Multiresolution monogenic signal analysis using the Riesz-Laplace wavelet transform

An explainable and interpretable model for attention deficit hyperactivity disorder in children using EEG signals

BACKGROUND

Attention deficit hyperactivity disorder (ADHD) is a neurodevelopmental disorder that affects a person's sleep, mood, anxiety, and learning. Early diagnosis and timely medication can help individuals with ADHD perform daily tasks without difficulty. Electroencephalogram (EEG) signals can help neurologists to detect ADHD by examining the changes occurring in it. The EEG signals are complex, non-linear, and non-stationary. It is difficult to find the subtle differences between ADHD and healthy control EEG signals visually. Also, making decisions from existing machine learning (ML) models do not guarantee similar performance (unreliable).

METHOD

The paper explores a combination of variational mode decomposition (VMD), and Hilbert transform (HT) called VMD-HT to extract hidden information from EEG signals. Forty-one statistical parameters extracted from the absolute value of analytical mode functions (AMF) have been classified using the explainable boosted machine (EBM) model. The interpretability of the model is tested using statistical analysis and performance measurement. The importance of the features, channels and brain regions has been identified using the glass-box and black-box approach. The model's local and global explainability has been visualized using Local Interpretable Model-agnostic Explanations (LIME), SHapley Additive exPlanations (SHAP), Partial Dependence Plot (PDP), and Morris sensitivity. To the best of our knowledge, this is the first work that explores the explainability of the model prediction in ADHD detection, particularly for children.

RESULTS

Our results show that the explainable model has provided an accuracy of 99.81%, a sensitivity of 99.78%, 99.84% specificity, an F-1 measure of 99.83%, the precision of 99.87%, a false detection rate of 0.13%, and Mathew's correlation coefficient, negative predicted value, and critical success index of 99.61%, 99.73%, and 99.66%, respectively in detecting the ADHD automatically with ten-fold cross-validation. The model has provided an area under the curve of 100% while the detection rate of 99.87% and 99.73% has been obtained for ADHD and HC, respectively.

CONCLUSIONS

The model show that the interpretability and explainability of frontal region is highest compared to pre-frontal, central, parietal, occipital, and temporal regions. Our findings has provided important insight into the developed model which is highly reliable, robust, interpretable, and explainable for the clinicians to detect ADHD in children. Early and rapid ADHD diagnosis using robust explainable technologies may reduce the cost of treatment and lessen the number of patients undergoing lengthy diagnosis procedures.

Motion Magnification of Vibration Image in Estimation of Technical Object Condition-Review

One of the most important features of the proper operation of technical objects is monitoring the vibrations of their mechanical components. The currently significant proportion of the research methods in this regard includes a group of research methods based on the conversion of vibrations using sensors providing data from individual locations. In parallel with the continuous improvement of these tools, new methods for acquiring information on the condition of the object have emerged due to the rapid development of visual systems. Their actual effectiveness determined the switch from research laboratories to actual industrial installations. In many cases, the application of the visualization methods can supplement the conventional methods applied and, under particular conditions, can effectively replace them. The decisive factor is their non-contact nature and the possibility for simultaneous observation of multiple points of the selected area. Visual motion magnification (MM) is an image processing method that involves the conscious and deliberate deformation of input images to the form that enables the visual observation of vibration processes which are not visible in their natural form. The first part of the article refers to the basic terms in the field of expressing motion in an image (based on the Lagrangian and Eulerian approaches), the formulation of the term of optical flow (OF), and the interpretation of an image in time and space. The following part of the article reviews the main processing algorithms in the aspect of computational complexity and visual quality and their modification for applications under specific conditions. The comparison of the MM methods presented in the paper and recommendations for their applications across a wide variety of fields were supported with examples originating from recent publications. The effectiveness of visual methods based on motion magnification in machine diagnosis and the identification of malfunctions are illustrated with selected examples of the implementation derived from authors’ workshop practice under industrial conditions.

Measuring local-directional resolution and local anisotropy in cryo-EM maps

The introduction of local resolution has enormously helped the understanding of cryo-EM maps. Still, for any given pixel it is a global, aggregated value, that makes impossible the individual analysis of the contribution of the different projection directions. We introduce MonoDir, a fully automatic, parameter-free method that, starting only from the final cryo-EM map, decomposes local resolution into the different projection directions, providing a detailed level of analysis of the final map. Many applications of directional local resolution are possible, and we concentrate here on map quality and validation.

It is important to analyse the local resolution of cryo-EM maps. Here the authors present MonoDir, a fully automatic and parameter free method for the directional local resolution analysis of cryo-EM maps that requires only the final map as input and they also propose indicators for assessing map quality.

Phase estimation of a 2D fringe pattern using a monogenic-based multiscale analysis

In this paper, a multiscale monogenic analysis is applied to 2D interference fringe patterns. The monogenic signal was originally developed as a 2D generalization of the well-known analytic signal in the 1D case. The analytic and monogenic tools are both useful to extract phase information, which can then be directly linked with physical quantities. Previous studies have already shown the interest in the monogenic signal in the field of interferometry. This paper presents theoretical and numerical illustrations of the connection between the physical phase information and the phase estimated with the monogenic tool. More specifically, the ideal case of pure cosine waves is deeply studied, and then the complexity of the fringe patterns is progressively increased. One important weakness of the monogenic transform is its singularity at the null frequency, which makes the phase estimations of low-frequency fringes diverge. Moreover, the monogenic transform is originally designed for narrowband signals, and encounters difficulties when dealing with noised signals. These problems can be bypassed by performing a multiscale analysis based on the monogenic wavelet transform. Moreover, this paper proposes a simple strategy to combine the information extracted at different scales in order to get a better estimation of the phase. The numerical tests (synthetic and real signals) show how this approach provides a finer extraction of the geometrical structure of the fringe patterns.

Design of textured multi-layered structures via magnetically assisted slip casting

Multi-layered composites in nature often show functional properties that are determined by the specific orientation of inorganic building blocks within each layer. The shell of bivalve molluscs and the exoskeleton of crustaceans constitute prominent examples. An effective approach to artificially produce textured microstructures inspired by such complex composites is magnetically assisted slip casting (MASC). MASC is a colloidal process in which anisotropic particles are magnetically oriented at arbitrarily defined angles and collected at the surface of a porous mould to grow the material in an additive manner. Whereas a number of proof-of-concept studies have established the potential of the technique, the full design space available for MASC-fabricated structures, and the limits of the approach, have so far not been explored systematically. To fill this gap, we have studied both theoretically and experimentally the various torques that act on the particles at the different stages of the assembly process. We define the boundary conditions of the MASC process for magnetically responsive alumina platelets suspended in a low-viscosity aqueous suspension, considering the composition of the colloidal suspension and the dynamics of the particle alignment process under a rotating magnetic field. These findings lead to design guidelines for the fabrication of bio-inspired composites with customized multi-scale structures for a broad range of applications.

Characterization of Color Images with Multiscale Monogenic Maxima

Can we build a feature-based analysis that fully characterizes images? The literature answers with edge-based reconstruction methods inspired by Marr's paradigm but limited to the greyscale case. This paper studies the color case. A new sparse representation is carried out with the monogenic concept and the Mallat-Zhong wavelet maxima method. Our monogenic maxima provide efficient contour shape and color characterization, as a sparse set of local features including amplitude, phase, orientation and ellipse parameters. This rich description takes the wavelet maxima representation further towards the wide topic of keypoint analysis. We propose a reconstruction process that retrieves the image from its monogenic maxima. While known works all rely on constrained optimization, implying an iterative use of the filterbank, we propose to interpolate the data in the feature domain by exploiting the visual knowledge from the feature-set. This direct retrieval is accurate enough so that no iteration is required. The main question is finally answered with comparative experiments. It is shown that a reasonably small amount of features is sufficiently informative for visually appealing image retrieval. The features appear numerically stable to rotation, and can be intuitively simplified to perform image regularization.

MonoRes: Automatic and Accurate Estimation of Local Resolution for Electron Microscopy Maps

Since the beginning of electron microscopy, resolution has been a critical parameter. In this article, we propose a fully automatic, accurate method for determining the local resolution of a 3D map (MonoRes). The foundation of this algorithm is an extension of the concept of analytic signal, termed monogenic signal. The map is filtered at different frequencies and the amplitude of the monogenic signal is calculated, after which a criterion is applied to determine the resolution at each voxel. MonoRes is fully automatic without compulsory user parameters, with great accuracy in all tests, and is computationally more rapid than existing methods in the field. In addition, MonoRes offers the option of local filtering of the original map based on the calculated local resolution.

Revealing chiral cell motility by 3D Riesz transform-differential interference contrast microscopy and computational kinematic analysis

Left–right asymmetry is a fundamental feature of body plans, but its formation mechanisms and roles in functional lateralization remain unclear. Accumulating evidence suggests that left–right asymmetry originates in the cellular chirality. However, cell chirality has not yet been quantitatively investigated, mainly due to the absence of appropriate methods. Here we combine 3D Riesz transform-differential interference contrast (RT-DIC) microscopy and computational kinematic analysis to characterize chiral cellular morphology and motility. We reveal that filopodia of neuronal growth cones exhibit 3D left-helical motion with retraction and right-screw rotation. We next apply the methods to amoeba Dictyostelium discoideum and discover right-handed clockwise cell migration on a 2D substrate and right-screw rotation of subcellular protrusions along the radial axis in a 3D substrate. Thus, RT-DIC microscopy and the computational kinematic analysis are useful and versatile tools to reveal the mechanisms of left–right asymmetry formation and the emergence of lateralized functions.

The lack of an appropriate method has hampered quantitative measurements of cell chirality. Here, the authors combine Riesz transform-differential interference contrast microscopy and computational kinematic analysis to reveal chiral cell motility of neuronal growth cone filopodia and cellular slime mold.

Classification via Sparse Representation of Steerable Wavelet Frames on Grassmann Manifold: Application to Target Recognition in SAR Image

Automatic target recognition has been widely studied over the years, yet it is still an open problem. The main obstacle consists in extended operating conditions, e.g.., depression angle change, configuration variation, articulation, and occlusion. To deal with them, this paper proposes a new classification strategy. We develop a new representation model via the steerable wavelet frames. The proposed representation model is entirely viewed as an element on Grassmann manifolds. To achieve target classification, we embed Grassmann manifolds into an implicit reproducing Kernel Hilbert space (RKHS), where the kernel sparse learning can be applied. Specifically, the mappings of training sample in RKHS are concatenated to form an overcomplete dictionary. It is then used to encode the counterpart of query as a linear combination of its atoms. By designed Grassmann kernel function, it is capable to obtain the sparse representation, from which the inference can be reached. The novelty of this paper comes from: 1) the development of representation model by the set of directional components of Riesz transform; 2) the quantitative measure of similarity for proposed representation model by Grassmann metric; and 3) the generation of global kernel function by Grassmann kernel. Extensive comparative studies are performed to demonstrate the advantage of proposed strategy.

3D Solid Texture Classification Using Locally-Oriented Wavelet Transforms

Many image acquisition techniques used in biomedical imaging, material analysis, and structural geology are capable of acquiring 3-D solid images. Computational analysis of these images is complex but necessary since it is difficult for humans to visualize and quantify their detailed 3-D content. One of the most common methods to analyze 3-D data is to characterize the volumetric texture patterns. Texture analysis generally consists of encoding the local organization of image scales and directions, which can be extremely diverse in 3-D. Current state-of-the- art techniques face many challenges when working with 3-D solid texture, where most approaches are not able to consistently characterize both scale and directional information. 3-D Riesz- wavelets can deal with both properties. One key property of Riesz filterbanks is steerability, which can be used to locally align the filters and compare textures with arbitrary (local) orientations. This paper proposes and compares three novel local alignment criteria for higher-order 3-D Riesz-wavelet transforms. The estimations of local texture orientations are based on higher- order extensions of regularized structure tensors. An experimental evaluation of the proposed methods for the classification of synthetic 3-D solid textures with alterations (such as rotations and noise) demonstrated the importance of local directional information for robust and accurate solid texture recognition. These alignment methods improved the accuracy of the unaligned Riesz descriptors up to 0.63, from 0.32 to 0.95 over 1 in the rotated data, which is better than all other techniques that are published and tested on the same database.

Improvement of displacement estimation of breast tissue in ultrasound elastography using the monogenic signal

BACKGROUND

In breast ultrasound elastography, tissues displacements estimation is obtained through a technique that follows the evolution of tissues under stress. However, during the acquisition of B-mode images, tissue displacements are often contaminated with multiplicative noise caused by changes in the speckle pattern in the tissue. Thus, the application of monogenic signal technique on the B-mode image in order to estimate displacement tissue, result in a presence of amplified noise in the deformation tissue image, which severely obscures the useful information. In this paper, we propose a new method based on the monogenic features, that is to improve the old monogenic signal (OMS) technique by improving the filtering step, so that the use of an effective denoising technique is enough to ensure a good estimation of displacement tissue. Our proposed method is based on the use of a robust filtering technique combined with the monogenic model.

METHODS

Two models of phantom elasticity are used in our test validation sold by CIRS company. In-vivo testing was also performed on the sets of clinical B-mode images to 20 patients including malignant breast tumors. Shrinkage wavelets has been used to eliminate the noise according to the threshold, then a guided filter is introduced to completely filter the image, the monogenic model is used after excerpting the image feature and estimating analytically the displacement tissue.

RESULTS

Accurate and excellent displacement estimation for breast tissue was observed in proposed method results. By adapting our proposed approach to breast B-mode images, we have shown that it demonstrated a higher performance for displacement estimation; it gives better values in term of standard deviation, higher contrast to noise ratio, greater peak signal-to-noise ratio, excellent structural similarity and much faster speed than OMS and B-spline techniques. The results of the proposed model are encouraging, allowing quick and reliable estimations.

CONCLUSION

Although the proposed approach is used in ultrasound domains, it has never been used in the estimation of the breast tissue displacement. In this context, our proposed approach could be a powerful diagnostic tool to be used in breast displacement estimation in ultrasound elastography.

A 3-D Riesz-Covariance Texture Model for Prediction of Nodule Recurrence in Lung CT

This paper proposes a novel imaging biomarker of lung cancer relapse from 3-D texture analysis of CT images. Three-dimensional morphological nodular tissue properties are described in terms of 3-D Riesz-wavelets. The responses of the latter are aggregated within nodular regions by means of feature covariances, which leverage rich intra- and inter-variations of the feature space dimensions. When compared to the classical use of the average for feature aggregation, feature covariances preserve spatial co-variations between features. The obtained Riesz-covariance descriptors lie on a manifold governed by Riemannian geometry allowing geodesic measurements and differentiations. The latter property is incorporated both into a kernel for support vector machines (SVM) and a manifold-aware sparse regularized classifier. The effectiveness of the presented models is evaluated on a dataset of 110 patients with non-small cell lung carcinoma (NSCLC) and cancer recurrence information. Disease recurrence within a timeframe of 12 months could be predicted with an accuracy of 81.3-82.7%. The anatomical location of recurrence could be discriminated between local, regional and distant failure with an accuracy of 78.3-93.3%. The obtained results open novel research perspectives by revealing the importance of the nodular regions used to build the predictive models.

Enhanced synthetic aperture radar automatic target recognition method based on novel features

This paper proposes a set of uncommonly rich feature representations for automatic target recognition (ATR) in synthetic aperture radar (SAR) images. The proposed novel feature representations capture both the spatial and spectral properties of a target in a unified framework, while simultaneously offering discrimination and robustness to aspect variations. Specifically, the proposed features are mainly derived from the ideas of the monogenic signal and polar mapping. The applicability of the monogenic signal within the field of SAR target recognition is demonstrated by its capability of capturing both the broad spectral information and spatial localization with compact support. Further, to reduce the influence of inevitable variations due to aspect changes in SAR images, the monogenic components are transformed from Cartesian to polar coordinates through polar mapping. Additionally, a new target-shadow feature is also presented to compensate for the important discriminative information about target geometry, which exists in the shadow area. Finally, the proposed features are jointly considered into a unified multiple kernel learning framework for target recognition. Experiments on the moving and stationary target acquisition and recognition (MSTAR) public dataset demonstrate the strength and applicability of the proposed representations to SAR ATR. Moreover, it is also shown that overall high recognition accuracy can be obtained by the established unified framework.

Wideband image demodulation via bi-dimensional multirate frequency transformations

Existing image demodulation approaches based on the two-dimensional (2D) multicomponent AM-FM model assume narrowband components that can be demodulated using energy operators, Hilbert transforms, or the monogenic image approaches. However, if the FM components are wideband, then these demodulation approaches incur significant errors. Recent work by the authors extended wideband FM demodulation in one dimension to accommodate large conversion factors using multirate frequency transformations. In this paper, we extend the multirate frequency transformations technique developed for one-dimensional signals to 2D and images in conjunction with a recently proposed 2D higher-order energy demodulation approach. This extension is applied to both synthetic and real images to demonstrate the efficacy of the approach.

Filter Design and Performance Evaluation for Fingerprint Image Segmentation

Fingerprint recognition plays an important role in many commercial applications and is used by millions of people every day, e.g. for unlocking mobile phones. Fingerprint image segmentation is typically the first processing step of most fingerprint algorithms and it divides an image into foreground, the region of interest, and background. Two types of error can occur during this step which both have a negative impact on the recognition performance: 'true' foreground can be labeled as background and features like minutiae can be lost, or conversely 'true' background can be misclassified as foreground and spurious features can be introduced. The contribution of this paper is threefold: firstly, we propose a novel factorized directional bandpass (FDB) segmentation method for texture extraction based on the directional Hilbert transform of a Butterworth bandpass (DHBB) filter interwoven with soft-thresholding. Secondly, we provide a manually marked ground truth segmentation for 10560 images as an evaluation benchmark. Thirdly, we conduct a systematic performance comparison between the FDB method and four of the most often cited fingerprint segmentation algorithms showing that the FDB segmentation method clearly outperforms these four widely used methods. The benchmark and the implementation of the FDB method are made publicly available.

Photometric invariant stereo matching method

A robust stereo matching method based on a comprehensive mathematical model for color formation process is proposed to estimate the disparity map of stereo images with noise and photometric variations. The band-pass filter with DoP kernel is firstly used to filter out noise component of the stereo images. Then the log-chromaticity normalization process is applied to eliminate the influence of lightning geometry. All the other factors that may influence the color formation process are removed through the disparity estimation process with a specific matching cost. Performance of the developed method is evaluated by comparing with some up-to-date algorithms. Experimental results are presented to demonstrate the robustness and accuracy of the method.

Identification of Buried Objects in GPR Using Amplitude Modulated Signals Extracted from Multiresolution Monogenic Signal Analysis

It is necessary to detect the target reflections in ground penetrating radar (GPR) images, so that surface metal targets can be identified successfully. In order to accurately locate buried metal objects, a novel method called the Multiresolution Monogenic Signal Analysis (MMSA) system is applied in ground penetrating radar (GPR) images. This process includes four steps. First the image is decomposed by the MMSA to extract the amplitude component of the B-scan image. The amplitude component enhances the target reflection and suppresses the direct wave and reflective wave to a large extent. Then we use the region of interest extraction method to locate the genuine target reflections from spurious reflections by calculating the normalized variance of the amplitude component. To find the apexes of the targets, a Hough transform is used in the restricted area. Finally, we estimate the horizontal and vertical position of the target. In terms of buried object detection, the proposed system exhibits promising performance, as shown in the experimental results.

Classification on the monogenic scale space: application to target recognition in SAR image

This paper introduces a novel classification strategy based on the monogenic scale space for target recognition in Synthetic Aperture Radar (SAR) image. The proposed method exploits monogenic signal theory, a multidimensional generalization of the analytic signal, to capture the characteristics of SAR image, e.g., broad spectral information and simultaneous spatial localization. The components derived from the monogenic signal at different scales are then applied into a recently developed framework, sparse representation-based classification (SRC). Moreover, to deal with the data set, whose target classes are not linearly separable, the classification via kernel combination is proposed, where the multiple components of the monogenic signal are jointly considered into a unifying framework for target recognition. The novelty of this paper comes from: the development of monogenic feature via uniformly downsampling, normalization, and concatenation of the components at various scales; the development of score-level fusion for SRCs; and the development of composite kernel learning for classification. In particular, the comparative experimental studies under nonliteral operating conditions, e.g., structural modifications, random noise corruption, and variations in depression angle, are performed. The comparative experimental studies of various algorithms, including the linear support vector machine and the kernel version, the SRC and the variants, kernel SRC, kernel linear representation, and sparse representation of monogenic signal, are performed too. The feasibility of the proposed method has been successfully verified using Moving and Stationary Target Acquiration and Recognition database. The experimental results demonstrate that significant improvement for recognition accuracy can be achieved by the proposed method in comparison with the baseline algorithms.

Experimental investigation on butane diffusion flames under the influence of magnetic field by using digital speckle pattern interferometry

In this paper, the effect of magnetic fields on the temperature and temperature profile of a diffusion flame obtained from a butane torch burner are investigated experimentally by using digital speckle pattern interferometry (DSPI). Experiments were conducted on a diffusion flame generated by a butane torch burner in the absence of a magnetic field and in the presence of uniform and nonuniform magnetic fields. A single DSPI fringe pattern was used to extract phase by using a Riesz transform and monogenic signal. Temperature inside the flame was determined experimentally both in the absence and in the presence of magnetic fields. Experimental results reveal that the maximum temperature of the flame is increased under the influence of an upward-decreasing magnetic gradient and decreased under an upward-increasing magnetic gradient while a negligible effect on temperature in a uniform magnetic field was observed.

2D Prony-Huang transform: a new tool for 2D spectral analysis

This paper provides an extension of the 1D Hilbert Huang transform for the analysis of images using recent optimization techniques. The proposed method consists of: 1) adaptively decomposing an image into oscillating parts called intrinsic mode functions (IMFs) using a mode decomposition procedure and 2) providing a local spectral analysis of the obtained IMFs in order to get the local amplitudes, frequencies, and orientations. For the decomposition step, we propose two robust 2D mode decompositions based on nonsmooth convex optimization: 1) a genuine 2D approach, which constrains the local extrema of the IMFs and 2) a pseudo-2D approach, which separately constrains the extrema of lines, columns, and diagonals. The spectral analysis step is an optimization strategy based on Prony annihilation property and applied on small square patches of the IMFs. The resulting 2D Prony–Huang transform is validated on simulated and real data.

3D steerable wavelets in practice

We introduce a systematic and practical design for steerable wavelet frames in 3D. Our steerable wavelets are obtained by applying a 3D version of the generalized Riesz transform to a primary isotropic wavelet frame. The novel transform is self-reversible (tight frame) and its elementary constituents (Riesz wavelets) can be efficiently rotated in any 3D direction by forming appropriate linear combinations. Moreover, the basis functions at a given location can be linearly combined to design custom (and adaptive) steerable wavelets. The features of the proposed method are illustrated with the processing and analysis of 3D biomedical data. In particular, we show how those wavelets can be used to characterize directional patterns and to detect edges by means of a 3D monogenic analysis. We also propose a new inverse-problem formalism along with an optimization algorithm for reconstructing 3D images from a sparse set of wavelet-domain edges. The scheme results in high-quality image reconstructions which demonstrate the feature-reduction ability of the steerable wavelets as well as their potential for solving inverse problems.

Steerable pyramids and tight wavelet frames in L2(R(d))

We present a functional framework for the design of tight steerable wavelet frames in any number of dimensions. The 2-D version of the method can be viewed as a generalization of Simoncelli's steerable pyramid that gives access to a larger palette of steerable wavelets via a suitable parametrization. The backbone of our construction is a primal isotropic wavelet frame that provides the multiresolution decomposition of the signal. The steerable wavelets are obtained by applying a one-to-many mapping (Nth-order generalized Riesz transform) to the primal ones. The shaping of the steerable wavelets is controlled by an M×M unitary matrix (where M is the number of wavelet channels) that can be selected arbitrarily; this allows for a much wider range of solutions than the traditional equiangular configuration (steerable pyramid). We give a complete functional description of these generalized wavelet transforms and derive their steering equations. We describe some concrete examples of transforms, including some built around a Mallat-type multiresolution analysis of L(2)(R(d)), and provide a fast Fourier transform-based decomposition algorithm. We also propose a principal-component-based method for signal-adapted wavelet design. Finally, we present some illustrative examples together with a comparison of the denoising performance of various brands of steerable transforms. The results are in favor of an optimized wavelet design (equalized principal component analysis), which consistently performs best.

Space-time stereo analysis combining local structure and modulation features in the monogenic wavelet domain

A multimodal cost function, based on the local multimodal image descriptors that combine local structure features (orientation, coherency) and modulation/localization information (amplitude, phase, and spatial frequency) of the monogenic wavelet transform, is proposed to estimate the time-varying disparity maps in the space-time stereo framework. The proposed cost function makes use of a constraint of local orientation, phase, and amplitude congruencies with the weighted coefficients, which are adapted to local image features and are insusceptible to level shift, scaling, and rotation and lighting invariance. Experiments on the synthetic and natural stereo sequences show the estimated results are more robust than the intensity-based sum of standard sum of squared difference cost function.

Steerable wavelet frames based on the Riesz transform

We consider an extension of the 1-D concept of analytical wavelet to n-D which is by construction compatible with rotations. This extension, called a monogenic wavelet, yields a decomposition of the wavelet coefficients into amplitude, phase, and phase direction. The monogenic wavelet is based on the hypercomplex monogenic signal which is defined using Riesz transforms and perfectly isotropic wavelets frames. Employing the new concept of Clifford frames, we can show that the monogenic wavelet generates a wavelet frame. Furthermore, this approach yields wavelet frames that are steerable with respect to direction. Applications to descreening and contrast enhancement illustrate the versatility of this approach to image analysis and reconstruction.

Wavelet steerability and the higher-order Riesz transform

Our main goal in this paper is to set the foundations of a general continuous-domain framework for designing steerable, reversible signal transformations (a.k.a. frames) in multiple dimensions ( d >or= 2). To that end, we introduce a self-reversible, Nth-order extension of the Riesz transform. We prove that this generalized transform has the following remarkable properties: shift-invariance, scale-invariance, inner-product preservation, and steerability. The pleasing consequence is that the transform maps any primary wavelet frame (or basis) of [Formula: see text] into another "steerable" wavelet frame, while preserving the frame bounds. The concept provides a functional counterpart to Simoncelli's steerable pyramid whose construction was primarily based on filterbank design. The proposed mechanism allows for the specification of wavelets with any order of steerability in any number of dimensions; it also yields a perfect reconstruction filterbank algorithm. We illustrate the method with the design of a novel family of multidimensional Riesz-Laplace wavelets that essentially behave like the N th-order partial derivatives of an isotropic Gaussian kernel.

Robust stereo image matching using a two-dimensional monogenic wavelet transform

A robust approach based on the 2D monogenic wavelet transform (MWT), which pairs the polyharmonic B-spline wavelet basis with its complex Riesz counterparts to specify a multiresolution monogenic signal analysis, is proposed to solve the general stereo image matching problem. The disparity is directly estimated by establishing correspondences between the monogenic signal components according to the suitable local properties, i.e., the monogenic wavelet annihilates antimonogenic signals, the MWT is phase-shift covariant and the transform magnitude is phase-shift invariant. This approach is different from the current matching techniques and is promising for image registration, flow estimation, and 3D reconstruction.