, A) (10)
When both Caught and Missed metrics are zero
the segmentation is perfect. Caught metric value is
related to the quality of the detector at detecting valid
pixels (true positives) and avoiding non-desired pix-
els (false positives). Analogously, the lower the value
of Missed, the less desired information was missed
(false negatives).
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