## 1 Introduction
### Related work  
Given the clinical importance of polyp detection, the subject of CTC-CAD has received considerable interest in the literature. Since there is no universally agreed upon set of a data and evaluation methodology for comparing CAD systems, clear and accurate comparisons between different CADsystems are not possible. However, a summary of many recent CTC-CAD systems, presented in Table 1, does provide a picture of the state-of-the-art in CTC-CAD. In the table, where possible, we list the data sources (number of centers from which the data came), training and testing data sets (in terms of patients), per-polyp sensitivity, false positives (FPs)/volume, and runtime when executed on a new volume (note different computers and CPUs are used so exact comparisons are not available). More statistical confidence would be attributed to CAD systems tested on larger sets of data. For the reasons mentioned earlier, in the table we focus on the detection of lesions 6 mm and larger. Note that each patient is scanned twice; once in a prone position and once in a supine position. Throughout the paper, we report FPs/volume, as is the convention in CTC. Progress on the problem of CTC-CAD started with the pioneering work of [PERSON] _et al_. [25], who identified polyps as an abnormal thickening of the colonic wall. Subsequent work by others focuses primarily on use of curvature as a mechanism for detecting polyps. The work of [PERSON] _et al_. [13] utilizes mean curvature computed at voxels on the mucosal surface. [PERSON] and [PERSON] [26] apply shape index and curvedness [27] to find initial candidates, which are then clustered and classified using quadratic discriminant analysis, whereas [PERSON] _et al_. [18] rely on eigenvalues of the Hessian matrix. Instead of computing curvature information on the voxel grid, [PERSON] _et al_. [12] apply a geometry processing approach directly on a mesh-based representation of the colon. [PERSON] _et al_. [24] use the second principle curvature in a differential equation that is solved explicitly on a mesh or implicitly on the image grid to identify polyp candidates. [PERSON] _et al_. [15] extend [14] by analyzing the symmetry of curvature patterns of raised objects in the colonic lumen. Others have deduced analytic shape models [16] \begin{table}
\begin{tabular}{c c c c c c c} \hline Reference & Data cen & Train & Test & Per-polyp sens. & FP / vol. (or 2x per patient) & Runtime / vol. \\ \hline [PERSON] _et al_. [10] & & 71 & 71 & 90\% (5 mm+) & 1 & \\ [PERSON] _et al_. [11, 12] & 2 & 8 & 35 & \(<\)35\% (5 mm+) & 2.5 & \\ [PERSON] _et al_. [13] & 3 & 394 & 792 & 61.3\% (6 mm+) & 3.95 & 10 min \\ [PERSON] _et al_. [14] & 2 & 88 & 62 & 90.5\% (5 mm+) & (3 median) & 4 min \\ [PERSON] _et al_. [15] & & 96 & 72 & 94\% (5 mm+) & 4.3 & \\ [PERSON] _et al_. [16] & & & 42 & 81.6\% (\(>\)6 mm) & 5.2 & \\ [PERSON] _et al_. [17] & & 117 & 35 & 92\% (6 mm+) & 5.8 & \\ [PERSON] _et al_. [18] & & 15 & 35 & 77.1\% (6 mm+) & 3.1 & \\ [PERSON] _et al_. [19] & 2 & 169 & 201 & 86\% (n/a) & 5 & 4 min 43 sec \\ [PERSON] _et al_. [20] & 2 & 10 & 98 & 100\% (n/a) & 3 & several minutes \\ [PERSON] _et al_. [12] & 2 & & 35 & 76\% (5 mm+) & 2.5 & 2 min 20 sec \\ [PERSON] _et al_. [21] & 1 & 10 & 73 & 96.4\% (5 mm+) & 0.55 & \\ [PERSON] _et al_. [22] & 3 & 394 & 793 & 77.4\% (6 mm+) & 2.91 & \\ [PERSON] _et al_. [23] & 3 & 395 & 791 & 75\% (6 - 9 mm) & 2.5 & \\ [PERSON] _et al_. [24] & 4 & 86 & 307 & 85 - 100\% (6 mm+) & 4-6 & \\ This paper & 8 & 180 & 3106 & 90.1\% (6 mm+) & 4.7 (3 median) & 1 min 23 sec \\ \hline \end{tabular}
\end{table} 
Table 1: CTC-CAD results reported in the literature. From left to right: columns refer to the reference, number of different data centers from which the data came, number of patients used in training, number of patients used in testing, per polyp sensitivity, false positives per volume (per patient false positive rates can be determined by multiplying the per volume false positive rate by \(2\)), and algorithm runtime per volume.