Directional derivatives along the gradient

Although it was suggested in Section 1.2 that isosurfaces are not always sufficient for visualizing objects in real world volume data, the method presented in this thesis still indirectly employs them as an indicator of object shape.

Figure 3.2: Isosurfaces tend to conform to object shape. For a cylindrical object (a) with a soft boundary, a small patch of an isosurface (b) is shown within the boundary, together with a line segment (c) which follows the gradient direction, perpendicular to the isosurface patch and to the object boundary.

That is, based on the mathematical property that the gradient vector at some position always points perpendicular to an isosurface through that position, we use the gradient vector as a way of finding the direction which passes perpendicularly through the object boundary. Figure 3.2 demonstrates, for a simple cylindrical object with a soft boundary, how a line segment along the gradient direction lies perpendicular to a patch of the boundary isosurface, and how the isosurface patch conforms to the shape of the boundary. However, in real datasets, the isosurfaces don't always perfectly conform to the local shape of the underlying object. But on average, over the whole volume, the gradient vector does tend to point perpendicular to the object boundary. Therefore, our approach will be to rely on the statistical properties of the histogram to provide the overall picture of the boundary characteristics throughout the entire volume.

The directional derivative of a scalar field $f$ along a vector $\mathbf{v}$, denoted $\mathbf{D}_\mathbf{v}f$, is the derivative of $f$ as one moves along a straight path in the $\mathbf{v}$ direction. This thesis studies $f$ and its derivatives along a path which is normal to the object boundary -- moving along the gradient direction -- in order to create an opacity function. Because the direction along which we're computing the directional derivative is always that of the gradient, we employ a mild abuse of notation, using $f'$ and $f''$ to signify the first and second directional derivative along the gradient direction, even though these would be more properly denoted by $\mathbf{D}_{\widehat{\nabla f}}f$ and $\mathbf{D}^{2}_{\widehat{\nabla f}}f$, where $\widehat{\nabla f}$ is the gradient direction. We treat $f$ as if it were a function of just one variable, keeping in mind that the axis along which we analyze $f$ always follows $\widehat{\nabla f}$, which constantly changes orientation depending on position. Figure 3.3 shows how the gradient direction changes with position to stay normal to the isosurfaces of a simple object.

Figure 3.3: The gradient direction $\nabla f$ (c) changes with position in the data (a), but it maintains its perpendicular orientation with respect to the isosurfaces (b).