Sparse Signal Separation with an Off-line Learned ...

2014 IEEE 28-th Convention of Electrical and Electronics Engineers in Israel

Sparse Signal Separation with an Off-line Learned Dictionary for Clutter Reduction in Echocardiography Javier S. Turek, Michael Elad and Irad Yavneh Department of Computer Science - Technion, Israel Institute of Technology Technion City, Haifa 32000, Israel {javiert,elad,irad}@cs.technion.ac.il Abstract—Clutter is an artifact in cardiac ultrasound that obscures parts of the heart. A cluttered signal is seen as a superposition of tissue, clutter and noise components. In this work, we introduce two novel methods for reducing clutter by separating these components using Morphological Component Analysis, where each component has a sparse representation under some dictionary. The clutter dictionary is trained using data acquired from right side of the chest, overcoming any assumption about the clutter behavior. The tissue dictionary is trained from off-line tissue data in one method, and adaptively from the patient data in the other. These methods are shown to be robust to the input data characteristics and yield state-of-the-art performance.

I. I NTRODUCTION In ultrasound cardiac imaging, clutter is an image artifact that appears mainly as a quasi-static cloud of echo signals obscuring tissue areas [1]. As a consequence, clutter artifacts reduce the contrast and readability of the image and can corrupt diagnostic information like in myocardium strain evaluation [2], tracking techniques for functioning diagnosis [3], or visualization of cardiac abnormalities [4]. Sources of clutter in ultrasound cardiac imaging include multi-path reverberations and off-axis reflexions from structures such as the ribs cage, the sternum and the lungs, which are in the proximity of the path of the acoustic waves to the heart [1]. As the energy of the waves decays with the distance covered and the number of bounces, the clutter effect is mostly visible in the near-field regions such as the myocardium. A typical strategy to reducing clutter involves separating the tissue echoes from the clutter ones by a linear decomposition, and then filtering out the artifacts. The echo data is transformed to a new coordinate system under which the clutter and the tissue signals can be separated along different bases or dictionaries. Several methods were developed using unitary a-priori bases, such as Discrete Fourier Transform [5] and Wavelets [6], or unitary adaptive bases learned from the echo data, such as Principal Component Analysis [7], [8]. A recent approach [9] separates the signals using Morphological Component Analysis (MCA) [10] with an overcomplete dictionary trained fully on-line from the echo data. In this work, we present two novel techniques based on MCA for clutter reduction. These techniques train a clutter and a tissue dictionary separately one from each other. The clutter

dictionary is learned off-line from a clutter only sequence obtained by scanning from the right side of the chest. In one method, the tissue dictionary is trained off-line from a clutterfree sample sequence. In the other method the tissue dictionary is learned on-line from the patient echo data, which requires further steps to remove from the dictionary unwanted clutter information. Then, these dictionaries are joined together to separate the signal and selectively remove clutter. This paper is organized as follows. Sparse Representations are described in Section II. The algorithms are introduced in Section III. In Section IV, we quantify the performance of the current methods on simulations and we demonstrate its feasibility on echocardiography images while comparing to other methods. We conclude this work in Section V. II. S PARSE R EPRESENTATIONS OF S IGNALS A signal t ∈ Cn is said to have a sparse representation over a known overcomplete bases, or “dictionary”, D ∈ Cn×m , if there exists a sparse vector x ∈ Cm , such that t = Dx. The representation vector x is assumed to be sparse with a number of non-zeros kxk0 = k that is much smaller than its length, m. The k·k0 represents the `0 quasi-norm, which measures the number of non-zero elements in a vector. In other words, vector x describes how to construct the signal t as linear combination of a few columns (also called “atoms”) from the dictionary D. In practice, an observed signal s is obtained by measuring a signal of interest t contaminated with noise n, which is often assumed to be additive, i.e. s = t + n. Additionally, it is frequently assumed that the noise n ∈ Cn is i.i.d. with noise level σ. Hence, the objective is to reconstruct signal t from the noisy observation s. In our case, we assume a sparse representation prior on the signal of interest t as described above. To reconstruct a signal t, first the sparse representation x is computed by finding the sparsest vector that yields an approximation of t ≈ Dx min kxk0 x

s.t.

2

ks − Dxk2 ≤ ε2 ,

(1)

where ε is the desired bound from the signal and proportional to the noise level σ. An alternative formulation to (1) is to minimize the fidelity data term while constraining the number

Fig. 1: Illustration of the extraction of a patch si from a sequence of complex echo data. The bold columns in the ultrasound image frames represent M axial elements in each frame that are concatenated to form the vectorized patch si . of non-zeros in the representation, 2

min ks − Dxk2 x

s.t.

kxk0 ≤ k,

(2)

where k is maximum sparsity allowed in the representation. The resulting approximate sparse representation x ˆ from solving (1) or (2), is then multiplied by the dictionary D to reconstruct the clean signal, i.e., ˆ t = Dˆ x. To approximate a solution x ˆ to Problems (1) and (2) a greedy pursuit algorithm called the Orthogonal Matching Pursuit (OMP) [11] is usually employed. The Basis Pursuit (BP) algorithm [12] is another method that relaxes the `0 -norm with the `1 -norm, and solves a convex optimization problem. The dictionary D is an important link between the signal of interest and the sparse representation. The dictionary can be selected a-priori [13], [14] or learned adaptively from the data [15]. The K-SVD [15] algorithm is a commonly used method for dictionary learning. This method iteratively updates each atom in the dictionary by fitting the data with the sparsest possible representations. Consequently, it requires multiple data samples to learn a dictionary. These data samples are usually obtained by extracting overlapping patches from the signals and processing them instead of the entire image. III. C LUTTER R EDUCTION WITH S PARSE S IGNAL S EPARATION Clutter artifacts originate from multi-path reverberations of the transmitted echoes. Due to the linearity of the echo waves, the acquired signal is seen as a superposition of the clutter echoes and tissue echoes. Therefore, we assume a model for the observed signal s that reflects this fact with additional i.i.d. white Gaussian noise n with level σ: s = t + c + n,

where the subscript i represents the signal or a component for the patch i. We further assume a sparsity prior for the tissue and the clutter signal patches with sparse representations xci and xti , under the dictionaries Dt and Dc , respectively. Therefore, Equation (4) can be rewritten as follows,   xti + ni si = Dt xti + Dc xci + ni = [Dt |Dc ] · xci = Dxi + ni , (5) where D is a dictionary merged with all the atoms of Dt and Dc together, and xi contains the sparse representations xti and xci within a larger sparse vector. Then, for a given dictionary D of the form in (5), clutter is reduced by first computing xi by solving Problem (1) or (2) with OMP, and then removing the clutter part Dc xci from si : sˆi = si − Dc xci = Dt xti + ni ,

(6)

where sˆi is the clutter-reduced version of si . Eventually, the clean patches sˆi are averaged together to reconstruct the clean signal ˆ s. The filtering approach described above using sparse representations is part of a general framework known as Morphological Component Analysis (MCA) [10], where a mixed signal can be decomposed into different morphological components (dictionaries) and each source is sparse under its respective dictionary. To success with the signal separation, the dictionaries Dt and Dc should describe their respective signals well, that is, they should have morphological differences. In [9], the dictionaries are obtained by learning fully on-line the dictionary D from the patient echo data and then partitioning its atoms into Dt and Dc based on the quasi-static behavior of clutter in the temporal domain [1] and the motion behavior of the tissue as the heart beats. In this work, we learn the dictionaries separately, hence eliminating the partitioning method. Additionally, the temporal pattern of the clutter is learned automatically from a clutter-like sequence, which avoids misclassifying atoms. As the clutter and tissue characteristics are in the temporal domain, we divide the echo data s into two-dimensional overlapping patches taken from the axial and temporal dimensions1 and apply the separation scheme to these small patches. That is, a signal si is a columnized version of a patch of M axial elements and across N frames (see Fig. 1). The number of frames N controls the amount of motion that is captured by the patches and it depends on the acquisition frame rate of the sequence – the faster the rate is, the bigger the value of N would be to capture the same amount of motion.

(3) A. Learning the Clutter Dictionary

where t is the tissue signal and c represents the clutter artifacts. In ultrasound, the linear model assumption in Equation (3) is satisfied only by RF or IQ signals. Moreover, the assumption holds when the signal is treated by patches, such that Equation (3) is transcribed to each observed sample patch si as follows si = ti + ci + ni , (4)

We suggest a novel approach to acquiring a clutter-like sequence that allows us to train a clutter dictionary Dc separately from the tissue one. As the clutter artifacts are 1 The echo data can include information in the lateral direction by adding consecutive lateral lines to every signal si and obtaining three dimensional elements.

these algorithms while comparing to state-of-the-art methods. Although simulations differ from real data, they are useful to analyze the performance of the methods under varying conditions of the input data and the algorithm parameters. A. Simulations Fig. 2: Example of an clutter image obtained by scanning the right side of the chest. caused mainly by multi-path reverberations from the ribs cage and sternum, we scan the right side of the chest to obtain a sequence that contains clutter echo data alone. The right side of the chest has a similar physical structure to that of the left side of the chest, but with the exception that there are no echoes returning from the heart. Fig. 2 shows such a sequence where the artifacts can be appreciated clearly and without tissue information. We train the clutter dictionary Dc with K-SVD on a sequence acquired from the right side of the chest. The echo data from a such a sequence does not vary extremely among patients, and can be used to train a clutter dictionary off-line that fits many patients. B. Learning the Tissue Dictionary For the tissue dictionary Dt , we propose two alternatives for the training process. The first option trains the dictionary from a clutter-free sequence, obtained from other patients that do not present clutter artifacts during the scanning session. We use K-SVD with patches from such clean echo data to obtain the dictionary Dt . Therefore, the tissue dictionary is learned off-line like the clutter dictionary. As reference, we name this method Off-line MCA (OFF-MCA). The OFFMCA method has a runtime advantage as the dictionaries are trained beforehand. On the other hand, the OFF-MCA has an important limitation. The tissue motion of a patient may be different from the motion in the training data, and the tissue dictionary may fail to represent important information and hence OFF-MCA may perform poorly. An alternative option is to learn the tissue dictionary adaptively to the patient data. The disadvantage of such data is that clutter artifacts are present, and learning Dt from them yields a dictionary with unwanted clutter atoms, which require special treatment. To clean the dictionary, we search such clutter atoms in Dt by checking the correlation of the atoms in Dt with those in Dc , and remove the highly correlated ones. Among the ways to correlate the atoms, we assume each atom in Dt to be a signal and find a sparse representation under the dictionary Dc using OMP with cardinality k < M . If the representation error is below a threshold τ , then the atom is considered clutter and thus, removed from Dt . As a reference, we name this method Tissue-Adaptive MCA (TA-MCA). IV. E XPERIMENTS Next, we demonstrate the OFF-MCA and TA-MCA methods and provide both simulation and real data results for

We perform a Field II simulation [16] of an hypoechoic lesion to evaluate the performance of the OFF-MCA and TA-MCA methods for reducing clutter. We reproduce the simulation in [9] with the same parameters. Clutter artifacts were simulated as a small region of scatterers with reflection amplitudes 20dB above the tissue. This scatterers were simulated separately and located in the center of the hypoechoic lesion. Tissue scatterers were simulated several times to obtain distinct frames. These scatterers were added echo decorrelation and axial motion across frames. Clutter scatterers were assigned a small axial motion different from the motion of the tissue. Eventually, the hypoechoic tissue echoes were summed with the clutter artifacts and with electronic noise with level σ (chosen for SNR of −30dB) to obtain the final frames according to (3). We refer the reader to [7], [9] for detailed information on the simulation. The performance of the OFF-MCA and TA-MCA methods was compared to that of SVF [7], and of the On-line MCA (ON-MCA) [9]. The size of the patches was selected to be N = 9 frames and M = 33 axial elements by default. OMP for solving Problem (1) was used to compute the √ sparse representations of the signals {si } with ε = 2.3σ 2N M . All other parameters were selected for best performance. Performance was measured using Peak-Signal-to-Noise-Ratio (PSNR). The experiments were repeated over 50 realizations and the resulting PSNR performances were averaged. For reference, we included the PSNR of the unfiltered image (“Unfiltered”) and the clutter-free image (“Perfect Filtering”). Examples of the simulations and the filtering results for each method are shown in Fig. 3. Figs. 4a and 4b show the performance of the algorithms as a function of the simulation parameters: the echo correlation and the tissue axial shift motion, respectively. The plots show that OFF-MCA yields low performance results when trained with off-line tissue data with different characteristics to those of the actual tissue in the signal s. Hence, we included in Figs. 4a and 4b an “improved OFF-MCA” plot that trains the tissue dictionary from tissue examples with similar characteristics of those in the input s. The performance differences between OFF-MCA and its improved version describe the influence on the tissue data used for training the dictionary. In addition, the mean PSNR performance of the algorithms is depicted in Figures 4c and 4d as a function of the patch size in the temporal and axial dimension, respectively. In general, the OFF-MCA obtained a lower performance than the other methods. However, in extreme cases such as high echo correlation values, low tissue motion, or small axial patch sizes, its performance is above that of ON-MCA and SVF. Despite its performance disadvantage, OFF-MCA obtains reasonable

(a) Unfiltered

(d) OFF-MCA

(b) Perfect filt.

(e) TA-MCA

(a)

(b)

(c)

(d)

(c) SVF

(f) ON-MCA

Fig. 3: Example of the filtering results obtained by each method (c)-(f) on the simulation image (a). The expected result is given in (b). and stable results without setting a separation parameter value as in the other methods. Furthermore, the TA-MCA shows the best performance of the overall methods. Using the offline clutter dictionary, it enjoys the stability of the OFFMCA and the boosted adaptivity of ON-MCA. In some cases its performance reached up to 3dB over the other methods’ performance. Additionally, it can be seen that it does not suffer from high variance as the ON-MCA algorithm. B. Real Data In this section we demonstrate the presented methods using frames of IQ echo data acquired from various apical views containing one heart cycle (40-50 frames). A Vivid S6 (GE Medical Systems, Israel) ultrasound scanner operating at 3.3MHz was used to acquire the sequences. Clutter artifact was present due to multi-path reverberations mainly from the thoracic cage and sternum. Nine data sets were acquired from three male volunteers, 30-55 years old. The performance of the algorithms was measured averaging the Contrast-to-Noise-Ratio (CNR) over the frames in the sequence and compared to the mean CNR of the unfiltered sequence. The CNR of a frame is defined as:   |µi − µo | CN R = 20 log10 , (7) σo where µi and µo are the mean envelope-detected quantities in regions with clutter artifact and without artifacts, respectively, and σo is the standard deviation in the clutter-empty region. It was suggested in [9] that the CNR measure has a drawback as it does not compare the filtered signal with the original

Fig. 4: Mean PSNR measurements for OFF-MCA, TA-MCA, ON-MCA, and SVF when varying (a) the echo correlation factor, (b) the tissue axial motion, (c) the temporal size of the patches, and (d) the axial size of the patches. signal leading to improved CNR values when tissue is being removed. Hence, we chose the parameters of the algorithms such that the PSNR of the clutter-empty region of the original image and the algorithms’ output is approximately the same. The optimal parameters for SVF were τ = 0.425, α = 30, for ON-MCA β = 0.45, and for TA-MCA τ = 0.35. The offline clutter dictionary was trained using a sequence acquired from the right side of the chest of one of the volunteers (an example is shown in Fig. 2). The noise level is not known for these data sets and hence, the MCA-based methods were allowed a maximum sparsity k of 20% of the patch size for solving (2). The data sets were treated using fully overlapping patches of 15 axial elements by 15 frames. The mean CNR performance improvement of the algorithms over all the data sets is summarized in Table I. The best performer was the TA-MCA method while the OFF-MCA obtained similar performance. The ON-MCA and SVF methods obtained results slightly lower than the other methods. Fig. 5 shows visual examples of the filtering results of each method. Clearly, the SVF has removed less clutter and a bit more tissue than the MCA methods. All the MCA methods managed to remove a the artifacts in great extent, but still removed a bit of tissue, with TA-MCA the best tissue preserver method. TABLE I: Mean CNR improvement comparison of real data over the unfiltered echo data. Method CNR (dB)

OFF-MCA 1.14

TA-MCA 1.15

ON-MCA 1.06

SVF 1.08

V. C ONCLUSIONS In this work, we presented two novel methods for clutter reduction in echocardiography. Both are MCA-based and learn

MCA this limitation is solved by learning the tissue dictionary adaptively from the patient. Hence, TA-MCA yields the best performance over state-of-the-art methods in both simulations and real echo data. ACKNOWLEDGMENTS

(a) Original

(b) OFF-MCA

The research leading to these results has received funding from the European Research Council under European Union’s Seventh Framework Programme, ERC Grant agreement no. 320649. This work was partially supported by MAGNETON project from the Office of the Chief Scientist [OCS] in the Israeli Ministry of Economy. R EFERENCES

(c) TA-MCA

(d) ON-MCA

(e) SVF

Fig. 5: Frame example of the filtering results on the frame in (a) obtained by applying each method (b)-(e) to one of the data sets. In (a), the lower white box corresponds to the artifact region and the upper black box corresponds to the clutter-free region. The arrows point to some regions of tissue incorrectly filtered. a clutter dictionary off-line. The first learns a tissue dictionary also off-line from clean echo data. The second learns the tissue dictionary from the patient echo data and removes unwanted atoms that are highly similar to those in the clutter dictionary. Learning the clutter dictionary from echo data acquired from the right side of the chest of a patient, imitating the characteristics of the clutter instead of looking for assumptions about it. The OFF-MCA eliminates the needs for a separation parameter and thus the separation becomes more stable to the variation of the data. However, it needs input data similar to the patient data to obtain acceptable filtering results. In TA-

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