Bit-shift Operations Research Articles

Low-complexity JPEG quantization table requiring only bit-shift operations

ABSTRACT In this paper, a low-complexity quantization table is proposed for the baseline JPEG encoder. The proposed scheme does not require any multiplications or additions; only bit-shift operations are involved. The computational complexity should be drastically reduced without almost incurring any significant performance loss. The resulting approximate quantization table has extremely low arithmetic cost and is compared to state-of-the-art schemes in terms of computational complexity. Comparisons showed that our proposal exhibits better or similar results. The proposed low-complexity quantization table is also investigated in image compression context. It could outperforms other related schemes in low and high-compression scenarios. Comparisons with respect to average PSNR and SSIM are conducted. Experiment outcomes show up to 4.03 dB improvement in average PSNR over the state-of-the-art competitor. The proposed quantization table has a considerably lower computational cost while maintaining much higher image quality.

Weighted zigzag decodable fountain codes for unequal error protection

By combining bit-shift and exclusive-or operations, a weighted zigzag decodable fountain code is proposed to achieve an unequal error protection property. In the proposed scheme, the input symbols of different importance levels are first pre-coded into variable nodes using low-density parity-check codes. Then, bit-shift operations prior to exclusive-or are performed on the non-uniformly selected variable nodes to generate encoded symbols. An analysis of the erasure probabilities based on the and-or tree is further introduced. Simulation results show that by appropriately choosing the maximum bit-shift amount, the proposed weighted zigzag decodable fountain codes can successfully recover the more important input symbols prior to the less important ones.

Approximate calculation of 8-point DCT for various scenarios of practical applications

In this paper, based on the parametric model of the matrix of discrete cosine transform (DCT), and using an exhaustive search of the parameters’ space, we seek for the best approximations of 8-point DCT at the given computational complexities by taking into account three different scenarios of practical usage. The possible parameter values are selected in such a way that the resulting transforms are only multiplierless approximations, i.e., only additions and bit-shift operations are required. The considered usage scenarios include such cases where approximation of DCT is used: (i) at the data compression stage, (ii) at the decompression stage, and (iii) both at the compression and decompression stages. The obtained results in effectiveness of generated approximations are compared with results of popular known approximations of 8-point DCT of the same class (i.e., multiplierless approximations). In addition, we perform a series of experiments in lossy compression of natural images using popular JPEG standard. The obtained results are presented and discussed. It should be noted, that in the overwhelming number of cases the generated approximations are better than the known ones, e.g., in asymmetric scenarios even by more than 3 dB starting from entropy of 2 bits per pixel. In the last part of the paper, we investigate the possibility of hardware implementation of generated approximations in Field-Programmable Gate Array (FPGA) circuits. The results in the form of resource and energy consumption are presented and commented. The experiment outcomes confirm the assumption that the considered class of transformations is characterized by low resource utilization.

A novel and efficient 8-point DCT approximation for image compression

The Discrete Cosine Transform is widely used in the field of still image compression. Many integer approximations are given in the literature whereas the most of these transforms requires bit shift operations. This paper presents an efficient and low complexity integer approximation of the DCT for image compression. Our new approach involves replacing the bit shift elements of a variant of the Signed DCT transform by zeros, in order to eliminate the bit shift operations. As a result, all elements of the proposed transform are zeros and ± 1. Indeed, the proposed transform retains all the characteristics of its original transform, such as orthogonality and high energy compaction capabilities, while generating computing cost savings. Experiments show that the proposed transform has a good compromise performance-computational complexity as well as state-of the-art DCT approximations. Moreover, an efficient algorithm primarily involving a small amount of arithmetical computation is well developed as no multiplications and bit-shift operations are required, with only 16 additions being involved.

Symmetric Quantization Scheme and Approximate Batch Normalization for Quantized Neural Networks Training

With the advancements of circuit design and manufacturing processes of hardware, Deep Neural Networks (DNN) have been developed to be deeper and larger to achieve remarkable results in many aspects. Despite this, problems such as excessive memory usage and severe resource consumption are still unavoidable. We proposed a symmetric quantization scheme, which quantizes the weights from a certain kernel to be symmetric to zero along with a scaling factor and bias term in order to make full use of the whole quantization space and reduce the multiplication operations when convolution happens. We also came up with an Approximate Batch Normalization (ABN) method, which simplifies the traditional BN to linear operations lightly. Besides, activations and gradients are quantized in our scheme with fine distinctions, so that most floating-point calculations can be converted into fixed-point calculations and bit-shift operations in both training and inference phase. Our proposed scheme greatly alleviates the problems mentioned above theoretically with fewer computations and smaller storage theoretically. We carried out various experiments to prove its feasibility.

Zigzag Decodable codes: Linear-time erasure codes with applications to data storage

An erasure code is said to be k-reliable if it maps k source packets into n coded packets, and any k out of the n coded packets allow recovery of the original k source packets. Codes of k-reliability achieve the best reliability-storage tradeoff, which are useful for fault tolerance in data storage systems. Zigzag Decodable (ZD) codes are k-reliable erasure codes. Its encoding and decoding (per information bit) can be done in linear time, involving only XOR and bit-shift operations. Two classes of ZD codes are constructed, and compared with Cacuhy-RS codes, the state-of-the-art general-purpose MDS codes. Numerical results show that ZD codes outperform Cauchy-RS codes over a wide range of coding parameters.

Fast algorithm for computing cosine number transform

A fast algorithm for the computation of a cosine-like number-theoretic transform is presented. The method, which corresponds to a finite field extension of a method originally designed for computing real-valued discrete cosine transforms, is recursive and suitable for VLSI implementation. A general flow diagram for the proposed algorithm is given and shows that, in some specific cases, it can be evaluated using additions and bit-shift operations only.

DCT-like transform for image compression requires 14 additions only

A low-complexity 8-point orthogonal approximate discrete cosine transform (DCT) is introduced. The proposed transform requires no multiplications or bit-shift operations. The derived fast algorithm requires only 14 additions, less than any existing DCT approximation. Moreover, in several image compression scenarios, the proposed transform could outperform the well-known signed DCT, as well as state-of-the-art algorithms.

A DCT Approximation for Image Compression

An orthogonal approximation for the 8-point discrete cosine transform (DCT) is introduced. The proposed transformation matrix contains only zeros and ones; multiplications and bit-shift operations are absent. Close spectral behavior relative to the DCT was adopted as design criterion. The proposed algorithm is superior to the signed discrete cosine transform. It could also outperform state-of-the-art algorithms in low and high image compression scenarios, exhibiting at the same time a comparable computational complexity.

An image multiresolution representation for lossless and lossy compression

We propose a new image multiresolution transform that is suited for both lossless (reversible) and lossy compression. The new transformation is similar to the subband decomposition, but can be computed with only integer addition and bit-shift operations. During its calculation, the number of bits required to represent the transformed image is kept small through careful scaling and truncations. Numerical results show that the entropy obtained with the new transform is smaller than that obtained with predictive coding of similar complexity. In addition, we propose entropy-coding methods that exploit the multiresolution structure, and can efficiently compress the transformed image for progressive transmission (up to exact recovery). The lossless compression ratios are among the best in the literature, and simultaneously the rate versus distortion performance is comparable to those of the most efficient lossy compression methods.