![](bg2.png)
we will consider the following matching measures:
MF (Matching Function), i.e. the approach recently
proposed in (Tombari et al., 2007), GC (Gradient
Correlation), i.e. the approach initially proposed in
(Scharstein, 1994) according to the formulation pro-
posed in (Crouzil et al., 1996) for the pattern matching
problem, OC (Orientation Correlation), proposed in
(Fitch et al., 2002). Moreover, we will consider the
SSD (Sum of Squared Differences) and NCC (Nor-
malized Cross Correlation) measures applied on gra-
dient norms (G-NCC and G-SSD), which showed
good robustness against illumination changes in the
experimental comparison described in (Martin and
Crowley, 1995). The approach relying on gradi-
ent orientation only, proposed in (Seitz, 1989) and
successively modified in (Aschwanden and Guggen-
buhl, 1992), has not been taken into account since, as
pointed out in (Crouzil et al., 1996), it is prone to er-
rors in the case of gradient vectors with small mag-
nitude. Considered traditional measures are NCC,
ZNCC (Zero-mean NCC) and SSD: NCC and ZNCC
showed good robustness with respect to brightness
changes, on the other hand SSD showed good insen-
sibility toward noise (Aschwanden and Guggenbuhl,
1992), (Martin and Crowley, 1995).
All the considered proposals are tested on 3
datasets which represent a challenging framework
for what regards the considered distortions. These
dataset are publicly available
1
and they might serve
as a testbed for future evaluations of robust match-
ing measures. Before reporting on the experimen-
tal evaluation, all compared measures are briefly de-
scribed. Furthermore, for what regards MF, the paper
also proposes some modifications to the original ap-
proach proposed in (Tombari et al., 2007), and ex-
ploits the proposed experimental evaluation to per-
form a behavioural analysis of this class of functions.
2 TRADITIONAL MATCHING
CRITERIA
Matching measures traditionally adopted in order to
compute the similarity between two pixel sets are
typically subdivided into two groups, given they are
based on an affinity or distortion criterion. Affinity
measures are often based on correlation, withthe most
popular metric being the Normalized Cross Correla-
tion (NCC). In a pattern matching scenario, being P
the pattern vector, sized M × N (width × height), I the
image vector, sized W × H, and I
xy
the image subwin-
dow at coordinates (x, y) and having the same dimen-
1
Available at: www.vision.deis.unibo.it/pm-eval.asp
sions as P, the NCC function at (x, y) is given by:
NCC(x, y) =
M
∑
i=1
N
∑
j=1
P(i, j) ·I
xy
(i, j)
v
u
u
t
M
∑
i=1
N
∑
j=1
P
2
(i, j) ·
v
u
u
t
M
∑
i=1
N
∑
j=1
I
2
xy
(i, j)
(1)
As it can be seen, the cross-correlation between
P and I
xy
is normalized by the L
2
norms of the two
vectors, in order to render the measure robust to any
spatially constant multiplicative bias. By subtracting
the mean intensity value of the pattern and of the im-
age subwindow we get an even more robust matching
measure:
ZNCC(x, y) =
M
∑
i=1
N
∑
j=1
(P(i, j) −
¯
P) · (I
xy
(i, j) −
¯
I
xy
)
s
M
∑
i=1
N
∑
j=1
(P(i, j) −
¯
P)
2
·
s
M
∑
i=1
N
∑
j=1
(I
xy
(i, j) −
¯
I
xy
)
2
(2)
where
¯
P and
¯
I
xy
represent respectively the mean in-
tensity of P and I
xy
. This measure is referred to as
Zero-mean NCC (ZNCC) and it is robust to spatially
constant affine variations of the image intensities.
As regards the distortion criterion, the classical
measures are based on the L
p
distance between P and
I
xy
. In particular, with p = 2 we get the Sum of Squared
Differences (SSD):
SSD(x, y) =
M
∑
i=1
N
∑
j=1
P(i, j) − I
xy
(i, j)
2
(3)
While all these measures are usually computed di-
rectly on the pixel intensities of the image sets, in
(Martin and Crowley, 1995) it was shown that by
computing these measures on the gradient norm of
each pixel a higher robustness is attained, i.e. for what
concerns insensitivity to illumination changes G-SSD
and G-NCC showed to perform well. In particular, if
we denote with G
P
(i, j) the gradient of the pattern at
pixel (i, j):
G
P
(i, j) =
∂P(i, j)
∂i
,
∂P(i, j)
∂j
T
=
h
G
P
i
(i, j), G
P
j
(i, j)
i
T
(4)
and similarly with G
I
xy
(i, j) the gradient of the image
subwindow at pixel (i, j):
G
I
xy
(i, j) =
∂I
xy
(i, j)
∂i
,
∂I
xy
(i, j)
∂ j
T
=
h
G
I
xy
i
(i, j), G
I
xy
j
(i, j)
i
T
(5)
the gradient norm, or magnitude, in each of the
two cases is computed as:
||G
P
(i, j)|| =
q
G
P
i
(i, j)
2
+ G
P
j
(i, j)
2
(6)
||G
I
xy
(i, j)|| =
q
G
I
xy
i
(i, j)
2
+ G
I
xy
j
(i, j)
2
(7)
i.e. || · || represents the L
2
norm of a vector. Hence the
G-NCC function can be defined as:
G− NCC(x, y) =
M
∑
i=1
N
∑
j=1
||G
P
(i, j)|| · ||G
I
xy
(i, j)||
s
M
∑
i=1
N
∑
j=1
||G
P
(i, j)||
2
·
s
M
∑
i=1
N
∑
j=1
||G
I
xy
(i, j)||
2
(8)
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