Image reconstruction

The method of moment matching (Section 1.2.3), as described for the reconstruction of non-orthogonal moments is also applicable to reconstruction of an image by orthogonal moments. However, the orthogonality condition enables a faster, more direct approach. Teague [18] showed that, for orthogonal Legendre moments, if all moments of a Cartesian function $f(x,y)$ up to a given order $N_{max}$ are known, then it is possible to reconstruct a discrete function $\hat{f}(x,y)$, whose moments match those of the original function $f(x,y)$, up to the order $N_{max}$. This relationship is due to the orthogonality condition of the Legendre moments, while the accuracy of the reconstructed function improves as $N_{max}$ approaches infinity. Khotanzad [7] expressed this relationship in terms of Zernike moments, shown here in radial coordinates:

$$\hat{f}(r,\theta) = \sum_{m=0}^{N_{max}} \sum_{n} A_{mn}V_{mn}(r,\theta)$$ (66)

and $n$ is constrained by Equation 1.50. Expanding this using real-valued functions produces:

$$\hat{f}(r,\theta) = \sum_{m=0}^{N_{max}} \sum_{n>0} (C_{mn}\... ...ta + S_{mn}\sin n\theta)~R_{mn}(r) + \frac{C_{m0}}{2}R_{m0}(r)$$ (67)

composed of their real ($Re[.]$) and imaginary ($Im[.]$) parts:

$$C_{mn} = 2 Re\left[A_{mn}\right] = \frac{2m+2}{\pi} \sum_x \sum_y f(r,\theta) R_{mn}(r) \cos n\theta$$ (68)

$$S_{mn} = -2 Im\left[A_{mn}\right] = \frac{-2m-2}{\pi} \sum_x \sum_y f(r,\theta) R_{mn}(r) \sin n\theta$$ (69)

bounded by $x^2+y^2\leq1$. Here, each Zernike moment simply adds its own contribution to the function $\hat{f}(r,\theta)$, unlike the Cartesian reconstruction case discussed in Section 1.2.3. Figure 1.8b shows the result of order $2$ through $12$ reconstruction on a $64\times 64$ image, while Figure 1.8a is the original image. Orders $0$ and $1$ are discarded due to the scale and translation mapping used, Equations 1.58 and 1.59. This makes $\mid A_{11} \mid$ zero, while $\mid A_{00} \mid$ (the shape's area) is set to a known value, $\beta$. Due to the nature of the function $\hat{f}(r,\theta)$, the Gibbs phenomena [16] will affect the final result (as already mentioned for the moment matching case in Section 1.2.3). However, the effects are less apparent in Figure 1.8b due to faster convergence of the final function $\hat{f}(r,\theta)$.

Figure 1.8: Zernike moment reconstruction example.

$\parbox{7cm}{ \center{ \fbox{\scalebox{1.0}{\includegraphics{images/theory/squiggle_in_circle.ps}}} }}$$\parbox{7cm}{ \center{ \fbox{\scalebox{1.0}{\includegraphics{images/theory/squiggle_out_in_circle.ps}}} }}$
$\parbox{7cm}{\center{(a) The original image.}}$$\parbox{7cm}{\center{(b)} Order $2-12$\ reconstruction}$