Fast Splitting$alpha$-Rooting Method of Image Enhancement: Tensor Representation

In the tensor representation, a two-dimensional (2-D) image is represented uniquely by a set of one-dimensional (1-D) signals, so-called splitting-signals, that carry the spectral information of the image at frequency-points of specific sets that cover the whole domain of frequencies. The image enhancement is thus reduced to processing splitting-signals and such process requires a modification of only a few spectral components of the image, for each signal. For instance, the$alpha$-rooting method of image enhancement can be fulfilled through processing separately a maximum of$3N/2$splitting-signals of an image$(Ntimes N)$, where$N$is a power of two. In this paper, we propose a fast implementation of the$alpha$-rooting method by using one splitting-signal of the tensor representation with respect to the discrete Fourier transform (DFT). The implementation is described in the frequency and spatial domains. As a result, the proposed algorithms for image enhancement use two 1-D$N$-point DFTs instead of two 2-D$Ntimes N$-point DFTs in the traditional method of$alpha$-rooting.     

VOIR: a volumetric image reconstruction algorithm based on Fouriertechniques for inversion of the 3-D Radon transform

A novel volumetric image reconstruction algorithm known as VOIR is presented for inversion of the 3-D Radon transform or its radial derivative. The algorithm is a direct implementation of the projection slice theorem for plane integrals. It generalizes one of the most successful methods in 2-D Fourier image reconstruction involving concentric-square rasters to 3-D; in VOIR, the spectral data, which is calculated by fast Fourier techniques, lie on concentric cubes and are interpolated by a bilinear method on the sides of these concentric cubes. The algorithm has great computational advantages over filtered-backprojection algorithms; for images of side dimension N, the numerical complexity of VOIR is O(N(3) log N) instead of O(N (4)) for backprojection techniques. An evaluation of the image processing performance is reported by comparison of reconstructed images from simulated cone-beam scans of a contrast and resolution test object. The image processing performance is also characterized by an analysis of the edge response from the reconstructed images.     

Fast computation of the two-dimensional generalised Hartleytransforms

The two-dimensional generalised Hartley transforms (2-D GDHTs) are various half-sample generalised DHTs, and are used for computing the 2-D DHT and 2-D convolutions. Fast computation of 2-D GDHTs is achieved by solving (n₁+(n₀₁/2))k₁+(n₂+(n₀₂ /2))k₂=(n+(½))k mod N, n₀₁, n₀₂ =1 or 0. The kernel indexes on the left-hand side and on the right-hand side belong to the 2-D GDHTs and the 1-D H₃, respectively. This equation categorises N×N-point input into N groups which are the inputs of a 1-D N-point H₃. By decomposing to 2-D GDHTs, an N×N-point DHT requires a 3N/2ⁱ 1-D N/2ⁱ-point H₃, i=1, ..., log₂N-2. Thus, it has not only the same number of multiplications as that of the discrete Radon transform (DRT) and linear congruence, but also has fewer additions than the DRT. The distinct H ₃ transforms are independent, and hence parallel computation is feasible. The mapping is very regular, and can be extended to an n-dimensional GDHT or GDFT easily     

A fundamental theorem of algebra, spectral factorization, and stability of two-dimensional systems

In his doctoral dissertation in 1797, Gauss proved the fundamental theorem of algebra, which states that any one-dimensional (1-D) polynomial of degree n with complex coefficients can be factored into a product of n polynomials of degree 1. Since then, it has been an open problem to factorize a two-dimensional (2-D) polynomial into a product of basic polynomials. Particularly for the last three decades, this problem has become more important in a wide range of signal and image processing such as 2-D filter design and 2-D wavelet analysis. In this paper, a fundamental theorem of algebra for 2-D polynomials is presented. In applications such as 2-D signal and image processing, it is often necessary to find a 2-D spectral factor from a given 2-D autocorrelation function. In this paper, a 2-D spectral factorization method is presented through cepstral analysis. In addition, some algorithms are proposed to factorize a 2-D spectral factor finely. These are applied to deriving stability criteria of 2-D filters and nonseparable 2-D wavelets and to solving partial difference equations and partial differential equations.     

Realizations of fast 2-D/3-D image filtering and enhancement

This paper proposes a novel algorithm for multidimensional image enhancement based on a fuzzy domain enhancement method, and an implementation of a recursive and separable low-pass filter. Considering a smoothed image as a fuzzy data set, each pixel in an image is processed independently, using fuzzy domain transformation and enhancement of both the dynamic range and the local gray level variations. The algorithm has the advantages of being fast and adaptive, so it can be used in real-time image processing applications and for multidimensional data with low computational cost. It also has the ability to reduce noise and unwanted background that may affect the visualization quality of two-dimensional (2-D)/three-dimensional (3-D) data. Examples for the applications of the algorithm are given for mammograms, ultrasound 3-D images, and photographic images.     

Significance of image representation for face verification

In this paper we discuss the significance of representation of images for face verification. We consider three different representations, namely, edge gradient, edge orientation and potential field derived from the edge gradient. These representations are examined in the context of face verification using a specific type of correlation filter, called the minimum average correlation energy (MACE) filter. The different representations are derived using one-dimensional (1-D) processing of image. The 1-D processing provides multiple partial evidences for a given face image, one evidence for each direction of the 1-D processing. Separate MACE filters are used for deriving each partial evidence. We propose a method to combine the partial evidences obtained for each representation using an auto-associative neural network (AANN) model, to arrive at a decision for face verification. Results show that the performance of the system using potential field representation is better than that using the edge gradient representation or the edge orientation representation. Also, the potential field representation derived from the edge gradient is observed to be less sensitive to variation in illumination compared to the gray level representation of images.     

An OFDM scheme with a half complexity

The paper deals with an OFDM (orthogonal frequency division multiplexing) system based on filter-bank architecture. The known implementation uses a DFT (discrete Fourier transform) processor and a polyphase network (PPN). Even if it is based on complex components, in the final step it operates the real part extraction of the incoming signal. This leads to redundant operations in the DFT processor and in the PPN. Specifically, for the transmission of N complex symbol sequences at a given rate ½F, an N-point DFT processor and an N-branch PPN, both working at the rate F, are required. This implementation can be improved with a complexity reduction by a factor of two. In fact, in the paper an architecture is presented based on an N-point DFT processor and N-branch PPN at the rate F, for the transmission of 2N (in place of N) complex symbol sequences at the rate ½F     

Efficient algorithms for computing the 2-D hexagonal Fouriertransforms

In this paper, representations of the two-dimensional (2-D) signals are presented that reduce the computation of 2-D discrete hexagonal Fourier transforms (2-D DHFTs). The representations are based on the concept of the covering that reveals the mathematical structure of the transforms. Specifically, a set of unitary paired transforms is derived that splits the 2-D DHFT into a number of smaller one-dimensional (1-D) DFTs. Examples for the 8×4 and 16×8 hexagonal lattices are described in detail. The number of multiplications required for computing the 8×4- and 16×8-point DHFTs are equal 20 and 136, respectively. In the general N⩾8 case, the number of multiplications required to compute the 2N×N-point DHFT by the paired transforms equals N² (log N-1)+N     

Real-time computation of two-dimensional moments on binary imagesusing image block representation

This work presents a new approach and an algorithm for binary image representation, which is applied for the fast and efficient computation of moments on binary images. This binary image representation scheme is called image block representation, since it represents the image as a set of nonoverlapping rectangular areas. The main purpose of the image block representation process is to provide an efficient binary image representation rather than the compression of the image. The block represented binary image is well suited for fast implementation of various processing and analysis algorithms in a digital computing machine. The two-dimensional (2-D) statistical moments of the image may be used for image processing and analysis applications. A number of powerful shape analysis methods based on statistical moments have been presented, but they suffer from the drawback of high computational cost. The real-time computation of moments in block represented images is achieved by exploiting the rectangular structure of the blocks.     

Simplified square to hexagonal lattice conversion based on 1-D multirate processing

Hexagonal image sampling and processing are theoretically superior to the most commonly used square lattice based sampling and processing, but due to the lack of commercial image sensors, current research mainly relies on virtually hexagonally sampled data through square to hexagonal lattice conversion, which is a typical 2-D interpolation problem. This paper presents a simplified and efficient square to hexagonal lattice conversion method. The method firstly utilizes the separable nature of the interpolation kernel to simplify the original 2-D interpolation into 1-D interpolation along the horizontal direction only, and then it applies the 1-D multirate technique to further simplify the shift-variant 1-D interpolation into shift-invariant 1-D convolutions. Compared with the original 2-D interpolation version, the proposed method becomes both simple and computationally efficient, and it is also suitable for implementation with parallel processing and hardware. Finally, experiments are performed and the results are consistent with the analysis.     

Duality of log-polar image representations in the space andspatial-frequency domains

In this paper, we study the result of applying a lowpass variant filtering using scaling-rotating kernels to both the spatial and spatial-frequency representations of a two-dimensional (2-D) signal (image). It is shown that if we apply this transformation to a Fourier pair, the two resulting signals can also form a Fourier pair when the filters used in each domain maintain a dual relationship. For a large class of “self-dual” filters, a perfect symmetry exists, so that the lowpass scaling-rotating variant filtering (SRVF) is the same in both domains, thus commuting with the Fourier transform operator. The lowpass SRVF of an image is often referred to as a “foveated” image, whereas its Fourier pair (the lowpass SRVF of its spectrum) can be realized as a local spectrum estimation around the point of attention. This lowpass SRVF is equivalent to a log-polar warping of the image representation followed by a lowpass invariant filtering and the corresponding inverse warping. The use of the log-polar warped representation allows us to extend the one-dimensional (1-D) scale transform to higher dimensions, in particular to images, for which we have defined a scale-rotation invariant representation. We also present an efficient implementation using steerable filters to compute both the foveated image and the local spectrum     

Nonlinear anisotropic filtering of MRI data

In contrast to acquisition-based noise reduction methods a postprocess based on anisotropic diffusion is proposed. Extensions of this technique support 3-D and multiecho magnetic resonance imaging (MRI), incorporating higher spatial and spectral dimensions. The procedure overcomes the major drawbacks of conventional filter methods, namely the blurring of object boundaries and the suppression of fine structural details. The simplicity of the filter algorithm permits an efficient implementation, even on small workstations. The efficient noise reduction and sharpening of object boundaries are demonstrated by applying this image processing technique to 2-D and 3-D spin echo and gradient echo MR data. The potential advantages for MRI, diagnosis, and computerized analysis are discussed in detail.     

Semi-Supervised Bilinear Subspace Learning

Recent research has demonstrated the success of tensor based subspace learning in both unsupervised and supervised configurations (e.g., 2-D PCA, 2-D LDA, and DATER). In this correspondence, we present a new semi-supervised subspace learning algorithm by integrating the tensor representation and the complementary information conveyed by unlabeled data. Conventional semi-supervised algorithms mostly impose a regularization term based on the data representation in the original feature space. Instead, we utilize graph Laplacian regularization based on the low-dimensional feature space. An iterative algorithm, referred to as adaptive regularization based semi-supervised discriminant analysis with tensor representation (ARSDA/T), is also developed to compute the solution. In addition to handling tensor data, a vector-based variant (ARSDA/V) is also presented, in which the tensor data are converted into vectors before subspace learning. Comprehensive experiments on the CMU PIE and YALE-B databases demonstrate that ARSDA/T brings significant improvement in face recognition accuracy over both conventional supervised and semi-supervised subspace learning algorithms.     

NixSiO2（1—x）颗粒膜的光学和磁光特性研究

用磁控溅射法制备了一系列NixSiO2(1-x)样品,并对部分样品作快速退火处理,室温下采用椭圆偏振光谱仪和磁光谱仪分别为1．5－4．5eV的光子能量区了样品的复介常数谱和极向复磁光克尔谱,研究了这种金属－绝缘体型颗粒膜的化学和磁光性质,发现调整适合的金属含量或对样品作退火处理,可以观察到复介电常数的实部从正到负的连续变化,而且在一定光子能量区,其值为零;介电张量的非对角元和光学常数对其磁光克尔角的增强起重要作用。     

The design of two-dimensional gradient estimators based onone-dimensional operators

Two-dimensional (2-D) gradient estimators are some of the most useful tools in image processing. A computational procedure for the extension of one-dimensional (1-D) gradient estimators to two dimensions (2-D) is presented. The procedure is equivalent to the surface fitting method. It is, however, simpler in design, as the design is 1-D rather than 2-D. Higher order derivative estimators can also be constructed by the same procedure.     

The design of 2-D nonseparable directional perfect reconstruction filter banks

In this paper, we develop a directional 2-D nonseparable filter bank that can perfectly reconstruct the downsampled subband signals. The filter bank represents two powerful image and video processing tools: directional subband decomposition and perfect reconstruction. The directional filter banks consist of (1) the input signal and the subband signals modulation, (2) diamond shape prefilter, and (3) four different parallelogram shape prefilters. This paper addresses the design and implementation of a two-band filter bank that is proved to be able to provide perfect reconstruction of the downsampled subband signals. Finally, we use a conventional 1-D half-band filter as a prototype and then apply the McClellan transform for the specific 2-D diamond shape and parallelogram shape subfilters. This method is extremely simple in designing the analysis/synthesis subfilters for the filter bank.     

Two-dimensional affine generalized fractional Fourier transform

As the one-dimensional (1-D) Fourier transform can be extended into the 1-D fractional Fourier transform (FRFT), we can also generalize the two-dimensional (2-D) Fourier transform. Sahin et al. (see Appl. Opt., vol.37, no. 11, p.2130-41, 1998) have generalized the 2-D Fourier transform into the 2-D separable FRFT (which replaces each variable 1-D Fourier transform by the 1-D FRFT, respectively) and the 2-D separable canonical transform (further replaces FRFT by the canonical transform). Sahin et al., (see Appl. Opt., vol.31, no.23, p.5444-53, 1998), have also generalized it into the 2-D unseparable FRFT with four parameters. In this paper, we introduce the 1-D affine generalized fractional Fourier transform (AGFFT). It has even further extended the 2-D transforms described above. It is unseparable, and has, in total, ten degrees of freedom. We show that the 2-D AGFFT has many wonderful properties, such as the relations with the Wigner distribution, shifting-modulation operation, and the differentiation-multiplication operation. Although the 2-D AGFFT form seems very complex, in fact, the complexity of the implementation will not be more than the implementation of the 2-D separable FRFT. Besides, we also show that the 2-D AGFFT extends many of the applications for the 1-D FRFT, such as the filter design, optical system analysis, image processing, and pattern recognition and will be a very useful tool for 2-D signal processing     

Directionlets: anisotropic multidirectional representation with separable filtering.

In spite of the success of the standard wavelet transform (WT) in image processing in recent years, the efficiency of its representation is limited by the spatial isotropy of its basis functions built in the horizontal and vertical directions. One-dimensional (1-D) discontinuities in images (edges and contours) that are very important elements in visual perception, intersect too many wavelet basis functions and lead to a nonsparse representation. To efficiently capture these anisotropic geometrical structures characterized by many more than the horizontal and vertical directions, a more complex multidirectional (M-DIR) and anisotropic transform is required. We present a new lattice-based perfect reconstruction and critically sampled anisotropic M-DIR WT. The transform retains the separable filtering and subsampling and the simplicity of computations and filter design from the standard two-dimensional WT, unlike in the case of some other directional transform constructions (e.g., curvelets, contourlets, or edgelets). The corresponding anisotropic basis unctions (directionlets) have directional vanishing moments along any two directions with rational slopes. Furthermore, we show that this novel transform provides an efficient tool for nonlinear approximation of images, achieving the approximation power O(N(-1.55)), which, while slower than the optimal rate O(N(-2)), is much better than O(N(-1)) achieved with wavelets, but at similar complexity.     

VLSI implementation of a focal plane image processor-a realizationof the near-sensor image processing concept

The near-sensor image processing concept, which has earlier been theoretically described, is here verified with an implementation. The NSIP describes a method to implement a two-dimensional (2-D) image sensor array with processing capacity in every pixel. Traditionally, there is a contradiction between high spatial resolution and complex processor elements, In the NSIP concept we have a nondestructive photodiode readout and we can thereby process binary images without loosing gray-scale information. The global image processing is handled by an asynchronous Global Logical Unit. These two features makes it possible to have efficient image processing in a small processor element. Electrical problems such as power consumption and fixed pattern noise are solved. All design is aimed at a 128×128 pixels NSIP in a 0.8 μm double-metal single-poly CMOS process. We have fabricated and measured a 32×32 pixels NSIP. We also give examples of image processing tasks such as gradient and maximum detection, histogram equalization, and thresholding with hysteresis. In the NSIP concept automatic light adaptivity within a 160 dB range is possible     

Shape-based normalization of the corpus callosum for DTI connectivity analysis

The continuous medial representation (cm-rep) is an approach that makes it possible to model, normalize, and analyze anatomical structures on the basis of medial geometry. Having recently presented a partial differential equation (PDE)-based approach for 3-D cm-rep modeling [1], here we present an equivalent 2-D approach that involves solving an ordinary differential equation. This paper derives a closed form solution of this equation and shows how Pythagorean hodograph curves can be used to express the solution as a piecewise polynomial function, allowing efficient and robust medial modeling. The utility of the approach in medical image analysis is demonstrated by applying it to the problem of shape-based normalization of the midsagittal section of the corpus callosum. Using diffusion tensor tractography, we show that shape-based normalization aligns subregions of the corpus callosum, defined by connectivity, more accurately than normalization based on volumetric registration. Furthermore, shape-based normalization helps increase the statistical power of group analysis in an experiment where features derived from diffusion tensor tractography are compared between two cohorts. These results suggest that cm-rep is an appropriate tool for normalizing the corpus callosum in white matter studies.