According to Dr. Math the Z-transform can create closed-form solutions for 1D series defined by difference equations (e.g. the Fibonacci series).
My 3D surface $z=LC(x,y)$ is defined by difference equations:
\begin{eqnarray} t_0 &=& N^2 \\ t_1 &=&(N-y)^2 \\ t_i &=&2*t_{i-1}-t_{i-2}+2y^2 \end{eqnarray}
I am interested in preserving the absolute minima of this surface iff it lies on an integer boundary. My first question is: how do I modify the closed-form solution for a 2D signal?
My second question is: should I expect the closed-form solution to be different from the original formula? { The original formula is $t=(N-xy)^2$ }
POSTSCRIPT
The difference equations come from the relation:
\begin{eqnarray} (1) & (N - (i+2)*y)^2 - (N-iy)^2 &=& -4Ny - 4y^2i - 4y^2\\ (2) & (N - (i+1)*y)^2 - (N-iy)^2 &=& -2Ny - 2y^2i - y^2\\ (1)-(2) & t(i+2)-t(i) &=& \hspace{1em}2 ( t(i+1) - t(i) ) - 2y^2 \end{eqnarray}
The function has an absolute minima at zero in t. When N=UV, it is the same absolute minima that occurs for the difference equation:
\begin{eqnarray} (1) & (N - (x+2U)*y)^2 - (N-xy)^2 & = & -4UNy - 4xUy^2 - 4(Uy)^2\\ (2) & (N - (x+U)*y)^2 - (N-xy)^2 & = & -2UNy - 2xUy^2 - (Uy)^2\\ (1),(2) & t_{i+2}-t_i & = & \hspace{1em} 2 ( t_{i+1} - t_i ) - 2U^2y^2 \end{eqnarray}
The initial conditions and the value of U are critical to the problem definition. My feeling is that even though the inverse z-transform is not unique, the set of inverse transforms will not converge at x=U, y=V, t=0 without prior knowledge of U.
z-transform
So, if I understand the process, the generalized z-transform for t along the x axis is:
\begin{eqnarray} Z[ - 2(Uy)^2 ] & = & Z[ t_{i+2} - 2 t_{i+1} + t_{i} ] \\ \frac{-2(Uy)^2z}{z-1}&=& z^2(T_z - t(0) -t(U)/z) - 2z(T_z - t(0)) + T_z \\ \frac{-2(Uy)^2z}{z-1}&=& T_z(z^2 - 2z + 1) -z^2t(0)-z(t(U)-2t(0)) \\ -2(Uy)^2z&=& T_z(z- 1)^3 -z^3t(0)-z^2(t(U)-3t(0))-z(t(U)-2t(0)) \\ -2(Uy)^2z&=& T_z(z- 1)^3 -z^3N^2-z^2((N-Uy)^2-3N^2)-z((N-Uy)^2-2N^2) \\ T_z(z- 1)^3 & = & z^3N^2+z^2((N-Uy)^2-3N^2)-z((N-Uy)^2-2N^2-2(Uy)^2) \\ T_z(z- 1)^3 & = &z^3 N^2 - z^2(2N^2+2NUy-(Uy)^2) + z((Uy)^2+2NUy+N^2 )\\[2em] \frac{T_z}{z} & = &\frac{z^2 N^2 - z(2N^2+2NUy-(Uy)^2) + ((Uy)^2+2NUy+N^2 )}{(z- 1)^3}\\ \frac{T_z}{z} & = N^2&\frac{z^2 - z(2+2y/V-(y/V)^2) + ((y/V)^2+2y/V+1 )}{(z- 1)^3}\\ \frac{T_z}{z} & = N^2&\frac{z^2 -2z +1 - z(2y/V-(y/V)^2) + ((y/V)^2 + 2y/V )}{(z- 1)^3}\\ \frac{T_z}{z} & = N^2&\frac{z^2 -2z +1 - (z-1)(2y/V) + (z-1)(y/V)^2 + 2(y/V)^2}{(z- 1)^3}\\ \frac{T_z}{z} & = N^2&\left(\frac{1}{z- 1}-\frac{2y/V-(y/V)^2}{(z- 1)^2}+\frac{2(y/V)^2}{(z- 1)^3}\right)\\ \frac{T_z}{z} & = N^2&\left(\frac{1}{z- 1}-\frac{2y/V}{(z- 1)^2}+\frac{(y/V)^2*(z-1)+2(y/V)^2}{(z- 1)^3}\right)\\ \frac{T_z}{z} & = N^2&\left(\frac{1}{z- 1}-\frac{2y/V}{(z- 1)^2}+\frac{(y/V)^2z+(y/V)^2}{(z- 1)^3}\right)\\ \frac{T_z}{z} & = N^2&\left(\frac{1}{z- 1}-\frac{2y/V}{(z- 1)^2}+\frac{(y/V)^2(z+1)}{(z- 1)^3}\right) \end{eqnarray}
normalization along y
The final form of the z-transform is dependent on the ratio $\Upsilon$ of $y$ to $V$, which we know will vary from $0$ to $1$. Along the cross-section when $y=V, \Upsilon=1$, the trivial case occurs with a sequence $N^2, 0, N^2, 4N^2, ...$.
\begin{eqnarray} \frac{T_z}{z} & = N^2&\left(\frac{1}{z- 1}-\frac{2y/V-(y/V)^2}{(z- 1)^2}+\frac{2(y/V)^2}{(z- 1)^3}\right)\\ \frac{T_z}{z} & = N^2&\left(\frac{1}{z- 1}-\frac{2\Upsilon-\Upsilon^2}{(z- 1)^2}+\frac{2\Upsilon^2}{(z- 1)^3}\right)\\ \left.\frac{T_z}{z}\right\rvert_{\Upsilon=1} & = N^2&\left(\frac{1}{z- 1}-\frac{1}{(z- 1)^2}+\frac{2}{(z- 1)^3}\right) \end{eqnarray}
inverse z-transform
A unique time-domain function can be obtained using lookup tables and numerical methods such as those found at Wolfram:
\begin{eqnarray} t_i & = N^2(i\Upsilon - 1)^2 \end{eqnarray}
The next step is to use Direct Inversion to calculate the inverse transform. This method involves calculating a closed integral as defined by:
$$t=\frac{1}{2{\pi}j}\oint_H Tz^{x-1} \,dz$$
For $t$ to be unique, $H$ must encircle all poles and zeros (aka ROC), however here it is desired to investigate what happens when $H$ is smaller than the ROC. Specifically, we desire to find a contour $H$ so that the inverse transform is equivalent to spline interpolation of our point cloud.
