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A loss function has two crucial roles in training a conventional discriminant deep neural network (DNN): (i) it measures the goodness of classification and (ii) generates the gradients that drive the training of the network. In this paper, we approximate the gradients of cross-entropy loss which is the most often used loss function in the classification DNNs. The proposed approximations are noise-free, which means they depend only on the labels of the training set. They have a fixed length to avoid the vanishing gradient problem of the cross-entropy loss. By skipping the forward pass, the computational complexities of the proposed approximations are reduced to O(n) where n is the batch size. Two claims are established based on the experiments of training DNNs using the proposed approximations: (i) It is possible to train a discriminant network without explicitly defining a loss function. (ii) The success of training does not imply the convergence of network parameters to fixed values. The experiments show that the proposed gradient approximations achieve comparable classification accuracy to the conventional loss functions and can accelerate the training process on multiple datasets.

This paper considers binary classification. We assess a classifier in terms of the Area Under the ROC Curve (AUC). We estimate three important parameters, the conditional AUC (conditional on a particular training set) and the mean and variance of this AUC. We derive, as well, a closed form expression of the variance of the estimator of the AUC. This expression exhibits several components of variance that facilitate an understanding for the sources of uncertainty of that estimate. In addition, we estimate this variance, i.e., the variance of the conditional AUC estimator. Our approach is nonparametric and based on general methods from U-statistics; it addresses the case where the data distribution is neither known nor modeled and where there are only two available data sets, the training and testing sets. Finally, we illustrate some simulation results for these estimators.

Fractal dimension (FD) is a feature which is widely used to characterize medical images. Previously, researchers have shown that FD separates important classes of images and provides distinctive information about texture. The authors analyze limitations of two principal methods of estimating FD: box-counting (BC) and power spectrum (PS). BC is ineffective when applied to data-limited, low-resolution images; PS is based on a fractional Brownian motion (fBm) model-a model which is not universally applicable. The authors also present background information on the use of fractal interpolation function (FIF) models to estimate FD of data which can be represented in the form of a function. They present a new method of estimating FD in which multiple FIF models are constructed. The mean of the FD's of the FIF models is taken as the estimate of the FD of the original data. The standard deviation of the FD's of the FIF models is used as a confidence measure of the estimate. The authors demonstrate how the new method can be used to characterize fractal texture of medical images. In a pilot study, they generated plots of curvature values around the perimeters of images of red blood cells from normal and sickle cell subjects. The new method showed improved separation of the image classes when compared to BC and PS methods.

The covariance matrices associated with each state of health or disease from a previous study are used as the basis of an image staining display technique for aid in quantitative differential diagnosis. A state of health or disease is chosen by the clinician: this selects the covariance matrix from the data base. A region of interest (ROI) is then scrolled through an abdominal B-scan. For each position of the ROI a point in the four-dimensional feature space is calculated. A natural measure of the distance of this point from the center of mass (multivariate mean) of the disease class is calculated in terms of the covariance matrix of this class; this measure is the Mahalanobis distance. The confidence level for acceptance or rejection of the hypothesized disease class is obtained from the probability distribution of this distance, the T/sup 2/ probability law. This confidence level is color coded and used as a color stain that overlays the original scan at that position. The variability of the calculated features is studied as a function of ROI size, or the spatial resolution of the color coded image, and it is found that for an ROI in the neighborhood of 4 cm/sup 2/ most of the variability due to the finite number of independent samples (speckles) is averaged out, leaving the "noise floor" associated with inter- and intra-patient variability. ROIs on the order of 1 cm/sup 2/ may result with technical advances in B-scan resolution. A small number of points on organ boundaries are entered by the user, to fit with arcs of ellipses to be used to switch between organ (liver and kidney) data bases as the ROI encounters the boundary. By selecting in turn various state-of-health or state-of-disease databases, such images of confidence levels may be used for quantitative differential diagnosis. The method is not limited to ultrasound, being applicable in principle to features obtained from any modality or multimodality combination. >

A combined-transform coding (CTC) scheme to reduce the blocking effect of conventional block transform coding and hence to improve the subjective performance is presented. The scheme is described, and its information-theoretic properties are discussed. Computer simulation results for a chest X-ray image are presented. The CTC scheme, the JPEG baseline scheme, and the conventional discrete Walsh-Hadamard transform (DWHT) are compared to demonstrate the performance improvement for the CTC scheme. The advantages of the CTC scheme include no ringing effect as there is no error propagation across the boundary, no additional computation, and distortion always held within a certain level. >

The authors present an efficient algorithm and the results of its application in simulating the three-dimensional (3-D) projection data resulting from a 3-D distribution of radioactivity. The algorithm was applied to a series of geometrical mathematical phantoms and to a realistic mathematical brain phantom. The authors simulated the projection data from a multidetector single-photon emission computed tomography (SPECT) system with point focusing collimators. The simulated projection data were then reconstructed using the manufacturer's software. The objects simulated included simple geometrical solids such as spheres and sheets, as well as the distribution of muscarinic cholinergic receptors in a realistic brain slice. Spheres were chosen as a model for brain structures such as caudate nucleus, thalamus, and cerebellum; sheets were selected as representing lateral cortical gray matter regions. The results of these simulations indicate the existence of significant qualitative and quantitative artifacts in reconstructed human brain images. >

The probabilistic distribution properties of a set of medical images are studied. It is shown that the generalized Gaussian function provides a good approximation to the distribution of AP chest radiographs. Based on this result and a goodness-of-fit test, a generalized Gaussian autoregressive model (GGAR) is proposed. Its properties and limitations are also discussed. It is expected that the GGAR model will be useful in describing the stochastic characteristics of some classes of medical images and in image data compression and other applications. >

The application of a procedure for classifying tissue types from unlabeled acoustic measurements using unsupervised analysis is reviewed and evaluated. Unsupervised learning techniques are applied to the problems of detecting tumors within an organ and of discriminating between tissue types of two neighboring organs such as the liver and the kidney. >