Scale Invariant Loss Function Research Articles

Depth completion towards different sensor configurations via relative depth map estimation and scale recovery

Depth completion, which combines additional sparse depth information from the range sensors, substantially improves the accuracy of monocular depth estimation, especially using the deep-learning-based methods. However, these methods can hardly produce satisfactory depth results when the sensor configuration changes at test time, which is important for real-world applications. In this paper, the problem is tackled by our proposed novel two-stage mechanism, which decomposes depth completion into two subtasks, namely relative depth map estimation and scale recovery. The relative depth map is first estimated from a single color image with our designed scale-invariant loss function. Then the scale map is recovered with the additional sparse depth. Experiments on different densities and patterns of the sparse depth input show that our model always produces satisfactory depth results. Besides, our approach achieves state-of-the-art performance on the indoor NYUv2 dataset and performs competitively on the outdoor KITTI dataset, demonstrating the effectiveness of our method.

Improved estimators for functions of scale parameters in mixture models

Estimation of the scale parameter of the scale mixture of a location–scale family under the scale-invariant loss function is considered. The technique of Strawderman (Ann Stat 2(1):190–198, 1974) is used to obtain a class of estimators improving upon the best affine equivariant estimator of the scale parameter under certain conditions. Further, integral expressions of risk difference approach of Kubokawa (Ann Stat 22(1):290–299, 1994) is used to derive similar improvements for the reciprocal of the scale parameter. Using the improved estimator of the scale parameter and the improved estimator of the reciprocal of the scale parameter, classes of improved estimators for the ratio of scale parameters of two populations have been derived. In particular, Stein type and Brewster–Zidek type estimators are provided for the ratio of scale parameters of two mixture models. These results are applied to the scale mixture of exponential distributions, this includes the multivariate Lomax and the modified Lomax distributions.

Deep Convolutional Network for Stereo Depth Mapping in Binocular Endoscopy

Depth mapping from binocular endoscopy images plays an important role in stereoscopic surgical treatment. Owing to the development of deep convolutional neural networks (CNNs), binocular depth estimation models have achieved many exciting results in the fields of autonomous driving and machine vision. However, the application of these methods to endoscopic imaging is greatly limited by the fact that binocular endoscopic images not only are rare, but also have unsatisfying features such as no texture, no ground truth, bad contrast, and high gloss. Aiming at solving the above-mentioned problems, we have built a precise gastrointestinal environment by the open-source software blender to simulate abundant binocular endoscopy data and proposed a 23-layer deep CNNs method to generate real-time stereo depth mapping. An efficient scale-invariant loss function is introduced in this paper to accommodate the characteristics of endoscope images, which improves the accuracy of achieved depth mapping results. Regarding the considerable training data for typical CNNs, our method requires only a few images ($960\times 720$ resolution) at 45 frames per second on an NVIDIA GTX 1080 GPU module, then the depth mapping information is generated in real-time with satisfactory accuracy. The effectiveness of the developed method is validated by comparing with state-of-the-art methods on processing the same datasets, demonstrating a faster and more accurate performance than other model frames.

Classes of improved estimators for parameters of a Pareto distribution

The problem of estimating parameters of a Pareto distribution is investigated under a general scale invariant loss function when the scale parameter is restricted to the interval (0, 1]. We consider the estimation of shape parameter when the scale parameter is unknown. Techniques for improving equivariant estimators developed by Stein, Brewster–Zidek and Kubokawa are applied to derive improved estimators. In particular improved classes of estimators are obtained for the entropy loss and a symmetric loss. Risk functions of various estimators are compared numerically using simulations. It is also shown that the technique of Kubokawa produces improved estimators for estimating the scale parameter when the shape parameter is known.

Estimating moments of a selected Pareto population under asymmetric scale invariant loss function

Consider independent random samples from \((k\ge 2)\) Pareto populations with the same known shape parameter but different scale parameters. Let \(X_i\) be the smallest observation of the ith sample. The natural selection rule which selects the population associated with the largest \(X_i\) is considered. In this paper, we estimate the moments of the selected population under asymmetric scale invariant loss function. We investigate risk-unbiased, consistency and admissibility of the natural estimators for the moments of the selected population. Finally, the risk-bias’s and risks of the natural estimators are numerically computed and compared for \(k=2,3.\)

Classification into Two-Parameter Exponential Populations with a Common Guarantee Time

SYNOPTIC ABSTRACTLet Π1, Π2, …, Πk be k (≥2) exponential populations with unknown scale parameters σ1, σ2, …, σk, respectively, and an unknown but common location parameter μ. First, we consider the estimation of scale parameters when an isotonic ordering among the scale parameters is present. We show the superiority of a class of mixed estimators over the maximum likelihood estimators of scale parameters under a scale-invariant loss function. Bayes and generalized Bayes estimates of scale parameters are obtained assuming proper and improper prior distributions, respectively. As an application of these new estimators, we have considered the problem of classifying an observation into one of k populations under order restrictions on scale parameters. Classification rules are proposed based on mixed estimators. We also derive plug-in Bayes classification rules and likelihood ratio-based classification rules. Extensive simulations are performed to compare these rules with respect to the expected probability of correct classification. An application of the classification rules is done on a real data set.

Estimation After Selection From Gamma Populations With Unequal Known Shape Parameters

Let be independent gamma populations, where the population has an unknown scale parameter and known shape parameter . We call the population associated with the best population. For the goal of selecting the best population, Misra and Arshad (2014) proposed a class of selection rules for the case of (possibly) unequal shape parameters. In this article, we consider the problem of estimating the mean of the population selected by a fixed selection rule , under a scale-invariant loss function. We derive the uniformly minimum variance unbiased estimator (UMVUE). Two other natural estimators and , which are respectively the analogs of the UMVUE and the best scale invariant estimators of for the component problem, are studied. We show that is generalized Bayes with respect to a noninformative prior distribution, and is also minimax when k = 2. The UMVUE and the natural estimator are shown to be inadmissible, and better estimators are obtained. A numerical study on the performance of various estimators indicates...

Estimating common standard deviation of two normal populations with ordered means

Independent random samples are taken from two normal populations with means \(\mu _1\) and \(\mu _2\) and a common unknown variance \(\sigma ^2.\) It is known that \(\mu _1\le \mu _2.\) In this paper, estimation of the common standard deviation \(\sigma \) is considered with respect to a scale invariant loss function. A general minimaxity result is proved and a class of minimax estimators is derived. An admissibility result is proved in this class. Further a class of equivariant estimators with respect to a subgroup of affine group is considered and dominating estimators in this class are obtained. The risk performance of some of these estimators is compared numerically.

Optimal rules and robust Bayes estimation of a Gamma scale parameter

For estimating an unknown scale parameter of Gamma distribution, we introduce the use of an asymmetric scale invariant loss function reflecting precision of estimation. This loss belongs to the class of precautionary loss functions. The problem of estimation of scale parameter of a Gamma distribution arises in several theoretical and applied problems. Explicit form of risk-unbiased, minimum risk scale-invariant, Bayes, generalized Bayes and minimax estimators are derived. We characterized the admissibility and inadmissibility of a class of linear estimators of the form $$cX\,{+}\,d$$ , when $$X\sim \varGamma (\alpha ,\eta )$$ . In the context of Bayesian statistical inference any statistical problem should be treated under a given loss function by specifying a prior distribution over the parameter space. Hence, arbitrariness of a unique prior distribution is a critical and permanent question. To overcome with this issue, we consider robust Bayesian analysis and deal with Gamma minimax, conditional Gamma minimax, the stable and characterize posterior regret Gamma minimax estimation of the unknown scale parameter under the asymmetric scale invariant loss function in detail.

Estimating the mean of the selected uniform population

Let πi(i=1,2,…K) be independent U(0,⊝i) populations. Let Yi denote the largest observation based on a random sample of size n from the i-th population. for selecting the best populaton, that is the one associated with the largest ⊝i, we consider the natural selection rule, according to which the population corresponding to the largest Yi is selected. In this paper, the estimation of M. the mean of the selected population is considered. The natural estimator is positively biased. The UMVUE (uniformly minimum variance unbiased estimator) of M is derived using the (U,V)-method of Robbins (1987) and its asymptotic distribution is found. We obtain a minimax estimator of M for K≤4 and a class of admissible estimators among those of the form cYmax. For the case K = 2, the UMVUE is improved using the Brewster-Zidek (1974) Technique with respect to the squared error loss function L1 and the scale-invariant loss function L2. For the case K = 2, the MSE'S of all the estimators are compared for selected values of n a...

Simultaneous estimation of parameters under entropy loss

The problem considered is simultaneous estimation of scale parameters and their reciprocals from p independent gamma distributions under a scale invariant loss function first introduced in James and Stein (1961). Under mild restrictions on the shape parameters, the best scale invariant estimators are shown to be admissible for p = 2. For p ≥ 3, a general technique is developed for improving upon the best scale invariant estimators. Improvement on the generalized Bayes estimators of a vector involving certain powers of the scale parameter is also obtained.