# Geometric Multi-Model Fitting with a Convex Relaxation Algorithm

## TL;DR

This paper introduces a convex relaxation algorithm for multi-model fitting and segmentation that improves scalability and flexibility over greedy methods, demonstrating superior results in image structure estimation tasks.

## Contribution

The paper presents a novel convex relaxation approach for multi-model fitting that handles multiple models efficiently and flexibly, outperforming existing methods.

## Key findings

- Accurate plane extraction from RGB-D data
- Effective homography estimation from image pairs
- Outperforms state-of-the-art methods on benchmark datasets

## Abstract

We propose a novel method to fit and segment multi-structural data via convex relaxation. Unlike greedy methods --which maximise the number of inliers-- this approach efficiently searches for a soft assignment of points to models by minimising the energy of the overall classification. Our approach is similar to state-of-the-art energy minimisation techniques which use a global energy. However, we deal with the scaling factor (as the number of models increases) of the original combinatorial problem by relaxing the solution. This relaxation brings two advantages: first, by operating in the continuous domain we can parallelize the calculations. Second, it allows for the use of different metrics which results in a more general formulation. We demonstrate the versatility of our technique on two different problems of estimating structure from images: plane extraction from RGB-D data and homography estimation from pairs of images. In both cases, we report accurate results on publicly available datasets, in most of the cases outperforming the state-of-the-art.