Cross correlation

Cross Correlation
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Cross correlation is a statistical measure used to assess the similarity between two waveforms as a function of a time-lag applied to one of them. It is a fundamental concept in signal processing, time series analysis, and various fields of medicine where it is used to analyze physiological signals.

Definition

Cross correlation is defined mathematically as the convolution of one signal with the time-reversed version of another signal. For two continuous signals, \( f(t) \) and \( g(t) \), the cross correlation \( R_{fg}(\tau) \) is given by:

\[ R_{fg}(\tau) = \int_{-\infty}^{\infty} f(t) g(t + \tau) \, dt \]

where \( \tau \) is the time-lag variable. For discrete signals, the cross correlation is given by:

\[ R_{fg}[n] = \sum_{m=-\infty}^{\infty} f[m] g[m+n] \]

Applications in Medicine

Cross correlation is widely used in medical imaging, neurophysiology, and cardiology.

Medical Imaging

In medical imaging, cross correlation is used to enhance image quality and to align images from different modalities. For example, in magnetic resonance imaging (MRI) and computed tomography (CT), cross correlation can be used to register images taken at different times or from different angles.

Neurophysiology

In neurophysiology, cross correlation is used to analyze electroencephalogram (EEG) and electromyogram (EMG) signals. It helps in identifying the synchronization between different brain regions or between muscle groups.

Cardiology

In cardiology, cross correlation is used to analyze electrocardiogram (ECG) signals. It can help in detecting abnormalities in heart rhythms by comparing the ECG signals from different leads or from different time periods.

Mathematical Properties

Cross correlation has several important mathematical properties:

• Symmetry: \( R_{fg}(\tau) = R_{gf}(-\tau) \)
• Linearity: If \( a \) and \( b \) are constants, then \( R_{af+bg,h}(\tau) = aR_{f,h}(\tau) + bR_{g,h}(\tau) \)
• Shift Invariance: Shifting a signal in time does not affect the shape of the cross correlation function.

Computation

Cross correlation can be computed efficiently using the Fast Fourier Transform (FFT). The cross correlation of two signals can be obtained by taking the inverse FFT of the product of the FFT of one signal and the complex conjugate of the FFT of the other signal.

Limitations

While cross correlation is a powerful tool, it has limitations. It assumes that the signals are stationary, which may not be the case in many physiological processes. Additionally, it can be sensitive to noise, which is common in medical signals.

See Also

• Autocorrelation
• Convolution
• Fourier Transform
• Signal Processing

External Links

• [Cross Correlation in Signal Processing]
• [Applications of Cross Correlation in Medicine]

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