The discrete quaternion Fourier transform

Phone cameras are currently used in various medical fields for processing color images. Today they can play an important role in facilitating access to some diagnostic services because they are available to many people, including doctors and patients who do not have access to advanced imaging systems. Image enhancement via cell phones is a highly desired preprocessing tool because many pictures and images are poor in quality. Processing of color images in one of the known color models, such as RGB, XYZ, or HSV15 is a very difficult task and has attracted much research interest in the last decade. In traditional color image processing, each color channel (red, green, and blue in the RGB color space, for example) is separately considered, then the color image is composed. In such a simple approach, the inherent correlation between the components are not taken into account; this fact results in color artifacts. For another approach that simultaneously processes the color components of the image,19 the most promising tool is the theory of quaternion algebra with 2D quaternion transforms, such as the Fourier, Hadamard, and cosine transforms. The 2D discrete Fourier transform (DFT) is still the primary transform used in image enhancement and filtration. Therefore, we will focus on an extension of this transform in quaternion arithmetic. The color image as a three-component image is mapped into pure quaternion data and can be called a color-in-quaternion image. This quaternion-transform-based approach can effectively be used in different applications of color science. First introduced by Hamilton,quaternion numbers now are used in bioinformatics, navigation systems,6 image and video processing, and in a number of studies on quaternions in color image processing. Here, we note that since quaternion multiplication is not commutative, different definitions of quaternion Fourier transforms can be used in color image processing.

Color Visual Secret Sharing (VSS) is an essential form of VSS. It is so because nowadays, most people like to share visual data as a color image. There are color VSS schemes capable of dealing with halftone color images or color images with selected colors, and some dealing with natural color images, which generate low quality of recovered secret. The proposed scheme deals with a color image in the RGB domain and generates gray shares for color images using color to gray and back through compression. These shares are encrypted into an innocent-looking gray cover image using a Discrete Cosine Transform (DCT) to make meaningful shares. Reconstruct a high-quality color image through the gray shares extracted from an innocent-looking gray cover image. Thus, using lower bandwidth for transmission and less storage.

The practice of using Fourier - processing of finite two-dimensional signals (including images), having confirmed its effectiveness, revealed a number of negative effects inherent in it. The purpose of this paper is to develop new discrete Fourier transforms for efficient and effective processing of two-dimensional signals. We have proposed a new method for splitting a rectangular discrete Fourier transform matrix into square matrices. Methods are based on the application of the modulus comparability relation to order the rows (columns) of the Fourier matrix. New discrete Fourier transforms with variable parameters were developed. The proposed discrete Fourier transforms with variable parameters are a generalization of the classical discrete Fourier transform. The developed discrete Fourier transforms with variable parameters can significantly reduce the cost of Fourier - processing of two-dimensional signals (including images), padded with zeros.

We propose a new convex regularizer, named the local color nuclear norm (LCNN), for color image recovery. The LCNN is designed to promote a property inherent in natural color images - in which their local color distributions often exhibit strong linearity - and is thus expected to reduce color artifact effectively. In addition, the very nature of LCNN allows us to incorporate it into various types of color image recovery formulations, with the associated convex optimization problems solvable using proximal splitting techniques. Applications of LCNN are demonstrated with illustrative numerical examples.

A recent trend in color image processing combines the quaternion algebra with dictionary learning methods. This paper aims to present a generalization of the quaternion dictionary learning method by using the octonion algebra. The octonion algebra combined with dictionary learning methods is well suited for representation of multispectral images with up to 7 color channels. Opposed to the classical dictionary learning techniques that treat multispectral images by concatenating spectral bands into a large monochrome image, we treat all the spectral bands simultaneously. Our approach leads to better preservation of color fidelity in true and false color images of the reconstructed multispectral image. To show the potential of the octonion based model, experiments are conducted for image reconstruction and denoising of color images as well as of extensively used Landsat 7 images.

The processing efficiency of the widely used time-space (t-x) domain vertical seismic profiling (VSP) wavefield separation methods (such as median and singular value decomposition filtering) via one-dimensional discrete Fourier transform (DFT) depends on the wavefield separation method (or algorithm with processing parameter) and the total number of samples in the input VSP data. Once the wavefield separation method is determined, its processing efficiency is set and cannot be optimized. Although the popular frequency-wavenumber (f-k) domain VSP wavefield separation method via two-dimensional DFT has higher processing efficiency than the t-x domain VSP wavefield separation methods, its processing precision is susceptible to the spatial alias and Gibbs effects. For efficiency and precision improvements, we introduced the discrete cosine transform (DCT) operation into VSP wavefield separation for the first time, and proposed a high-efficiency and precision frequency-space (f-x) domain VSP wavefield separation method via DFT and DCT, in which the wavefield separation efficiency and precision can be optimized using the effective bandwidth cutoff frequency of the input VSP data. Based on the relationship between DFT and DCT, we combined their operations in the proposed method into one step (referred to as DCT–DFT) for further efficiency enhancement, thereby designing a high-efficiency and precision f-x domain VSP wavefield separation method via DCT–DFT. Theory analysis and synthetic and field VSP data examples show that the proposed method is highly efficient and precise, and can be widely used for three-dimensional (3D)-VSP data wavefield separation, especially for large distributed acoustic sensing (DAS)-VSP data.

This paper deals with a novel steganographic technique which demonstrates the colour image authentication technique in frequency domain based on the Discrete Fourier Transform (DFT). The DFT is applied on sub-image block called mask of size 2 × 2 for frequency components of corresponding spatial component. This transforms process done from beginning to end mask in row major order of the carrier image. Image authentication is done by hiding secrete message/image into the transformed frequency components of carrier image. Four secrete message/image bits are fabricated within the transformed real frequency component of each carrier image byte except the LSB of first frequency component of each mask. After embedding, a delicate re-adjust phase is incorporated in all the frequency component of each mask, to keep the quantum value positive and non fractional in spatial domain. Robustness is achieved by hiding an authenticating or secretes message/image in the frequency component with positive and negatives both quantum values and invisibility is satisfied in spatial domain using delicate re-adjust phase. Inverse DFT (IDFT) is performed after embedding to transform embedded image in frequency domain to spatial domain. This novel technique is also applicable for secrete data transmission through carrier colour image by hiding secrete data. Experimental results conform that the proposed algorithm performs better than discrete cosine transformation (DCT), Quaternion Fourier Transformation (QFT) and Spatio Chromatic DFT (SCDFT) based schemes.

High-throughput screening in histology and analysis need a necessary automatic cell or nucleus counting. Current methods and systems based on grayscale or color images give results with counting errors. We suggest to use multispectral imaging (with more than three bands) rather than color one for nucleus counting. A traditional acquisition chains uses a source of white light and a CCD camera in addition to the optical microscope. To pass to a multispectral acquisition, we use a Programmable Light Source (PLS) in the place of the white light source. This PLS is capable of generating different wavelengths in the visible spectrum. So, one color image and four multispectral images have been acquired from histological slices. The four multispectral images contain respectively 3 bands, 5 bands, 7 bands and 10 bands. To make a proper comparison of data, several considerations have been taken, like camera linearity, intensity difference between the wavebands from the PLS and non uniformity of the light intensity range in the images. So, a set of measures were done for calibrating the system. An automatic detection method based on segmentation by expectation-maximization and ellipse fitting is used. An extension of this method is proposed in order to be applied to multispectral images. The original and the extended method are then applied to the data previously acquired to have first results regarding the effect of using multispectral images rather than color ones.

In wireless communication, multiple receive-antennas are used with orthogonal frequency division multiplexing (OFDM) to improve the system capacity and performance. The discrete Fourier transform (DFT) plays an important part in such a system since the DFTs are required to be performed for the output of all those antennas separately. This paper presents area-time efficient systolic structures for one-dimensional (1-D) and two-dimensional (2-D) DFTs of general lengths. A low-complexity recursive algorithm based on Clenshaw’s recurrence relation is formulated for the computation of 1-D DFT. The proposed algorithm is used further to derive a linear systolic array for the DFT. The concurrency of computation has been enhanced and complexity is minimized by the proposed algorithm where an N −point DFT is computed via four inner-products of real-valued data of length ≈ (N/2). The proposed 1-D structure offers significantly lower latency, twice the throughput, and involves nearly the same area-time complexity of the corresponding existing structures. The proposed algorithm for 1-D DFT is extended further to obtain a 2-D systolic structure for the 2-D DFT without involving any transposition operation.

Purpose: To automatically and efficiently count cell‐clusters of a clonogenic assay using color information and image processing techniques. Method and Materials: A computer algorithm was designed in the Matlab (Natick, MA) programming environment to automatically count cell colonies. In this implementation, six‐well tissue culture plates with stained colonies were scanned using a commercial flatbed scanner. These images were then read into the program as three‐channel (Red, Green, Blue) color images. Color windows were applied to each channel to isolate colony clusters from the background. Region labeling techniques were then used to identify individual cell clusters. Next, individual cluster statistics (area, eccentricity, etc.) were obtained and used to determine if a cluster were a viable colony (>50 cells) and whether it was a single, double or triple colony. The number of colonies counted was compared with the number determined manually. Results: The images obtained from the flatbed scanner were of high quality and were observed to have minimal shadowing effects. The time for the automated colony counting software to analyze and count individual wells was on the order of 7–20 sec. Comparison between manual and computer‐assisted counting showed agreement of 3–5%. The Wilcoxon signed‐rank test showed no statistically significant difference (p > 0.05) between the number of colonies counted manually or with the computer program. Conclusion:Images from the flatbed scanner effectively represent the required colony data, and no advanced equipment is required to record the images. Color range filtering removed the need for most post‐processing techniques and also is an effective method for dealing with errors from shadowing along the edges of the plates. Use of this automated technique will allow for a large quantity of clonogenic survival data to be analyzed in an efficient and consistent manner.

In this paper a novel coded multi-dimensional spreading technique using the Discrete Fourier Transform (DFT) is introduced. It exploits the DFT characteristic of spreading one symbol in one domain to all the symbols in the corresponding transform domain. Frequency domain channel estimation is performed at the receiver. The performance of a coded twodimensional (2D) DFT spreading communication system is investigated and simulation results are presented, using first a 2D DFT time-frequency system and then a 2D DFT space-frequency system. The proposed 2D DFT system shows good immunity to narrowband interference and impulse noise and exhibits a significant performance improvement when compared with a coded conventional OFDM system of equivalent size.

Sparse emission source microscopy (ESM) is an efficient method to identity radiating sources. With the purpose to minimize the number of required measurement points, the presented work investigates how numerical properties of sparse ESM affects the quality of source reconstruction. A simulation model of a simple printed circuit board was used instead of measurements to isolate the observed effect of the two-dimensional (2-D) discrete Fourier transformation (DFT) and the plane wave spectrum's numerical properties. The paper shows that sub-Nyquist is achievable and suggests uniform sampling is superior to nonuniform, in contrast to other reported uses of microwave imaging. Finally, the study shows that if the source reconstruction is based on uniform 2-D DFT care should be taken with the previously suggested intelligent selection of sparse samples based on real-time observation of the measured field.

This paper presents an approach to target detection using discrete Fourier transform (DFT) processing and a stepped frequency waveform. The approach utilizes an advanced mathematical two-way signal model to separate the non-linearity associated with the target returns. This model is based on a three dimensional third-order Taylor series expansion of the round trip delay about incremental azimuth, elevation and time. Once a range bin is established, proper time sampling removes the signal's time dependency. Thus, the two-way phase is proportional to azimuth and elevation increments. Detection is achieved on the basis of target's range, azimuth and elevation angels. Elevation (vertical strip) and azimuth (horizontal strip) profiles of the target's plane are synthesized through a two-dimensional DFT.

A novel region-based color mapping method is proposed to render fused image of visible and infrared (IR). The method is based on image segmentation, region recognition, image fusion and color transfer. Firstly visible image and IR image are fused based on the curvelet transform. At the same time, a color database is formed by grouping a set of natural color images according to their scene contents. A false color image obtained by assigning the source images to RGB channels is segmented with the multiscale normalized cuts method. Then each segmented region is recognized by support vector machine (SVM) and automatically associated with a natural color image. Finally the colors of the natural color image are transferred to the segmented region in a perceptually decorrelated color space. Experiments show that the image colorized by the proposed method resembles the natural color appearance and will help the observer interpreting the scene more intuitively than produced by other coloring methods.

Natural color image enhancement and evaluation algorithm based on human visual system