Digital Signal Processing Reference
In-Depth Information
Fig. 11.15
Self-similarity
matrix for Adriano
Celentano—“
Azzurro
”. Bright
45
◦
diagonals indicate a high
self-similarity [
31
]
250
200
150
100
50
50
100
150
200
250
x
. Further, beat-tracking as was described in Sect.
11.3
is used for
beat-grid alignment.
An
N
=
(
x
1
,...,
x
12
)
N
self-similarity matrix
S
is calculated (cf. Sect.
11.3
) based on the cosine
distance [
141
] as follows:
×
x
(
i
),
x
(
j
)
S
(
i
,
j
)
=
)
.
(11.33)
x
(
i
)
·
x
(
j
If this matrix is visualised as heat-map, one now searches 'bright' diagonal segments
parallel to the main diagonal at a 45 degree angle (cf. Fig.
11.15
)inmatrix
S
which
indicate highly self-similar segments in a musical piece. To locate these, an edge
filter can be used as given by
⎧
⎨
1for
i
=
j
F
Diag
(
i
,
j
)
=
c
for
0
<
|
i
−
j
|≤
b
(11.34)
⎩
0for
|
i
−
j
|
>
b
2
with 1
≤
i
,
j
≤
20,
b
=
5 and
c
=−
17
. Then, a normalisation follows and a
is subtracted from the filtered 'image'. This results in the matrix
S
and
is carried out for reduction of noise introduced by the edge filter. The threshold
threshold
δ
δ
can be chosen as the highest value exceeded by at least 10
·
N
values in the filtered
'image'. Next, a binary matrix
S
b
is computed by
1for
S
(
i
,
j
)>
0
S
b
(
i
,
j
)
=
.
(11.35)
0for
S
(
i
,
j
)
≤
0
