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The global uncentered correlation coefficient between two discrete
signals
and
can be defined as the functional
![\begin{displaymath}
\gamma = \frac{\displaystyle\sum_{i=1}^{N}\mathbf{a}_i \mat...
...m_{i=1}^{N}\mathbf{a}^2_i \sum_{i=1}^{N}\mathbf{b}^2_i}}\;,
\end{displaymath}](img11.png) |
(1) |
where N is the length of a signal. The global correlation in equation 1
supplies only one number for the whole signal. For measuring the
similarity between two signals locally, one can define the sliding-window
correlation coefficient
![\begin{displaymath}
\gamma_w(t) = \frac{\displaystyle\sum_{i=t-w/2}^{t+w/2} \ma...
...^2 \displaystyle\sum_{i=t-w/2}^{t+w/2} \mathbf{b}_i^2 }}\;,
\end{displaymath}](img12.png) |
(2) |
where
is window length.
Fomel (2007a) proposes the local correlation attribute that identifies
local changes in signal similarity in a more elegant way. In a linear
algebra notation, the correlation coefficient in equation 1 can be
represented as a product of two least-squares inverses
and
:
![\begin{displaymath}
\gamma^2 = \gamma_1 \gamma_2\;,
\end{displaymath}](img16.png) |
(3) |
![\begin{displaymath}
\gamma_1 = \textrm{arg} \min_{\gamma_1}\parallel \mathbf{b}...
... = (\mathbf{a}^T\mathbf{a})^{-1}(\mathbf{a}^T\mathbf{b})\;,
\end{displaymath}](img17.png) |
(4) |
![\begin{displaymath}
\gamma_2 = \textrm{arg} \min_{\gamma_2}\parallel \mathbf{...
...^2 = (\mathbf{b}^T\mathbf{b})^{-1}(\mathbf{b}^T\mathbf{a})\;,
\end{displaymath}](img18.png) |
(5) |
where
and
are vector notions for
and
. Let
and
be two diagonal
operators composed of the elements of a and b. Localizing
equations 4 and 5 amounts to adding regularization to
inversion. Using shaping regularization (Fomel, 2007b), scalars
and
turn into vectors
and
, defined as
![\begin{displaymath}
\mathbf{c}_1 = [\lambda^2 \mathbf{I} + \mathbf{S}(\mathbf{A...
...mbda^2 \mathbf{I})]^{-1}\mathbf{S}\mathbf{A}^T\mathbf{b}\;,
\end{displaymath}](img25.png) |
(6) |
![\begin{displaymath}
\mathbf{c}_2 = [\lambda^2 \mathbf{I} + \mathbf{S}(\mathbf{B...
...mbda^2 \mathbf{I})]^{-1}\mathbf{S}\mathbf{B}^T\mathbf{a}\;,
\end{displaymath}](img26.png) |
(7) |
where
scaling controls relative scaling of operators
and
and
where
is a shaping operator such as Gaussian smoothing with an
adjustable radius. The component-wise product of vectors
and
defines the local correlation measure. Local correlation is a measure
of the similarity between two signals.
An iterative, conjugate-gradient inversion for computing the inverse
operators can be applied in equations 6 and 7.
Interestingly, the output of the first iteration is equivalent to the
algorithm of fast local
cross-correlation proposed by Hale (2006).
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2013-03-02