Abstract: Cone regression is a particular case of quadratic programming that minimizes a weighted sum of squared residuals under a set of linear inequality constraints. Several important statistical problems such as isotonic, concave regression or ANOVA under partial orderings, just to name a few, can be considered as particular instances of the cone regression problem. Given its relevance in Statistics, this paper aims to address the fundamentals of cone regression from a theoretical and practical point of view. Several formulations of the cone regression problem are considered and, focusing on the particular case of concave regression as an example, several algorithms are analyzed and compared both qualitatively and quantitatively through numerical simulations. Several improvements to enhance numerical stability and bound the computational cost are proposed. For each analyzed algorithm, the pseudo-code and its corresponding code in Matlab are provided. The results from this study demonstrate that the choice of the optimization approach strongly impacts the numerical performances. It is also shown that methods are not currently available to solve efficiently cone regression problems with large dimension (more than many thousands of points). We suggest further research to fill this gap by exploiting and adapting classical multi-scale strategy to compute an approximate solution.

Abstract: BACKGROUND AND OBJECTIVE:
Fast intravascular ultrasound (IVUS) video processing is required for calcium volume computation during the planning phase of percutaneous coronary interventional (PCI) procedures. Nonlinear multiresolution techniques are generally applied to improve the processing time by down-sampling the video frames.
METHODS:
This paper presents four different segmentation methods for calcium volume measurement, namely Threshold-based, Fuzzy c-Means (FCM), K-means, and Hidden Markov Random Field (HMRF) embedded with five different kinds of multiresolution techniques (bilinear, bicubic, wavelet, Lanczos, and Gaussian pyramid). This leads to 20 different kinds of combinations. IVUS image data sets consisting of 38,760 IVUS frames taken from 19 patients were collected using 40 MHz IVUS catheter (Atlantis® SR Pro, Boston Scientific®, pullback speed of 0.5 mm/sec.). The performance of these 20 systems is compared with and without multiresolution using the following metrics: (a) computational time; (b) calcium volume; (c) image quality degradation ratio; and (d) quality assessment ratio.
RESULTS:
Among the four segmentation methods embedded with five kinds of multiresolution techniques, FCM segmentation combined with wavelet-based multiresolution gave the best performance. FCM and wavelet experienced the highest percentage mean improvement in computational time of 77.15% and 74.07%, respectively. Wavelet interpolation experiences the highest mean precision-of-merit (PoM) of 94.06 ± 3.64% and 81.34 ± 16.29% as compared to other multiresolution techniques for volume level and frame level respectively. Wavelet multiresolution technique also experiences the highest Jaccard Index and Dice Similarity of 0.7 and 0.8, respectively. Multiresolution is a nonlinear operation which introduces bias and thus degrades the image. The proposed system also provides a bias correction approach to enrich the system, giving a better mean calcium volume similarity for all the multiresolution-based segmentation methods. After including the bias correction, bicubic interpolation gives the largest increase in mean calcium volume similarity of 4.13% compared to the rest of the multiresolution techniques. The system is automated and can be adapted in clinical settings.
CONCLUSIONS:
We demonstrated the time improvement in calcium volume computation without compromising the quality of IVUS image. Among the 20 different combinations of multiresolution with calcium volume segmentation methods, the FCM embedded with wavelet-based multiresolution gave the best performance.