Geometry

Contains measurements expressing shape attributes.

Note: These measurements ignore the intensity input image.

Intercepts and diametral variation

Several geometric measurements, such as the Crofton perimeter, and the 3D area, are approximations based on the diametral variation concept.

For example, consider a set of lines $d(X,\alpha)$, parallel to a direction $\alpha$ and regularly spaced by $dx$.
The diametral variation is defined as: $$D_2(X,\alpha)=\int{N_\alpha(X)dx}$$ Where $N_\alpha(X)$ is the number of boundary points that intersect the set of lines $d(x,\alpha)$. This is equivalent to the sum of the lengths of the projections of the object in the $\alpha$ direction. Note that $D_2(X,\alpha)$ is symmetric:
The discrete case in 2D translates to the search for specific configurations, as defined in the following table. $$ \begin{array}{|c|c|c|c|c|} \hline Orientation & 0^\circ & 45^\circ & 90^\circ & 135^\circ\\ \hline Kernel & \begin{array}{ccc} \mbox{x} & \mbox{x} & \mbox{x}\\ \mbox{x} & 0 & 1\\ \mbox{x} & \mbox{x} & \mbox{x} \end{array} & \begin{array}{ccc} \mbox{x} & \mbox{x} & 1\\ \mbox{x} & 0 & \mbox{x}\\ \mbox{x} & \mbox{x} & \mbox{x} \end{array} & \begin{array}{ccc} \mbox{x} & 1 & \mbox{x}\\ \mbox{x} & 0 & \mbox{x}\\ \mbox{x} & \mbox{x} & \mbox{x} \end{array} & \begin{array}{ccc} 1 & \mbox{x} & \mbox{x}\\ \mbox{x} & 0 & \mbox{x}\\ \mbox{x} & \mbox{x} & \mbox{x} \end{array}\\ \hline \end{array}$$

Crofton perimeter

The Crofton formula computes the perimeter using the intercept count. Cauchy's formula shows that: $\alpha$, and regularly spaced by $dx$. $$L(X)=\int_{0}^{\pi}D_2(X,\alpha)d\alpha$$ The perimeter is then approximated when the above integral is replaced by a discrete summation. In the square grid case, the summation is computed for the four fundamental directions: $$ L(X)=\frac{\pi}{4}\times [a \times(N_0+N_{90})+\frac{a}{\sqrt2}\times(N_{45}+N_{135})] $$ Where:

• $N_\alpha$ represents the intercept count along the direction $\alpha$,
• $a$ represents the distance between two points on the grid and the distance between two lines in the $0^\circ$ or $90^\circ$ directions,
• $\frac{a}{\sqrt2}$ is the distance between two lines in the $45^\circ$ and $135^\circ$ directions.

Finally, this formula can be corrected for non-square pixels. In such a situation, the diagonals no longer fit the $45^\circ$ and $135^\circ$ directions. Let $M_i$ be the four neighbouring pixels on the grid, $a$ be the distance between horizontal lines, $b$ be the distance between vertical lines, and $c$ the distance between diagonal lines. The formula becomes: $$ L(X)=\alpha \times a \times N_0 + (\frac{\pi}{2}-\alpha)\times b \times N_{90} + \frac{\pi}{4}\times c \times (N_{\alpha}+N_{\pi-\alpha}) $$

3D area of object boundary

Computing the area of a 3D object boundary is not a trivial issue. An intuitive idea would be to sum the area of voxel faces connected to the background. If it gives a relevant response for objects made of few voxels or perfect cubes, it quickly overestimates the area when the object size grows, especially for uneven surfaces.

A better estimation consists of extending the Crofton perimeter concept to the 3D area by computing diametral variations in 13 directions and summing these values with weights appropriate to each direction. Unlike the previous method, this estimation tends to underestimate the surface area. It is especially imprecise for objects made of few voxels.