Generalizing one dimensional masks

The two masks discussed thus far ( $[-0.5~~0~~0.5]$ for first derivative and $[1~-2~~1]$ for second derivative) are only useful for measuring the derivatives of one dimensional data. These are now generalized to create masks for measuring first and second partial derivatives of sampled data in three dimensions.

Any reconstruction kernel $h(x)$ in one dimension can be trivially extended to a separable kernel $h_3(x,y,z)$ in three dimensions:

$$h_3(x,y,z) = h(x)h(y)h(z)$$ (44)

The function representing the sampled data in a volume is a three dimensional lattice of delta functions, scaled by the data values $v_{i,j,k}$, where $i, j, k$ all range over the integers:

$$g_3(x,y,z) = \sum_{i,j,k} v_{i,j,k} \delta (x-i, y-j, z-k)$$

The reconstructed function $g_3 \star h_3$ is a lattice of kernels scaled by the data values:

$$(g_3 \star h_3)(x,y,z) = \sum_{i,j,k} v_{i,j,k} h_3(x-i, y-j, z-k)$$

$$= \sum_{i,j,k} v_{i,j,k} h(x-i) h(y-j) h(z-k)$$

The partial derivatives of the reconstructed function are simple to express:

$$\frac{\partial (g_3 \star h_3)}{\partial x}(x_0,y_0,z_0) = \sum_{i,j,k} v_{i,j,k} h'(x_0-i) h(y_0-j) h(z_0-k)$$

$$\frac{\partial (g_3 \star h_3)}{\partial y}(x_0,y_0,z_0) = \sum_{i,j,k} v_{i,j,k} h(x_0-i) h'(y_0-j) h(z_0-k)$$

$$\frac{\partial (g_3 \star h_3)}{\partial z}(x_0,y_0,z_0) = \sum_{i,j,k} v_{i,j,k} h(x_0-i) h(y_0-j) h'(z_0-k)$$

As in the one dimensional case, these expressions can be rewritten when the location at which the derivative is taken is a point $(i_0, j_0, k_0)$ on the integer lattice:

$$\frac{\partial (g_3 \star h_3)}{\partial x}(i_0,j_0,k_0) = \sum_{i,j,k} v_{i_0-i,j_0-j,k_0-k} h'(i) h(j) h(k)$$ (45)

$$\frac{\partial (g_3 \star h_3)}{\partial y}(i_0,j_0,k_0) = \sum_{i,j,k} v_{i_0-i,j_0-j,k_0-k} h(i) h'(j) h(k)$$

$$\frac{\partial (g_3 \star h_3)}{\partial z}(i_0,j_0,k_0) = \sum_{i,j,k} v_{i_0-i,j_0-j,k_0-k} h(i) h(j) h'(k)$$

In this form, the expressions for the partial derivatives of the reconstructed function are a close analog to Equation B.4, the formulation of one dimensional derivative masks.

Whereas the one dimensional mask was a vector of evaluations of $h'(x)$, we now have a three dimensional lattice of evaluations of a partial derivative of $h_3$. These are the masks for derivative measurement in three dimensions. For example, the mask for calculating the first partial derivative along the $x$ axis, $\frac{\partial }{\partial x}$, is the lattice of values $h'(i) h(j) h(k)$, where $i, j, k$ all range over the integers. The mask for the second mixed partial derivative along the $z$ and $y$ axes, $\frac{\partial ^2}{\partial y \partial z}$, is the lattice of values $h(i) h'(j) h'(k)$. As before, we assume the kernel $h$ has finite support, thus there are only a finite number of non-zero values in the masks. Figure B.2 depicts the masks for $\frac{\partial }{\partial x}$ and $\frac{\partial ^2}{\partial y \partial z}$.

Figure B.2: Two examples of masks for volume derivative measurement

Even though these masks have an obvious three dimensional structure, the mathematics of their application is still that of a dot product: the mask is centered at the sample point of interest, and the products between corresponding data and mask values are summed to produce the final measurement. Having decided on the quintic kernel $q(x)$ for all mask generation, Equations B.11 can be used to generate $3 \times 3 \times 3$ masks for all the partial derivatives needed for any of the directional derivative measures.