Theoretical Basis

As the approximation of the derivative (equation 40) is not precise, it introduces a bias into the SPM estimation technique. This source of bias can be eliminated by using a finite difference calculation instead of the derivative-based one.

That is, the difference field is:

$$F_{\Delta}(\ensuremath{\mathbf{x}}) = F_S(\ensuremath{\mathbf{x}}+ \ensuremath{\mathbf{d}}) - F_S(\ensuremath{\mathbf{x}}).$$ (42)

The variance of this difference field is:

$$E \{ F_{\Delta}(\ensuremath{\mathbf{x}}) F_{\Delta}(\ensuremath{\mathbf{x}}) \} = E \{ F_S(\ensuremath{\mathbf{x}}+ \ensuremath{\mathbf{d}}) F_S(\e... ...nsuremath{\mathbf{x}}+ \ensuremath{\mathbf{d}}) F_S(\ensuremath{\mathbf{x}}) \}$$ (43)

$$= 2 E\{ F_S(\ensuremath{\mathbf{x}}) F_S(\ensuremath{\mathbf{x}}) \... ...nsuremath{\mathbf{x}}+ \ensuremath{\mathbf{d}}) F_S(\ensuremath{\mathbf{x}}) \}$$ (44)

Setting $\ensuremath{\mathbf{x_2}}= \ensuremath{\mathbf{x_1}}+ \ensuremath{\mathbf{d}}_x$ with $\ensuremath{\mathbf{d}}_x = (d,0,0)$, the expectations can be calculated using equation 14:

$$E \{ F_S(\ensuremath{\mathbf{x}}) F_S(\ensuremath{\mathbf{x}}+ \ensuremath{\mathbf{d}}_x) \} = \frac{1}{(4 \pi)^\frac{3}{2} \sigma_x \sigma_y \sigma_z} \exp\left( \frac{- d^{\, 2}}{4 {\sigma_x}^2} \right)$$ (45)

$$E \{ F_S(\ensuremath{\mathbf{x}}) F_S(\ensuremath{\mathbf{x}}) \} = \frac{1}{(4 \pi)^\frac{3}{2} \sigma_x \sigma_y \sigma_z}.$$ (46)

giving

$$E \{ F_{\Delta}(\ensuremath{\mathbf{x}}) F_{\Delta}(\ensurem... ...- \exp\left( \frac{- d^{\, 2}}{4 {\sigma_x}^2} \right) \right)$$ (47)

This enables the individual smoothing parameters, $\sigma_x, \sigma_y, \sigma_z$ to be found. That is:

$${\sigma_x}^2 = \frac{- d^{\, 2}}{4 \ln\left( 1 - \frac{E \{ ... ...emath{\mathbf{x}}) F_S(\ensuremath{\mathbf{x}}) \} } \right)}.$$ (48)