Fast Discrete Cosine Transform Research Articles

The Conservative and Efficient Numerical Method of 2-D and 3-D Fractional Nonlinear Schrödinger Equation Using Fast Cosine Transform

This paper introduces a novel approach employing the fast cosine transform to tackle the 2-D and 3-D fractional nonlinear Schrödinger equation (fNLSE). The fractional Laplace operator under homogeneous Neumann boundary conditions is first defined through spectral decomposition. The difference matrix Laplace operator is developed by the second-order central finite difference method. Then, we diagonalize the difference matrix based on the properties of Kronecker products. The time discretization employs the Crank–Nicolson method. The conservation of mass and energy is proved for the fully discrete scheme. The advantage of this method is the implementation of the Fast Discrete Cosine Transform (FDCT), which significantly improves computational efficiency. Finally, the accuracy and effectiveness of the method are verified through two-dimensional and three-dimensional numerical experiments, solitons in different dimensions are simulated, and the influence of fractional order on soliton evolution is obtained; that is, the smaller the alpha, the lower the soliton evolution.

A 34.7 µW Speech Keyword Spotting IC Based on Subband Energy Feature Extraction

In the era of the Internet of Things (IoT), voice control has enhanced human–machine interaction and the accuracy of keyword spotting (KWS) algorithms has reached 97%; however, the high power consumption of KWS algorithms caused by their huge computing and storage requirements has limited their application in Artificial Intelligence of Things (AIoT) devices. In this study, voice features are extracted by utilizing the fast discrete cosine transform (FDCT) for frequency-domain transformation and to shorten the process of calculating the logarithmic spectrum and cepstrum. The designed KWS system is a two-stage wake-up system, with a sound detection (SD) awakening KWS. The inference process of the KWS network is achieved using time-division computation, reducing the KWS clock to an ultra-low frequency of 24 kHz.At the same time, the implementation of a depthwise separable convolution neural network (DSCNN) greatly reduces the parameter quantity and computation. Under the GSMC 0.11 µm technology, post-layout simulation results show that the total synthesized area of the entire system circuit is 0.58 mm2, the power consumption is 34.7 µW, and the F1-score of the KWS is 0.89 with 10 dB noise, which makes it suitable as a KWS system in AIoT devices.

Light network high performance discrete cosine transform for digital images

This paper proposes a new high-performance Discrete Cosine Transform (DCT) for image and signal processing applications. Generally, DCT has been developed in a wide range of applications such as image compression and motion detection. However, the high complexity and required processing have effectively minimized the practical utilization of DCT. A New mathematical derivation for DCT is achieved in this paper with the purpose to reduce its complexity and speed up its processing. The new fast DCT version is mainly based on removing coefficients redundant calculations. The new developed model has shown a potential performance in comparison with standard DCT. Moreover, the proposed model has a high quality that has been investigated based on Mean Square Error (MSE) and Peak Signal to Noise Ratio (PSNR). Fast DCT requires only 0.15%–6% processing time in comparison with traditional DCT.

Developed JPEG Algorithm Applied in Image Compression

JPEG is most popular image compression and encoding, this technique is widely used in many applications (images, videos and 3D animations). Meanwhile, researchers are very interested to develop this massive technique to compress images at higher compression ratios with keeping image quality as much as possible. For this reason in this paper we introduce a developed JPEG based on fast DCT and removed most of zeros and keeps their positions in a transformed block. Additionally, arithmetic coding applied rather than Huffman coding. The results showed up, the proposed developed JPEG algorithm has better image quality than traditional JPEG techniques.

Real sparse fast DCT for vectors with short support

In this paper we present a new fast and deterministic algorithm for the inverse discrete cosine transform of type II for reconstructing a real input vector of length N with short support of length m from its discrete cosine transform of type II if an upper bound M on the support length m is known. The resulting algorithm only uses real arithmetic, has a sublinear runtime of O(Mlog⁡M+mlog2⁡NM) and requires O(M+mlog2⁡NM) samples. For m and M approaching the vector length, the runtime and sampling requirements approach those of a regular IDCT-II for vectors with full support.

Sparse fast DCT for vectors with one-block support

In this paper we present a new fast and deterministic algorithm for the inverse discrete cosine transform of type II that reconstructs the input vector $\mathbf{x}\in\mathbb{R}^{N}$, $N=2^{J-1}$, with short support of length $m$ from its discrete cosine transform $\mathbf{x}^{\widehat{\mathrm{II}}}=\mathbf{C}_N^{\mathrm{II}}\mathbf{x}$. The resulting algorithm has a runtime of $\mathcal{O}\left(m\log m\log \frac{2N}{m}\right)$ and requires $\mathcal{O}\left(m\log \frac{2N}{m}\right)$ samples of $\mathbf{x}^{\widehat{\mathrm{II}}}$. In order to derive this algorithm we also develop a new fast and deterministic inverse FFT algorithm that constructs the input vector $\mathbf{y}\in\mathbb{R}^{2N}$ with reflected block support of block length $m$ from $\widehat{\mathbf{y}}$ with the same runtime and sampling complexities as our DCT algorithm.

Design and Digital Implementation of Fast and Recursive DCT II–IV Algorithms

Using the proposed factorizations of discrete cosine transform (DCT) matrices, fast and recursive algorithms are stated. In this paper, signal flow graphs for the n-point DCT II and DCT IV algorithms are introduced. The proposed algorithms yield exactly the same results as with standard DCT algorithms but are faster. The arithmetic complexity and stability of the algorithms are explored, and improvements of these algorithms are compared with previously existing fast and stable DCT algorithms. A parallel hardware computing architecture for the DCT II algorithm is proposed. The computing architecture is first designed, simulated, and prototyped using a 40-nm Xilinx Virtex-6 FPGA and thereafter mapped to custom integrated circuit technology using 0.18- $$\upmu $$ m CMOS standard cells from Austria Micro Systems. The performance trade-off exists between computational precision, chip area, clock speed, and power consumption. This trade-off is explored in both FPGA and custom CMOS implementation spaces. An example FPGA implementation operates at clock frequencies in excess of 230 MHz for several values of system word size leading to real-time throughput levels better than 230 million 16-point DCTs per second. Custom CMOS-based results are subject to synthesis and place-and-route steps of the design flow. Physical silicon fabrication was not conducted due to prohibitive cost.

Lowest Complexity Self-Recursive Radix-2 DCT II/III Algorithms

This paper presents the lowest multiplication complexity, self-recursive, radix-2 DCT II/III algorithms for any $n=2^t; (t \geq 1)$ with implementations in signal-flow graphs and image compression. These algorithms are derived mainly using a matrix factorization technique to factor DCT II/III matrices into sparse and scaled orthogonal matrices. Although there are small length DCT II algorithms available only for $n=8$ and $n=16$, there is no existing self-recursive fast radix-2 DCT II/III algorithms for any $n$. To fill this gap, this paper presents the lowest multiplication complexity radix-2 self-recursive DCT II/III algorithms. We also attain the lowest theoretical multiplication bound for $n=8$ with the new DCT II algorithm and establish new lowest bounds for DCT III for any $n$ and for DCT II for any $n \geq 32$. The paper also establishes a novel relationship between DCT II an DCT IV having sparse factors. This enables one to see the connection of most traditional factorization of DCT II/III matrices with the proposed DCT II/III matrices.

Fast recovery of unknown coefficients in DCT-transformed images

The advancement of cryptography and cryptanalysis has driven numerous innovations over years. Among them is the treatment of cryptanalysis on selectively encrypted content as a recovery problem. Recent research has shown that linear programming is a powerful tool to recover unknown coefficients in DCT-transformed images. While the time complexity is polynomial, it is still too high for large images so faster methods are still desired. In this paper, we propose a fast hierarchical DCT coefficients recovery method by combining image segmentation and linear programming. In theory the proposed method can reduce the overall time complexity by a linear factor which is the number of image segments used. Our experimental results showed that, for 100 test images of different sizes and using a naive image segmentation method based on Otsu’s thresholding algorithm, the proposed method is faster for more than 92% cases and the maximum improvement observed is more than 19 times faster. While being mostly faster, results also showed that the proposed method can roughly maintain the visual quality of recovered images in both objective and subjective terms.

An Efficient Image Watermarking Method Based on Fast Discrete Cosine Transform Algorithm

This paper proposes an image watermarking method based on the fast discrete cosine transform (DCT) algorithm for implementation in digital signal processor. A digital watermark can be effectively embedded and efficiently extracted without the host image. The keys in watermarking process include four frequency coefficients in DCT, two random permutation vectors, and a quantization matrix for normalizing the watermark and the host image. The fast DCT algorithm has been shown to reduce the complexity of two-dimensional image transformation so that embedding/decoding an image watermark can be completed in real time within 0.33 seconds. The quality of both watermarked image and extracted (retrieved) watermark remains excellent. It is shown that the watermarking method is efficient in and robust to data cropping, transmission loss, and compression/decompression.

A Fast DCT Algorithm for Watermarking in Digital Signal Processor

Discrete cosine transform (DCT) has been an international standard in Joint Photographic Experts Group (JPEG) format to reduce the blocking effect in digital image compression. This paper proposes a fast discrete cosine transform (FDCT) algorithm that utilizes the energy compactness and matrix sparseness properties in frequency domain to achieve higher computation performance. For a JPEG image of8×8block size in spatial domain, the algorithm decomposes the two-dimensional (2D) DCT into one pair of one-dimensional (1D) DCTs with transform computation in only 24 multiplications. The 2D spatial data is a linear combination of the base image obtained by the outer product of the column and row vectors of cosine functions so that inverse DCT is as efficient. Implementation of the FDCT algorithm shows that embedding a watermark image of 32 × 32 block pixel size in a 256 × 256 digital image can be completed in only 0.24 seconds and the extraction of watermark by inverse transform is within 0.21 seconds. The proposed FDCT algorithm is shown more efficient than many previous works in computation.

Fast approximate DCT with GPU implementation for image compression

Recent developments in image and video processing employed in multimedia and communication systems require fast 2-D Discrete Cosine Transforms (DCT). The DCT is widely employed in image compression for its high power compaction property. Approximate DCT transforms have been developed to proceed faster than the original DCT while maintaining comparative levels of power compaction. This paper introduces a multiplierless efficient and low complexity 8-point approximate DCT. A flow diagram is provided for the fast implementation of the proposed transform. Only 17 additions are required for both forward and backward transformations. A fast and efficient Graphics Processing Unit (GPU) implementation for the proposed transform is provided. Performance evaluation shows that the proposed transform outperforms other approximate DCT transforms in JPEG-like image compression.

Fast DCT algorithms for EEG data compression in embedded systems

Electroencephalography (EEG) is widely used in clinical diagnosis, monitoring and Brain - Computer Interface systems. Usually EEG signals are recorded with several electrodes and transmitted through a communication channel for further processing. In order to decrease communication bandwidth and transmission time in portable or low cost devices, data compression is required. In this paper we consider the use of fast Discrete Cosine Transform (DCT) algorithms for lossy EEG data compression. Using this approach, the signal is partitioned into a set of 8 samples and each set is DCT-transformed. The least-significant transform coefficients are removed before transmission and are filled with zeros before an inverse transform. We conclude that this method can be used in real-time embedded systems, where low computational complexity and high speed is required.

Visual Object Tracking Using Fast Three-Dimensional Discrete Cosine Transform

A novel visual object tracking method for color video stream based on traditional particle filter is proposed in this paper. Feature vectors are extracted from coefficient matrices of fast three-dimensional Discrete Cosine Transform (fast 3-D DCT). The feature, as experiment showed, is very robust to occlusion and rotation and it is not sensitive to scale changes. The proposed method is efficient enough to be used in a real-time application. The experiment was carried out on some common used datasets in literature. The results are satisfied and showed the estimated trace follows the target object very closely.

Review on Fractal Image Compression based on Fast DCT Algorithm

Fractal image compressions is a lossy compression technique for digital images. It is based on the assumptions that the image redundancies can be efficiently exploited by means of block self-affine transformations .Unlike other compression techniques it considers the interrelations between local (range blocks) and global (domain blocks) data. Use of contractive transform on the space of images encompasses a wide variety of coding scheme. However, the high computational complexity of fractal image encoding greatly restricts its application. This review gives a study of different speed ups using DCT to reduce the searching time . Index terms Iterated function system, Discrete cosine transform, Self similarity, contractive mapping,

A Blind Watermarking Algorithm Research Based on Fast DCT

This article exploits a fast algorithm to watermark implant and extract, which first transform N×N(N=2n) two dimension DCT into one dimension DCT with N numbers of points. And then the adopted algorithm implants watermark using matrix decomposition and masterly adjusting interrelation of DCT coefficients, thus to make the visual distortion to be minima after a watermark being implanted in the image, in addition, this method is a blind approach. The results indicate that the algorithm enjoy high image reducing property and robustness to several common attacks when ensuring invisibility of watermark. Finally, it is also superior to R-C, index-mapping like matrix cut algorithms and can also improve calculating speed, which make it possible to apply for image protection.

An algebraic method for synthesizing fast algorithms of discrete cosine transform of arbitrary size

An algebraic method for synthesizing fast algorithms of the discrete cosine transform (DCT) of arbitrary size is proposed. The method is based on the polynomial algebra \(\mathbb{F}{{[x]} \mathord{\left/ {\vphantom {{[x]} {p(x)}}} \right. \kern-\nulldelimiterspace} {p(x)}}\) associated with the DCT. The fast DCT algorithm comes as a result of the step-by-step decomposition of this algebra. In turn, the decomposition requires step-by-step factorization of the polynomial p(x). This problem is solved using Galois’s theory, which allows finding all the subfields of the splitting field of the polynomial p(x) where p(x) can be factorized.

FAST DCT ALGORITHM FOR VIDEO TRANSMISSION

This paper presents a fast DCT algorithm by using convolution method. Multipliers will be a crucial process in designing a fast DCT algorithm because they take a long time to complete their process. Reducing a number of multipliers is the one way to make a fast DCT algorithm. This paper shows how to reduce multipliers by using two steps: the first step is by grouping transformation matrix to be two groups in which this process uses the folding concept. The second step is by using fast convolution by inspection to minimize multipliers. Keywords : multiplier, fast DCT, fast convolution, folding.

FAST DCT ALGORITHM FOR VIDEO TRANSMISSION

This paper presents a fast DCT algorithm by using convolution method. Multipliers will be a crucial process in designing a fast DCT algorithm because they take a long time to complete their process. Reducing a number of multipliers is the one way to make a fast DCT algorithm. This paper shows how to reduce multipliers by using two steps: the first step is by grouping transformation matrix to be two groups in which this process uses the folding concept. The second step is by using fast convolution by inspection to minimize multipliers. Keywords : multiplier, fast DCT, fast convolution, folding.

Modified VLSI implementation of DA-DWT for image compression

The wavelet transformation is a widely used technique for image processing application. Hence compared to traditional transforms such as the Fast Fourier Transform (FFT) and Discrete Cosine Transform (DCT), wavelet transform holds both time and frequency information, based on a multi-resolution analysis framework. There is a growing need to embed it into a real system. Field Programmable Gate Array (FPGA) implementation of Discrete Wavelet Transform (DWT) results in higher processing speed and lower costs when compared to other implementation methods. In this work a modified DA which performs low-pass and high-pass filtering improves the computational speed of DWT. A 9/7 filter is designed using modified DA architecture to perform low-pass and high-pass filtering. By using optimised DA architecture the LUT size is reduced from 29 to 24 locations in 9th order filter and 27 to 24 in 7th order filter. The design is verified in a Verilog HDL simulator and implemented on XILINX Virtex5 FPGA board.