Aviles 2017 ]. **In** this context, many challenges have to be resolved **in** order to achieve the objective of reliable **cardiac** function assessment. For example, there is still **a** need for deriving accurate **and** reproducible quantitative measures of **motion** to overcome the current state of inter-vendor variability of left ventricular (LV) longitudinal strains. Furthermore, the assessment of the **cardiac** function is still limited to global measure- ments [ Alessandrini 2015 ] **and** undergoes great amounts of smoothing, causing loss of clinically valuable local information [ Mirea 2016 ]. An accurate local analysis of the car- diac deformation has **a** major impact on the diagnosis, treatment choice **and** timing of surgical interventions **in** many clinical cases, e.g., ischemia, valvular heart disease **and** early detection of adverse **cardiac** effect of chemotherapy **in** oncology. Therefore, new **motion** **estimation** strategies that limit the loss of structural **and** local information are needed **in** the process of endorsing regional strains [ Mirea 2016 ]. **In** particular, de- veloping more adaptive alternatives to the purely geometrical regularizations used for **cardiac** **motion** **estimation** is still an open challenge. **In** the context of UI, 2D **cardiac** **motion** **estimation** is still **a** difficult task. **In** particular, US **images** are characterized by **a** poor signal-to-noise ratio caused by the speckle noise. Another drawback of UI is the presence of acquisition related artefacts that can affect **cardiac** **motion** **estimation**. Moreover, **in** the case of 2D UI, out-of-plane motions cause discrepancies **in** the speckle pattern, leading also to erroneous **motion** estimates. More generally, the smoothness assumptions typically used for **cardiac** **motion** **estimation** can be violated, e.g., **in** the case of anatomical boundaries. These shortcomings still call for new robust **motion** esti- mation strategies that mitigate the effects of outliers **in** **cardiac** UI. Furthermore, several recent works have attempted to address the spatio-temporal nature of **cardiac** **motion** [ De Craene 2012 , Zhijun 2014 , McLeod 2015 ]. However, many current methods either suffer from problems of large motions between distant frames, or do not process the image sequences as **a** whole. Therefore, it is still an open challenge to efficiently incor- porate temporal aspects into the problem of **cardiac** **motion** **estimation** from US image sequences.

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2 Universit´e de Toulouse, IRIT, CNRS, Toulouse, France
ABSTRACT
**Cardiac** **motion** **estimation** from **ultrasound** **images** is an ill- posed problem that needs regularization to stabilize the solu- tion. **In** this work, regularization is achieved by exploiting the sparseness of **cardiac** **motion** fields when decomposed **in** an appropriate **dictionary**, as well as their smoothness through **a** classical total variation term. The main contribution of this work is to robustify the **sparse** coding step **in** order to handle anomalies, i.e., **motion** patterns that significantly deviate from the expected model. The proposed approach uses an ADMM-based optimization algorithm **in** order to simultaneously recover the **sparse** representations **and** the outlier components. It is evaluated **using** two realistic simu- lated datasets with available ground-truth, containing native outliers **and** corrupted by synthetic attenuation **and** clutter artefacts.

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where K is the maximum number of non-zero coefﬁcients of
α p . Typical algorithms designed for solving (2) include the
K-SVD [11] **and** online DL (ODL) algorithms [10].
At this point, it is interesting to mention that **a** few recent attempts to use **sparse** representations **and** DL for **motion** es- timation have been investigated **in** the literature. **In** [12], the authors included **a** sparsity prior to an OF **estimation** prob- lem **and** used the wavelet basis for the **sparse** coding step. This approach was also investigated **in** [13] **using** **a** learned **motion** **dictionary**. The method proposed **in** this paper com- bines **a** speciﬁc similarity measure for UI with spatial smooth- ness **and** **sparse** regularizations. This strategy exploits jointly the statistical properties of the speckle noise **and** the smooth **and** **sparse** properties of the **cardiac** **motion**. More precisely, we consider **a** multiplicative Rayleigh noise model introduced **in** [14] 1 , **a** spatial regularization based on the l

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118 Route de Narbonne 31062 Toulouse Cedex 9, France
ABSTRACT
This paper investigates **a** new method for **cardiac** **motion** es- timation **in** 2D **ultrasound** **images**. The **motion** **estimation** problem is formulated as an energy minimization with spa- tial **and** **sparse** regularizations. **In** addition to **a** classical spa- tial smoothness constraint, the proposed method exploits the **sparse** properties of the **cardiac** **motion** to regularize the so- lution via an appropriate **dictionary** **learning** step. The pro- posed method is evaluated **in** terms of **motion** **estimation** **and** strain accuracy **and** compared with state-of-the-art algorithms **using** **a** dataset of realistic simulations. These simulation re- sults show that the proposed method provides very promising results for myocardial **motion** **estimation**.

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For future work, it would be necessary to investigate possi- ble extensions of the algorithm to 3D UI. **In** this work, we have addressed the problem of 2D **motion** **estimation**, which can present some shortcomings, such as out-of-plane **motion** **and** limited geometrical information, that could be overcome **in** 3D. Nevertheless, it should be pointed out that **in** contrast with 2D imagery, 3D UI is affected by the problems of frame rate **and** image spatial resolution **in** the azimuthal direction **and** thus, does not necessarily provide better **motion** **estimation** results. Furthermore, it is worth mentioning that the data fidelity **and** regularization terms used **in** the actual formulation are not inherently limited to 2D **and** could be extended to 3D. **In** the same way, the dictionaries could be learned separately for each direction or jointly for the 3 dimensions. The differences between these two strategies of **learning** the **dictionary** have not been investigated **in** this paper, but would also deserve consideration **in** future work. Another research prospect would be to study the interest of adaptive **dictionary** **learning** techniques for applications **in** which the training database is updated periodically. Furthermore, the proposed approach has not exploited the temporal properties of **cardiac** **motion**. Integrating this aspect could be performed by **using** more than two consecutive frames or by **learning** **motion** dictionaries that take into account the sparsity of the **motion** versus time. Another possible prospect concerns the problem of outliers. Considering potential model deviations or viola- tions of smoothness assumptions (e.g., **motion** boundaries) **in** the current approach for robust **motion** **estimation** is clearly an interesting prospect.

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V. D ISCUSSION **AND** P ERSPECTIVES
This paper presented **a** new **motion** **estimation** method for robust 2D **cardiac** US **images**. The main objective of this method was to robustify the **cardiac** **motion** **estimation** algorithm of [21] (based on spatial **and** **sparse** regularizations) **in** order to mitigate the effects of outliers. The obtained fully robust approach allowed us to deal with the problem of native outliers, e.g., **motion** boundaries or background motions, as well as UI artefacts **and** image noise. It is worth mentioning at this point that other strategies have been proposed **in** the literature to address the problem of **cardiac** **motion** **estimation** outliers (see Section I). For example, **in** [32] the myocardium was segmented prior to the **motion** **estimation**, allowing to down-weight the displacements located at the epicardial bor- ders, **and** thus, to prevent over-smoothing **in** this area. **In** con- trast with the method studied **in** [32], the proposed method addressed the problem of spatial outliers for the entire **motion** field (i.e., **using** pixel-wise weights). It allowed us to deal not only with discontinuities at the contours, but also with outliers located inside the myocardium. **In** addition, the proposed strategy did not require **a** beforehand segmentation (which may be difficult to obtain **in** some practical applications), allowing spatial discontinuities to be directly compensated from the estimated motions. More generally, the proposed approach showed the interest of jointly robustifying the data fidelity **and** regularization terms **in** **a** variational approach.

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Other prospects include the robustification of the **dictionary** **learning** step. **A** robust **learning** of the **cardiac** **motion** dictio- nary can be especially useful when **using** corrupted **learning** data. For example, this is the case when the training set contains patterns far from typical **cardiac** motions. If the **dictionary** is learnt **in** an adaptive way, i.e., **using** the esti- mated motions themselves, **a** robust **learning** approach would allow us to discard erroneous **motion** estimates. Furthermore, it would be worth to take advantage of the **sparse** codes, the **dictionary** atoms **and** the robust weights that are obtained **using** the proposed method. For example, **a** joint robust **motion** **estimation** **and** segmentation could be obtained by combining the information provided by the weights of the spatial **and** **sparse** regularizations. Taking into account the increased use of 3D UI, it is also worth mentioning that the proposed method could be extended to 3D. However, the limitations of frame rate **and** spatial resolution **in** the azimuthal direction imply that the use of 3D US **images** does not lead necessarily to **a** more accurate **estimation** when compared to 2D UI.

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V. D ISCUSSION **AND** P ERSPECTIVES
This paper presented **a** new **motion** **estimation** method for robust 2D **cardiac** US **images**. The main objective of this method was to robustify the **cardiac** **motion** **estimation** algorithm of [21] (based on spatial **and** **sparse** regularizations) **in** order to mitigate the effects of outliers. The obtained fully robust approach allowed us to deal with the problem of native outliers, e.g., **motion** boundaries or background motions, as well as UI artefacts **and** image noise. It is worth mentioning at this point that other strategies have been proposed **in** the literature to address the problem of **cardiac** **motion** **estimation** outliers (see Section I). For example, **in** [32] the myocardium was segmented prior to the **motion** **estimation**, allowing to down-weight the displacements located at the epicardial bor- ders, **and** thus, to prevent over-smoothing **in** this area. **In** con- trast with the method studied **in** [32], the proposed method addressed the problem of spatial outliers for the entire **motion** field (i.e., **using** pixel-wise weights). It allowed us to deal not only with discontinuities at the contours, but also with outliers located inside the myocardium. **In** addition, the proposed strategy did not require **a** beforehand segmentation (which may be difficult to obtain **in** some practical applications), allowing spatial discontinuities to be directly compensated from the estimated motions. More generally, the proposed approach showed the interest of jointly robustifying the data fidelity **and** regularization terms **in** **a** variational approach.

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Abstract—This paper studies **a** new **motion** **estimation** method based on convolutional **sparse** coding. The **motion** **estimation** problem is formulated as the minimization of **a** cost function composed of **a** data fidelity term, **a** spatial smoothness constraint, **and** **a** regularization based on convolution **sparse** coding. We study the potential interest of **using** **a** convolutional **dictionary** **in**- stead of **a** standard **dictionary** **using** specific examples. Moreover, the proposed method is evaluated **in** terms of **motion** **estimation** accuracy **and** compared with state-of-the-art algorithms, showing its interest for **cardiac** **motion** **estimation**.

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4.2. Evaluation **using** **a** Synthetic Sequence
We evaluate the method **using** one synthetic time series of T = 30 **cardiac** im- age frames computed **using** the method described **in** Prakosa et al. (2013). The use of **a** synthetic sequence has the important advantage to provide **a** dense point correspondence field following the **motion** of the myocardium during the **cardiac** cycle which can be used to evaluate the accuracy of the tracking. Another op- tion could be to use point correspondence manually defined by experts, but they tend to be inconsistent **and** not reliable Tobon-Gomez et al. (2013). First, we compute the optimal references **using** the methodology described **in** Algorithm 2, giving us the three reference frames spanning the barycentric subspace: #1 is frame 1, #2 is frame 11 **and** #3 is frame 21. Then we register each frame i of the sequence **using** the method described above to get the deformations from each of the three references to the current **images** **using** both the standard method **and** our approach **using** Barycentric Subspaces as **a** prior. We deform each of the 3 ground truth meshes corresponding to the reference frames (1,11 **and** 21) with the deformation from the reference frame to the current frame. We compare our approach with the standard approach where the registration between one of the reference **and** the current frame is done directly. **In** Figure 8 (left), we show the point-to-point registration error of the deformed mesh **using** the 3 different deformations (one with respect to each references). Substantial reduction of the error (of about 30%) can be seen for the largest deformations (between end-systole **and** the first reference for the blue curve corresponding to the frame 1 chosen as reference). This comes at the cost of additional error for the small deformations evaluated at the frame near the respective references. **In** Figure 8 (right), we show the **estimation** of the volume curve (which is one of the most important **cardiac** feature used **in** clinical practice). Our better **estimation** of the large deformation leads to **a** substantial improvement of the volume curve **estimation**. **In** particular the **estimation** of the ejection fraction goes from 32% with the standard method to 38%, closer to the ground truth (43%), reducing the **estimation** error by half.

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k =1
e D,k 2 (4)
where e D ,k 2 , k = 1 , . . . , K D is the Euclidean norm, e D,k corre- sponds to the k th time-series of e D associated with the k th param- eter **and** b D is **a** regularization parameter that controls the level of sparsity of e D . The sparsity constraint for the anomaly signal re- ﬂects the fact that anomalies are rare **and** affect few parameters at the same time. Note that the discrete vector x D is constrained to belong to B, where B is the canonical or natural basis of R L , i.e., B = { l , l = 1 , · · · , L } , where l is **a** vector whose l th component equals 1 **and** whose other components equal 0. **In** other words, only one atom of the discrete **dictionary** D is chosen to repre- sent the discrete signal, this amounts to looking for the nearest neighbour of y D **in** the **dictionary**. This strategy has proved to be an effective method to reconstruct discrete signals (compared to **a** **representation** **using** **a** linear combination of atoms), which ex- plains this choice. Since x D belongs to **a** ﬁnite set, its **estimation** is combinatorial **and** can be solved for each atom φ D, l (where φ D, l is the l th column of D ) as follows

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This paper first describes **a** new tree-structured **dictionary** **learning** method called Tree K-SVD. Inspired from ITD **and** TSITD, each **dictionary** at **a** given level is learned from **a** sub- set of residuals of the previous level **using** the K-SVD algo- rithm. The tree structure enables the **learning** of more atoms than **in** **a** ”flat” **dictionary**, while keeping the coding cost of the index-coefficient pairs similar. Tests are conducted on fa- cial **images**, as **in** [1, 4, 5], compressed for multiple rates **in** **a** compress scheme. Thus, for **a** given bit rate, Tree K-SVD is shown to outperform ”flat” dictionaries (K-SVD, **Sparse** K- SVD **and** the predetermined (over)complete DCT **dictionary**) **in** terms of quality of reconstruction for **a** high sparsity, i.e. when the number of atoms used **in** the **representation** of **a** vector is low. Setting the sparsity constraint to only **a** few atoms limits the number of levels, **and** so of atoms, **in** the tree-structured **dictionary**. The paper then describes an adap- tive **sparse** coding method applied on the tree-structured dic- tionary to adapt the sparsity per level, i.e. to allow selecting more than 1 atom per level. It is shown to improve the quality of reconstruction.

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2
Over the past years, various approaches for semi-automatic **and** automatic segmentation of MS lesions have been proposed. **In** these methods, different im- age features, classification methods **and** models have been tried, but they usu- ally suffer from high sensitivity to the imaging protocols **and** so usually require tedious parameter tuning or specific normalized protocols [3]. More recently, **sparse** **representation** has evolved as **a** model to represent an important variety of natural signals **using** few elements of an overcomplete **dictionary**. Many pub- lications have demonstrated that **sparse** modeling can achieve state-of-the-art results **in** image processing applications such as denoising, texture segmentation **and** face recognition [4, 5]. **In** [5], given multiple **images** of individual subjects under varying expressions **and** illuminations, the **images** themselves were used as **dictionary** elements, for classification. Such **a** method uses **dictionary** **learning** to analyze image as **a** whole. Mairal et al [6] proposed to learn discriminative dictionaries better suited for local image descrimination tasks. **In** medical imag- ing, local image analysis is of prime importance **and** it could be interesting to see the performance of **sparse** **representation** **and** **dictionary** **learning** based classification methods **in** the context of disease detection. Some researchers have reported works on segmentation of endocardium **and** MS lesions **using** **dictionary** **learning** [7, 8]. Weiss et al. proposed an unsupervised approach for MS lesion seg- mentation, **in** which **a** **dictionary** learned **using** healthy brain tissue **and** lesion patches is used as basis for classification [7].

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5. Department of Radiology, Nanjing Hospital Affiliated to Nanjing Medical University , 210096, People’s Republic of China
Abstract —**In** abdomen computed tomography (CT), repeated radiation exposures are often inevitable for cancer patients who receive guided surgery or radiotherapy. Low-dose scans should thus be considered **in** order to avoid too high accumulative harm of radiation. This work is aimed at improving abdomen tumor CT **images** from low-dose scans by **using** **a** fast **dictionary** **learning** (DL) based processing. Stemming from **sparse** **representation** theory, the patch-based DL approach proposed **in** this paper allows effective suppression of both mottled noise **and** streak artifacts. The experiments carried out on clinical data show that the proposed method brings encouraging improvements **in** abdomen low-dose CT **images** with tumors.

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this work combines merits of Markovian-based methods **and** clustering-based methods by finding the local clusters characterized by partial trajectory segments, **and** making predictions **using** both the local **motion** models **and** the global Markovian transition dynamics. Multi-class inverse reinforcement **learning** (IRL) algorithms [16], [17], which are related to clustering-based methods (e.g. [7]), have also been applied to modeling **motion** patterns. Previous work based on clustering partial trajectory segments [18], [19] was limited to modeling local **motion** patterns as short straight line segments, whereas this work is more flexible since we do not constrain the shape of local **motion** patterns. Recent work [20] applied **sparse** coding to an image **representation** of hand gesture trajectories, but this work models each dimension independently, which would not be suitable for location- based applications as considered **in** this paper.

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This paper proposes an unsupervised CD method able to deal with **images** dissimilar **in** terms of modality **and** of spatial **and**/or spectral resolutions. The adopted methodology, similar to [17, 18], learns coupled dictionaries able to conveniently represent multi- modal remote sensing **images** of the same geographical location. The problem is formulated as **a** joint **estimation** of the coupled **dictionary** **and** **sparse** code for each observed image. Additionally, appropriate statistical models are used to better fit the modalities of the pair of observed **images**. Overlapping patches are also taken into account during the **estimation** process. Finally, to better couple im- ages with different resolutions, additional scaling matrices [19] are jointly estimated within the whole process. The overall **estimation** process is formulated as an inverse problem. Due to the nonconvex nature of the problem, it is solved iteratively **using** the proximal alternating linearized minimization (PALM) algorithm [20], which ensures convergence towards **a** critical point for some nonconvex non-smooth problems. CD is, then, envisaged through the differ- ences between **sparse** codes estimated for each image **using** the estimated coupled dictionaries. This paper is organized as follows.

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This paper proposes an unsupervised CD method able to deal with **images** dissimilar **in** terms of modality **and** of spatial **and**/or spectral resolutions. The adopted methodology, similar to [17, 18], learns coupled dictionaries able to conveniently represent multi- modal remote sensing **images** of the same geographical location. The problem is formulated as **a** joint **estimation** of the coupled **dictionary** **and** **sparse** code for each observed image. Additionally, appropriate statistical models are used to better fit the modalities of the pair of observed **images**. Overlapping patches are also taken into account during the **estimation** process. Finally, to better couple im- ages with different resolutions, additional scaling matrices [19] are jointly estimated within the whole process. The overall **estimation** process is formulated as an inverse problem. Due to the nonconvex nature of the problem, it is solved iteratively **using** the proximal alternating linearized minimization (PALM) algorithm [20], which ensures convergence towards **a** critical point for some nonconvex non-smooth problems. CD is, then, envisaged through the differ- ences between **sparse** codes estimated for each image **using** the estimated coupled dictionaries. This paper is organized as follows.

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F (4)
where D is the **dictionary**, **A** is the **sparse** code, **and** ¯ U is the ap- proximation of U derived from the **dictionary** **and** the code. Gen- erally, an over-complete **dictionary** is proposed as **a** basis for the image patches. **In** many applications, the **dictionary** D is •xed **a** priori, **and** corresponds to various types of bases constructed **using** atoms such as wavelets [21] or discrete cosine transform coef•cients [22]. However, these bases are not necessarily well matched to nat- ural or remote sensing **images** since they do not necessarily adapt to the nature of the observed **images**. As **a** consequence, **learning** the **dictionary** from the observed **images** instead of **using** prede•ned bases generally improves signal **representation** [23]. More precisely, the strategy advocated **in** this paper consists of **learning** **a** **dictionary** D from the high resolution MS image to capture most of the spatial information contained **in** this image. To learn **a** **dictionary** from **a** multi-band image, **a** popular method consists of searching for **a** dic- tionary whose columns (or atoms) result from the lexicographically vectorization of the HS 3D patches [16, 24]. However, this strat- egy cannot be followed here since the **dictionary** is learned on the MS image Y m ∈ R n λ ×n composed of n λ bands to approximate

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F (4)
where D is the **dictionary**, **A** is the **sparse** code, **and** ¯ U is the ap- proximation of U derived from the **dictionary** **and** the code. Gen- erally, an over-complete **dictionary** is proposed as **a** basis for the image patches. **In** many applications, the **dictionary** D is •xed **a** priori, **and** corresponds to various types of bases constructed **using** atoms such as wavelets [21] or discrete cosine transform coef•cients [22]. However, these bases are not necessarily well matched to nat- ural or remote sensing **images** since they do not necessarily adapt to the nature of the observed **images**. As **a** consequence, **learning** the **dictionary** from the observed **images** instead of **using** prede•ned bases generally improves signal **representation** [23]. More precisely, the strategy advocated **in** this paper consists of **learning** **a** **dictionary** D from the high resolution MS image to capture most of the spatial information contained **in** this image. To learn **a** **dictionary** from **a** multi-band image, **a** popular method consists of searching for **a** dic- tionary whose columns (or atoms) result from the lexicographically vectorization of the HS 3D patches [16, 24]. However, this strat- egy cannot be followed here since the **dictionary** is learned on the MS image Y m ∈ R n λ ×n composed of n λ bands to approximate the target image U composed of m e λ spectral bands. Conversely, to capture most of the spatial details contained **in** each band of the MS image, we propose to approximate each band of the target image U by **a** **sparse** decomposition on **a** dedicated **dictionary**. **In** this case, the regularization term (4) can be written as

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