A novel robust method for large numbers of gross errors

In computer vision tasks, it frequently happens that gross noise occupies the absolute majority of the data. Most robust estimators can tolerate no more than 50% gross errors. In this article, we propose a highly robust estimator, called MDPE (Maximum Density Power Estimator), employing density estimation and density gradient estimation techniques in the residual space. This estimator can tolerate more than 85% outliers. Experiments illustrate that the MDPE has a higher breakdown point and less errors than other recently proposed similar estimators: Least Median of Squares (LMedS), Residual Consensus (RESC), and Adaptive Least kth Order Squares(ALKS). 1.Introduction There has recently been a general recognition in computer vision community that algorithms should be robust because it is unavoidable that data are contaminated (due to faulty feature extraction, sensor noise, segmentation errors, etc). The outliers may include uniformly distributed, or clustered outliers or pseudo-outliers (arising from multiple structures). Thus outliers may occupy the absolute majority of the data. In this paper we introduce a new estimator (MDPE). The goals in designing MDPE were: it should be able to fit signals corresponding to less than 50% of the data points, and fit data with multi-structures. We assume the inliers occupy a relative majority (instead of the absolute majority which is assumed in general estimators such as the LMedS and the LTS [5]) of the data points. Probability Density estimation and Mean Shift techniques [13] are employed in MDPE. The mean shift vector always points towards the direction of the maximum increase in the probability density function. Through the mean shift iterations, the local maximum density, corresponding to the mode (or the center of the regions of high concentration) of data, can be found. Two criteria are considered in our objective function: • The density distribution of the data points provided by the density estimation technique. • The value of the residual corresponding to the local maximum of probability density. If the signal is correctly fitted, the densities of inliers should be as large as possible; at the same time, the value of the center of the high concentration of data should be as close to zero as possible in the residual space. The contributions of this paper can be summarized as follows : (1) We provide a novel estimator which can tolerate more than 85% outliers. (2) We apply nonparametric density estimation and density gradient estimation techniques in parametric estimation. Instead of considering residuals as the only feature, both the density distribution of data points and the residual corresponding to the local maximum density distribution are considered as features in our objective function. (3) MDPE can deal with the data with multistructured outliers. This paper is organized as follows: in section 2, we review previous methods and their limits. The density gradient estimation and mean shift method are introduced in section 3. In section 4, we describe MDPE method. Experimental results are contained in section 5. Finally, we conclude with a discussion of further possible work. 2. Previous methods and their limitations. Great efforts have been made in the search for high breakdown point estimators in recent decades. The maximum-likelihood-type estimators (M-estimators) [2][3] are well-known among the robust estimators. Although M-estimators can reduce the influence of outliers, they have breakdown points less than 1/(p+1), where p is the number of the parameters to estimate: robustness diminishes when the dimension p increases. Siegel [4] proposed the repeated median (RM) estimator with the breakdown point of 50%. However, the time complexity of the repeated median estimator is O(nlogn), which prevents the method being useful in applications where p is even moderately large. Rousseeuw [5] proposed the least median of squares (LMedS) method. The LMedS finds the parameters to estimate by minimizing the median of residuals corresponding to the data points. The LMedS also has a breakdown point of 50% (except in extreme situations where it may breakdown earlier). When the outliers are more than 50% of the data, the LMedS method will fail completely to fit a model. The RESC [8] method can tolerate more than 50% outliers. RESC uses a compressed histogram method to infer residual consensus. However, RESC needs the user to tune parameters in the procedure for compressing the histogram and in its objective function for optimal performance. The authors of MUSE [10] and those of ALKS [11] consider robust scale estimate and they both obtain a breakdown point higher than 50%. However, MUSE needs a lookup table for the scale estimator correction; ALKS is limited in its ability to handle extreme outliers. 3. Density Gradient Estimation and Mean shift Method When a model is correctly fitted, there are two criteria that should be satisfied: (1) Data points on or near the model (inliers) should be as many as possible, i.e., the probability density function (PDF) around the model should be as high as possible; (2) The residuals of inliers should be as small as possible. Our new method, MDPE, considers these two criteria in its objective function and it employs density estimation and density gradient estimation techniques. The accuracy of density estimation and density gradient estimation will directly affect the achievements of MDPE in fitting models. There are several nonparametric methods available for probability density estimation: histogram, naive method, the nearest neighbor method, and kernel estimation [12]. The kernel estimation method is one of the most popular techniques used in estimating density. Given a set of n data points {Xi}I=1,...,n in a d-dimensional Euclidian space R , the multivariate kernel density estimator with kernel K and window radius (band-width) h is defined as follows [12, p.76]