Phase correlation

Phase correlation is an approach to estimate the relative translative offset between two similar images (digital image correlation) or other data sets. It is commonly used in image registration and relies on a frequency-domain representation of the data, usually calculated by fast Fourier transforms. The term is applied particularly to a subset of cross-correlation techniques that isolate the phase information from the Fourier-space representation of the cross-correlogram.

Phase correlation is an approach to estimate the relative translative offset between two similar images (digital image correlation) or other data sets. It is commonly used in image registration and relies on a frequency-domain representation of the data, usually calculated by fast Fourier transforms. The term is applied particularly to a subset of cross-correlation techniques that isolate the phase information from the Fourier-space representation of the cross-correlogram. \ \mathbf{G}_a = \mathcal{F}\{g_a\}, \; \mathbf{G}_b = \mathcal{F}\{g_b\} Calculate the cross-power spectrum by taking the complex conjugate of the second result, multiplying the Fourier transforms together elementwise, and normalizing this product elementwise. \ R = \frac{ \mathbf{G}_a \circ \mathbf{G}_b^*} Where \circ is the Hadamard product (entry-wise product) and the absolute values are taken entry-wise as well. Written out entry-wise for element index (j,k) : \ R_{jk} = \frac{ G_{a,jk} \cdot G_{b,jk}^*} Obtain the normalized cross-correlation by applying the inverse Fourier transform. \ r = \mathcal{F}^{-1}\{R\} Determine the location of the peak in \ r . \ (\Delta x, \Delta y) = \arg \max_{(x, y)}\{r\} \ \Delta x = \frac{ r_{(1,0)} }{r_{(1,0)} \plusmn r_{(0,0)}} The Foroosh et al. method is quite fast compared to most methods, though it is not always the most accurate. Some methods shift the peak in Fourier space and apply non-linear optimization to maximize the correlogram peak, but these tend to be very slow since they must apply an inverse Fourier transform or its equivalent in the objective function. It is also possible to infer the peak location from phase characteristics in Fourier space without the inverse transformation, as noted by Stone. These methods usually use a linear least squares (LLS) fit of the phase angles to a planar model. The long latency of the phase angle computation in these methods is a disadvantage, but the speed can sometimes be comparable to the Foroosh et al. method depending on the image size. They often compare favorably in speed to the multiple iterations of extremely slow objective functions in iterative non-linear methods. Since all subpixel shift computation methods are fundamentally interpolative, the performance of a particular method depends on how well the underlying data conform to the assumptions in the interpolator. This fact also may limit the usefulness of high numerical accuracy in an algorithm, since the uncertainty due to interpolation method choice may be larger than any numerical or approximation error in the particular method. Subpixel methods are also particularly sensitive to noise in the images, and the utility of a particular algorithm is distinguished not only by its speed and accuracy but its resilience to the particular types of noise in the application. \ g_b(x,y) \ \stackrel{\mathrm{def}}{=}\ g_a((x - \Delta x) \bmod M, (y - \Delta y) \bmod N) (where the images are \ M \times N in size). Then, the discrete Fourier transforms of the images will be shifted relatively in phase: \mathbf{G}_b(u,v) = \mathbf{G}_a(u,v) e^{-2 \pi i (\frac{u \Delta x}{M} + \frac{v \Delta y}{N}) } One can then calculate the normalized cross-power spectrum to factor out the phase difference: \begin{align} R(u,v) &= \frac{ \mathbf{G}_a \mathbf{G}_b^*} \\ &= \frac{ \mathbf{G}_a \mathbf{G}_a^* e^{2 \pi i (\frac{u \Delta x}{M} + \frac{v \Delta y}{N}) }} \\ &= \frac{ \mathbf{G}_a \mathbf{G}_a^* e^{2 \pi i (\frac{u \Delta x}{M} + \frac{v \Delta y}{N}) }} \\ &= e^{2 \pi i (\frac{u \Delta x}{M} + \frac{v \Delta y}{N}) } \end{align} since the magnitude of an imaginary exponential always is one, and the phase of \ \mathbf{G}_a \mathbf{G}_a^* always is zero. The inverse Fourier transform of a complex exponential is a Dirac delta function, i.e. a single peak: \ r(x,y) = \delta(x + \Delta x, y + \Delta y) This result could have been obtained by calculating the cross correlation directly. The advantage of this method is that the discrete Fourier transform and its inverse can be performed using the fast Fourier transform, which is much faster than correlation for large images.

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