4.4 Gradient

From a volume of scalar values representing density at specific grid locations, a surface needs to be visualized. Taking the gradient of these densities gives information on surfaces. Density changes at boundaries and the changes appear in the gradient. Therefore, calculating the gradient for a sample of densities reveals the underlying surfaces. The presence of a surface is quantified by the magnitude of the gradient: the larger the gradient the more likely the presence of a surface. So a tool for deriving gradients is necessary. Moreover, the gradient direction computed at each point in the volume can be used to approximate the normal direction of an imaginary surface passing through the point. Once an approximate normal is known, volume shading can be derived. The gradient is the least biased way of deducing surface information without explicitly identifying surfaces. Furthermore, it does not introduce spurious effects that result from surface detection algorithms.

The gradient has some fundamental properties that have a major influence on extracting volume information. The gradient is orthogonal to iso-surface¹ and is large at boundaries. For a continuous density function, $\rho(\vec x)$, the gradient $\nabla \rho$ is defined as a directional derivative :

$\nabla \rho(\vec x) = \frac{\delta \rho}{\delta x}\hat x + \frac{\delta \rho}{\delta y}\hat y + \frac{\delta \rho}{\delta z}\hat z$

The algorithm to approximate gradient of sampled data will be discussed in section 4.4.

... iso-surface¹

Conceptual volume created by the locus of points in three space $\vec x$ that satisfy an equation $v = f(\vec x)$ for a continuous single-valued function $f$ and a specific value $v$ (called an iso-value).