Covariance of the Smoothed Derivative Field

Consider taking the spatial derivative of the smoothed field. That is, constructing three derivative fields, $F_{Dx}, F_{Dy}, F_{Dz}$, where:

$$F_{Dx}(\ensuremath{\mathbf{x}}) = \frac{\partial}{\partial x... ...}})}{\partial x} \right) \otimes F_W(\ensuremath{\mathbf{x}})$$ (15)

and similarly for $F_{Dy}$ and $F_{Dz}$.

Now the partial derivative of the smoothing filter is given by:

$$\frac{\partial G(\ensuremath{\mathbf{x}})}{\partial x} = \fr... ...{\sigma_x}^2} G_{\sigma_x}(x) G_{\sigma_y}(y) G_{\sigma_z}(z).$$ (16)

Therefore, the covariance can be computed as in the last section:

$$E \{ F_{Dx}(\ensuremath{\mathbf{x_1}}) F_{Dx}(\ensuremath{\mathbf{x_2}}) \} = E \left\{ \int \int \frac{\partial G(\ensuremath{\mathbf{x_1}}- \... ...athbf{p_2}}) \, d\ensuremath{\mathbf{p_1}}\, d\ensuremath{\mathbf{p_2}}\right\}$$ (17)

$$= \int \int \frac{\partial G(\ensuremath{\mathbf{x_1}}- \ensuremath... ...emath{\mathbf{p_2}}) \, d\ensuremath{\mathbf{p_1}}\, d\ensuremath{\mathbf{p_2}}$$ (18)

$$= \int \frac{\partial G(\ensuremath{\mathbf{x_1}}- \ensuremath{\mat... ...hbf{x_2}}- \ensuremath{\mathbf{x_0}})}{\partial x}\, d\ensuremath{\mathbf{x_0}}$$ (19)

$$= \left( \int \frac{(x_1 - x_0)(x_2 - x_0)}{{\sigma_x}^4} G_{\sigma_x}(x_1 - x_0) G_{\sigma_x}(x_2 - x_0) \, dx_0 \right) \times \cdots$$

$$\quad \left( \int G_{\sigma_y}(y_1 - y_0) G_{\sigma_y}(y_2 - y_0)... ...ht) \left( \int G_{\sigma_z}(z_1 - z_0) G_{\sigma_z}(z_2 - z_0) \, dz_0 \right)$$ (20)

The latter two integrals are given by equation 13. The former integral is:

$$I_2 = \int \frac{(x_1 - x_0)(x_2 - x_0)}{\sigma^4} G_{\sigma}(x_1 - x_0) G_{\sigma}(x_2 - x_0) \, dx_0$$ (21)

$$= \frac{1}{2 \pi \sigma^6} \exp\left( \frac{- (x_1 - x_2)^2}{4 \sig... ... \frac{-1}{\sigma^2} \left[ x_0 - \frac{x_1 + x_2}{2} \right]^2 \right) \, dx_0$$ (22)

Rewriting the last part using $w = x_0 - \frac{x_1 + x_2}{2}$ gives:

$$I_3 = \int (x_1 - x_0) (x_2 - x_0) \exp\left( \frac{-1}{\sigma^2} \left[ x_0 - \frac{x_1 + x_2}{2} \right]^2 \right) \, dx_0$$ (23)

$$= \int \left(w - \frac{x_1 - x_2}{2} \right) \left(w + \frac{x_1 - x_2}{2} \right) \exp\left( \frac{- w^2}{\sigma^2} \right) \, dw$$ (24)

$$= \int w^2 \exp \left( \frac{- w^2}{\sigma^2} \right) \, dw -\frac{(x_1 - x_2)^2}{4} \int \exp \left( \frac{- w^2}{\sigma^2} \right) \, dw$$ (25)

$$= \frac{\sigma^2}{2} \sqrt{\pi \sigma^2} -\frac{(x_1 - x_2)^2}{4} \sqrt{\pi \sigma^2}$$ (26)

$$= (\pi)^\frac{1}{2} \sigma \left( \frac{\sigma^2}{2} - \frac{(x_1 - x_2)^2}{4} \right)$$ (27)

Therefore, combining equations 20, 22 and 27 gives:

$$E \{ F_{Dx}(\ensuremath{\mathbf{x_1}}) F_{Dx}(\ensuremath{\mathbf{x_2}}) \} = E \{ F_{S}(\ensuremath{\mathbf{x_1}}) F_{S}(\ensuremath{\mathbf{x... ...}{{\sigma_x}^4} \left( \frac{{\sigma_x}^2}{2} - \frac{(x_1 - x_2)^2}{4} \right)$$ (28)