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Convolutional Neural Network (CNN) have been widely used in imageclassification. Over the years, they have also benefited from variousenhancements and they are now considered as state of the art techniques forimage like data. However, when they are used for regression to estimate somefunction value from images, fewer recommendations are available. In this study,a novel CNN regression model is proposed. It combines convolutional neurallayers to extract high level features representations from images with a softlabelling technique. More specifically, as the deep regression task ischallenging, the idea is to account for some uncertainty in the targets thatare seen as distributions around their mean. The estimations are carried out bythe model in the form of distributions. Building from earlier work, a specifichistogram loss function based on the Kullback-Leibler (KL) divergence isapplied during training. The model takes advantage of the CNN featurerepresentation and is able to carry out estimation from multi-channel inputimages. To assess and illustrate the technique, the model is applied to GlobalNavigation Satellite System (GNSS) multi-path estimation where multi-pathsignal parameters have to be estimated from correlator output images from the Iand Q channels. The multi-path signal delay, magnitude, Doppler shift frequencyand phase parameters are estimated from synthetically generated datasets ofsatellite signals. Experiments are conducted under various receiving conditionsand various input images resolutions to test the estimation performancesquality and robustness. The results show that the proposed soft labelling CNNtechnique using distributional loss outperforms classical CNN regression underall conditions. Furthermore, the extra learning performance achieved by themodel allows the reduction of input image resolution from 80x80 down to 40x40or sometimes 20x20.

Distributional loss for convolutional neural network regression and application to GNSS multi-path estimation