Abstract: Two dimensional layer data representing successive layers of objects to be 3D-printed is reduced in size to reduce storage requirements and transmission bottlenecks that increase costs and reduce the efficiency of 3D printers, by encoding the layer data in accordance with systems and methods that take into account 2D and 3D regularities inherent in such layer data, and allow for efficient and lossless reconstruction of the original 3D layer data when it is needed by the 3D printer.

Abstract: A system and method for interconnecting nodes and routing data packets in high radix networks includes constructing or redefining a network structure to provide improved performance. Computation and data storage nodes are connected to a network of switching nodes that provide near optimum bandwidth and latency for networks of any size. Specialized interconnection patterns and addressing methods ensure reliable data delivery in very large networks with high data traffic volume.

Abstract: A system and method for interconnecting nodes and routing data packets in high radix networks includes constructing or redefining a network structure to provide improved performance. Computation and data storage nodes are connected to a network of switching nodes that provide near optimum bandwidth and latency for networks of any size. Specialized interconnection patterns and addressing methods ensure reliable data delivery in very large networks with high data traffic volume.

Abstract: Methods and apparatuses for synchronizing displays of a local and a remote device using image compression to achieve efficiency gains are disclosed. A local device requests an update for a particular requested region of a remote device's display, receives an update region which is a portion of the requested region, and updates a stored local representation of the remote display based on the update region. A remote device compares, upon a request from a local device for an update, portions of the old and new stored images corresponding to the requested region. Compared pixels are then encoded to yield an update region and transmitted to the local device. In the encoding, pixels which changed between the old image and the new image are encoded opaque and with the color corresponding to the pixels in the new image, and at least one pixel which did not change between the old image and the new image is encoded transparent.

Abstract: Entropy encoders and decoders employ an entropy-pump operation. The entropy-pump operation includes classifying blocks of the input sequence's symbols in accordance with the probabilities of occurrence of those blocks' values. Entropy coding an output component that represents the sequence of classes to which the blocks belong tends to be profitable, because that component has an entropy density less than that of the input sequence. In contrast, little opportunity for compression tends to be lost by not entropy encoding an output component that represents the values of the blocks that belong to single class, because those values' probabilities of occurrence are similar.

Abstract: An enumerator employs “indexing volumes” as the add-on values used to compute indexes for n-item ordered sets such as symbol sequences. Each indexing volume is associated with a different class into which the allowed ordered sets are partitioned. The indexing volumes all equal or exceed the number of ordered sets that belong to their respective classes. Additionally, the indexing volume Vi associated with a given class of i-item equals or exceeds the sum of the indexing volumes associated with the classes that contain the (i?1)-item prefixes of the ordered sets that belong to the given class.