Steerable wavelet frames based on the Riesz transform

Separable Paravector Orientation Tensors for Enhancing Retinal Vessels

Robust detection of retinal vessels remains an unsolved research problem, particularly in handling the intrinsic real-world challenges of highly imbalanced contrast between thick vessels and thin ones, inhomogeneous background regions, uneven illumination, and complex geometries of crossing/bifurcations. This paper presents a new separable paravector orientation tensor that addresses these difficulties by characterizing the enhancement of retinal vessels to be dependent on a nonlinear scale representation, invariant to changes in contrast and lighting, responsive for symmetric patterns, and fitted with elliptical cross-sections. The proposed method is built on projecting vessels as a 3D paravector valued function rotated in an alpha quarter domain, providing geometrical, structural, symmetric, and energetic features. We introduce an innovative symmetrical inhibitory scheme that incorporates paravector features for producing a set of directional contrast-independent elongated-like patterns reconstructing vessel tree in orientation tensors. By fitting constraint elliptical volumes via eigensystem analysis, the final vessel tree is produced with a strong and uniform response preserving various vessel features. The validation of proposed method on clinically relevant retinal images with high-quality results, shows its excellent performance compared to the state-of-the-art benchmarks and the second human observers.

Wavelet Guided 3D Deep Model to improve Dental Microfracture Detection

Epidemiological studies indicate that microfractures (cracks) are the third most common cause of tooth loss in industrialized countries. An undetected crack will continue to progress, often with significant pain, until the tooth is lost. Previous attempts to utilize cone beam computed tomography (CBCT) for detecting cracks in teeth had very limited success. We propose a model that detects cracked teeth in high resolution (hr) CBCT scans by combining signal enhancement with a deep CNNbased crack detection model. We perform experiments on a dataset of 45 ex-vivo human teeth with 31 cracked and 14 controls. We demonstrate that a model that combines classical wavelet-based features with a deep 3D CNN model can improve fractured tooth detection accuracy in both micro-Computed Tomography (ground truth) and hr-CBCT scans. The CNN model is trained to predict a probability map showing the most likely fractured regions. Based on this fracture probability map we detect the presence of fracture and are able to differentiate a fractured tooth from a control tooth. We compare these results to a 2D CNN-based approach and we show that our approach provides superior detection results. We also show that the proposed solution is able to outperform oral and maxillofacial radiologists in detecting fractures from the hr-CBCT scans. Early detection of cracks will lead to the design of more appropriate treatments and longer tooth retention.

Two-dimensional iterative projection method for subsample speckle tracking of ultrasound images

Speckle tracking provides robust motion estimation necessary to create accurate post-processed images. These methods are known to be less accurate in the lateral dimension compared with the axial dimension due to the limitations on the lateral resolution of ultrasound scanning. This paper proposes a two-dimensional iterative projection (TDIP) algorithm using the Riesz transform to generate the analytic signals. The TDIP is an improvement to an already accurate speckle tracking algorithm called the phase coupled (PC) method. The PC method projects the intersection of gradients on the correlation map to the zero phase contour to estimate displacement. The TDIP method performs iterative projections and uses the aggregate of these projected locations to estimate the motion, in addition to rejecting inaccurate projections by checking them against the aggregate projection location. The TDIP additionally adopts the Riesz transform to generate two-dimensional analytic signals to improve lateral accuracy. The Riesz transform is a multidimensional extension of the Hilbert transform into the nD Euclidean space and therefore can be used to include data in both axial and lateral dimensions as opposed to the commonly used Hilbert transform which is one dimensional. The accuracy of the TDIP is quantitatively proven on simulated datasets from the Field II simulation program and on experimental data from two flow phantoms. At all cases, the TDIP is more accurate than the PC algorithm at two-dimensional displacement estimation. Graphical Abstract The lateral estimates from the Phase Coupled algorithm. This method uses the Hilbert transform for the analytic signal. There is large estimated motion within the flow blockage bounded by the red, fin shape in the center of the flow channel. The flow channel itself is also bounded by dashed, red lines. Graphical Abstract The lateral estimates from the TDIP method. This method is not tracking motion within the blockage in the center of the flow channel. The channel and the blockage are both bounded by dashed, red lines.

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.

Automatic quantification framework to detect cracks in teeth

Studies show that cracked teeth are the third most common cause for tooth loss in industrialized countries. If detected early and accurately, patients can retain their teeth for a longer time. Most cracks are not detected early because of the discontinuous symptoms and lack of good diagnostic tools. Currently used imaging modalities like Cone Beam Computed Tomography (CBCT) and intraoral radiography often have low sensitivity and do not show cracks clearly. This paper introduces a novel method that can detect, quantify, and localize cracks automatically in high resolution CBCT (hr-CBCT) scans of teeth using steerable wavelets and learning methods. These initial results were created using hr-CBCT scans of a set of healthy teeth and of teeth with simulated longitudinal cracks. The cracks were simulated using multiple orientations. The crack detection was trained on the most significant wavelet coefficients at each scale using a bagged classifier of Support Vector Machines. Our results show high discriminative specificity and sensitivity of this method. The framework aims to be automatic, reproducible, and open-source. Future work will focus on the clinical validation of the proposed techniques on different types of cracks ex-vivo. We believe that this work will ultimately lead to improved tracking and detection of cracks allowing for longer lasting healthy teeth.

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.

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.

Maximally Localized Radial Profiles for Tight Steerable Wavelet Frames

A crucial component of steerable wavelets is the radial profile of the generating function in the frequency domain. In this paper, we present an infinite-dimensional optimization scheme that helps us find the optimal profile for a given criterion over the space of tight frames. We consider two classes of criteria that measure the localization of the wavelet. The first class specifies the spatial localization of the wavelet profile, and the second that of the resulting wavelet coefficients. From these metrics and the proposed algorithm, we construct tight wavelet frames that are optimally localized and provide their analytical expression. In particular, one of the considered criterion helps us finding back the popular Simoncelli wavelet profile. Finally, the investigation of local orientation estimation, image reconstruction from detected contours in the wavelet domain, and denoising indicate that optimizing wavelet localization improves the performance of steerable wavelets, since our new wavelets outperform the traditional ones.

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.

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.

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.

Multiresolution monogenic signal analysis using the Riesz-Laplace wavelet transform

The monogenic signal is the natural 2-D counterpart of the 1-D analytic signal. We propose to transpose the concept to the wavelet domain by considering a complexified version of the Riesz transform which has the remarkable property of mapping a real-valued (primary) wavelet basis of L(2) (R(2)) into a complex one. The Riesz operator is also steerable in the sense that it give access to the Hilbert transform of the signal along any orientation. Having set those foundations, we specify a primary polyharmonic spline wavelet basis of L(2) (R(2)) that involves a single Mexican-hat-like mother wavelet (Laplacian of a B-spline). The important point is that our primary wavelets are quasi-isotropic: they behave like multiscale versions of the fractional Laplace operator from which they are derived, which ensures steerability. We propose to pair these real-valued basis functions with their complex Riesz counterparts to specify a multiresolution monogenic signal analysis. This yields a representation where each wavelet index is associated with a local orientation, an amplitude and a phase. We give a corresponding wavelet-domain method for estimating the underlying instantaneous frequency. We also provide a mechanism for improving the shift and rotation-invariance of the wavelet decomposition and show how to implement the transform efficiently using perfect-reconstruction filterbanks. We illustrate the specific feature-extraction capabilities of the representation and present novel examples of wavelet-domain processing; in particular, a robust, tensor-based analysis of directional image patterns, the demodulation of interferograms, and the reconstruction of digital holograms.