This article presents a novel variational model based on fuzzy clustering and total variation regularization for superpixel segmentation. Compared with the classical hard-labeled methodologies, our approach gives soft results via the fuzzy membership function, and moreover, the use of total variation provides additional information that can enhance the superpixel regularity, which in turn improves the segmentation performance. To efficiently minimize the energy functional of the proposed model, we adopt an alternating direction method of multipliers with the modified Chambolle’s fast duality projection algorithm. Our algorithm can generate regular and compact superpixels with high segmentation accuracy, satisfactory boundary adherence, and low computational cost. Comparative experimental results with the current state-of-the-art approaches reveal the superior performance of the proposed method.

BATS codes were proposed for communication through networks with packet loss. A BATS code consists of an outer code and an inner code. The outer code is a matrix generation of a fountain code, which works with the inner code that comprises random linear coding at the intermediate network nodes. In this paper, the performance of finite-length BATS codes is analyzed with respect to both belief propagation (BP) decoding and inactivation decoding. Our results enable us to evaluate efficiently the finite-length performance in terms of the number of batches used for decoding ranging from 1 to a given maximum number, and provide new insights on the decoding performance. Specifically, for a fixed number of input symbols and a range of the number of batches used for decoding, we obtain recursive formulae to calculate the stopping time distribution of BP decoding and the inactivation probability in inactivation decoding. We also find that both the failure probability of BP decoding and the expected number of inactivations in inactivation decoding can be expressed in a power-sum form where the number of batches appears only as the exponent. This power-sum expression reveals clearly how the decoding failure probability and the expected number of inactivation decrease with the number of batches. When the number of batches used for decoding follows a Poisson distribution, we further derive recursive formulas with potentially lower computational complexity for both decoding algorithms. For the BP decoder that consumes batches one by one, three formulae are provided to characterize the expected number of consumed batches until all the input symbols are decoded.