Closed loop low rank echocardiographic artifact removal

Abstract

Echocardiographic image sequences are frequently corrupted by quasi-static artifacts (“clutter”) superimposed on the moving myocardium. Conventionally, localized blind source separation methods exploiting local correlation in the clutter have proven effective in the suppression of these artifacts. These methods use spectral characteristics to distinguish clutter from tissue and background noise, and are applied exhaustively over the dataset. The exhaustive application results in high computational complexity and a loss of useful tissue signal. In this paper, we develop a closed loop algorithm in which the clutter is first detected using an adaptively determined weighting function and then removed using low rank estimation methods. We show that our method is adaptable to different low rank estimators, by presenting two such estimators: sparse coding in the principal component domain and nuclear norm minimization. We compare the performance of our proposed method (CLEAR) to two methods: singular value filtering (SVF) and morphological component analysis (MCA). The performance was quantified in silico by measuring the error with respect to a known ‘ground truth’ dataset with no clutter for different combinations of moving clutter and tissue. Our method retains more tissue with a lower error of 3.88 ± 0.093 dB (sparse coding), 3.47 ±0.78 (nuclear norm) compared to the benchmark methods 8.5 ± 0.7 dB (SVF), and 9.3 ± 0.5 dB (MCA) particularly in instances where the rate of tissue motion and artifact motion are small (≤ 0.25 periods of center frequency per frame) while producing comparable clutter reduction performance. CLEAR was also validated in vivo by quantifying tracking error over the cardiac cycle on five mouse heart datasets with synthetic clutter. CLEAR reduced the error by approximately 50%, compared to 25% for the SVF.

I. INTRODUCTION

ECHOCARDIOGRAPHY is the most frequently employed diagnostic imaging modality used in assessing cardiac anatomy and function [1]. Quasi-static artifacts, generally known as “clutter”, arise from off axis reflections and reverberant signals from the chest wall, ribs and other anatomical structures [2–4]. Conventional methods to reduce clutter in cardiac imaging with phased arrays include aperture apodization to reduce the impact of off-axis reflections, which widens the main lobe and consequently degrades lateral resolution, and lowering the frequency of ensonification at wider angles of steering to reduce the effect of off-axis artifacts [5]. While this approach reduces clutter, it also degrades axial resolution when the array is aggressively steered.

Tissue Harmonic Imaging (THI) suppresses near field clutter in echocardiography [6, 7]. However, it has limited application in imaging of small animals, due to the high frequencies used and shallow depth of propagation. Additionally, THI is unable to suppress clutter with significant nonlinear energy. The performance of THI is also dependent on the separation between linear and harmonic bands and this may necessitate the choice of narrower bandwidth pulses resulting in degraded axial resolution. [8–10].

Blind source separation methods, such as independent component analysis (ICA) [11, 12] or principal component analysis (PCA) [13], have shown efficacy in removing clutter and retaining the underlying tissue. These methods were improved on by Mauldin et al. [4] with an approach known as the singular value filter (SVF) using PCA where a local measure of rank one signal content known as the normalized singular spectral area (NSSA) was used to adaptively suppress the clutter and retain the underlying tissue, and at each point, the singular value decomposition (SVD) is repeatedly computed leading to high computational complexity.

Turek et al[14] partially address this problem by using a variant of the morphological component analysis (MCA) method developed by Starck et al. [15] In this method, the data are considered to be spanned by an overcomplete representation (dictionary), which is a union of bases representing tissue and clutter. The dictionary is learned from the data, and a single overcomplete dictionary is computed for the entire dataset.

Once the dictionary has been learned, the rank one content of the basis vectors (“atoms”) is used to discriminate atoms which contribute to the clutter, the clutter content is subsequently subtracted.

Both methods utilize local patches from each A-line through time on (2D + T) data, and are applied exhaustively on all local patches resulting in increased computational complexity. Additionally, the local rank of the clutter is in general greater than one, and the signal from multiple eigenvectors often needs to be suppressed. It may be appreciated that using a patch-wise approach, the maximum rank of the clutter possible is limited by the size of the patch which consequently limits the algorithm’s performance, as the rank of the clutter may rise to be comparable with the patch size owing to motion or multipath artifacts. This creates an inherent tradeoff when there is clutter from multiple sources and varying rank, such as from motion.

In this paper, we address these limitations using an approach that considers the ultrasound image as a whole, determines a weighting function using the temporal phase of the clutter signal, and pursues a combination of bases that estimates the clutter as shown in Fig.1 (b). The proposed method differs from current methods (Fig 1(a)) by using a ‘closed loop’ approach where clutter is iteratively detected, estimated and removed, thus precluding operation on the entire dataset [16]. This enables a reduced computational complexity, since we assume that the clutter is spatially sparse, occupying only a part of the image, furthermore, since we carry out the clutter removal procedure only on regions containing clutter, the remaining parts of the image are unaffected. Consequently, more tissue signal is retained. We term our method CLEAR, as an acronym for Closed loop Low rank Echocardiographic Artifact Removal.

Fig. 1.

(a) Schematic of current clutter removal methods (b) Schematic of proposed clutter removal method.

The paper is organized as follows: Firstly, we describe a spatially sparse and temporally low rank model of reverberant clutter. Next, we describe the method used to obtain the weighting function using a bootstrapped cross correlation approach, which is computationally inexpensive, as it does not use the spectral characteristics of the clutter and consequently does not require computing a new basis. We demonstrate the efficacy of this weighting procedure in reducing computational complexity and ensuring that regions not containing clutter are unaffected. We demonstrate the results of our clutter removal procedure on six simulated datasets in Field II [16], containing a single source of clutter, eight simulated datasets, with two clutter sources, moving independently, and five in vivo datasets. The effectiveness of our method is demonstrated by comparing the tracking information obtained before and after clutter suppression. Throughout the paper, we use the following notational convention, matrices are in boldface and are italicized. Vectors are represented in bold face, while scalars are only italicized.

II. SIGNAL MODEL

In this section, we describe a model of data contaminated by reverberation clutter, which serves as a rationale for our proposed closed loop method depicted in Fig 1 (b). Tissue (speckle) can be modelled as the convolution of the analytic ultrasound system point spread function (PSF) and a random distribution of scatterers, spaced closer than the wavelength of the ultrasound pulse [17–20].

Existing methods [4,14] treat all data points of the M-mode signal I (spatio-temporal (depth × time) patches in a local neighborhood) identically, either as piecewise low rank [4], or sparse in a representation spanning a high dimensional union of subspaces [14].

$$argmi{n}_{\mathit{D},\mathit{\gamma }}\parallel \mathit{I}-\mathit{D}\mathit{\gamma }{\parallel }_{\mathit{F}}^{\mathbf{2}}\text{subject}\text{to}rank(\mathit{D}\mathit{\gamma })\ll p$$ (2.1)

$$argmi{n}_{D,\gamma }\parallel \mathit{I}-\mathit{D}\mathit{\gamma }{\parallel }_{\mathit{F}}^{\mathbf{2}}\text{subject}\text{to}\parallel \mathit{\gamma }{\parallel }_{\mathbf{0}}\le K$$ (2.2)

Where $\parallel {\parallel }_0$ denotes the L0 ‘norm’.

Here the patch is of size $p\times p,\parallel {\parallel }_F$ and denotes the Frobenius norm. We assume a square patch here for simplicity, but the methods hold for rectangular patches as well without loss of generality. K is the sparsity denoting how many basis vectors are used to represent each of the data points. Both of these methods first project the data into a learned basis D, and subsequently discriminate between cluttered and non-cluttered data using spectral properties.

(2.2) can also be written as follows,

$$\begin{array}{l}argmi{n}_{\mathit{D},\mathit{\gamma }}\parallel \mathit{I}-\mathit{D}(\mathit{\alpha }\mathbf{+}\mathit{\beta }){\parallel }_{\mathit{F}}^{\mathbf{2}},\hfill \\ \text{subject}\text{to}\parallel \mathit{\beta }{\parallel }_{\mathbf{0}}\le K_1,\parallel \mathit{\alpha }{\parallel }_{\mathbf{0}}\le K_2\hfill \end{array}$$ (2.3)

Where

$$\mathit{\gamma }=(\mathit{\alpha }+\mathit{\beta })$$ (2.4)

Where α and β are the coefficients of tissue and clutter with sparsity K₁ and K₂ respectively. However, if it is possible to discriminate between regions containing tissue and clutter before removal, we can treat the data and tissue differently and apply different constraints on each in the learning of the basis D. Additionally, due to the reasons outlined in the previous section, we treat the data globally instead of patch wise. Furthermore, tissue is generally modelled a randomly distributed ensemble of closely spaced scatterers [17–20], which suggests that the imposition of a sparse prior on tissue in [14] is not representative of underlying physical reality.

Hence we formulate a new problem that estimates an axially sparse clutter component C in some basis D which remains low rank over time as the clutter is quasi-static, and is axially sparse owing to the reverberation model described as follows. In this work, we focus on reverberant clutter that occludes part of the useful signal in the image such as reflections off the ribs or sternum, and is not distributed over the entire myocardium. Consequently, reverberant clutter in one A-line (as a vector) can be modelled as a weighted sum of the PSF at different locations in the image plane. This which can be written as the convolution (⊗) of the ultrasound system PSF (P) with a suitable spatial impulse response (H).

$$R=P\otimes H$$ (2.5)

The impulse response H is a sparse vector, and consequently, the reverberant clutter R has sparse support, denoted by $\parallel H{\parallel }_0\le K$, where K is small compared to the number of samples in the image, and $\parallel {\parallel }_0$ denotes the l0 ‘norm’. Using this, we can rewrite (2.1) as

$$R=\mathit{M}H,subjectto\parallel H{\parallel }_0\le K$$ (2.6)

Where M is a matrix of translated ultrasound system PSFs to all possible locations of reverberant clutter in an A-line. Additionally, it may be appreciated, M is overcomplete and redundant (few of the columns of M relative to its size contribute to R) as the system PSF is spatially variant.

We can treat a sequence of ultrasound B-mode image frames as an ensemble of M-mode images as in Mauldin et al. in [4] and Turek et al.[14]. For a single M-mode image (matrix) (at a constant lateral location), comprised of F frames, the support of H can be time varying, as follows (2.2) can be written as

$$\begin{gathered} R\left(t+n{t}_F\right)=\mathit{M}H\left(t+n{t}_F\right), \\ subjectto\parallel H{\left(t+n{t}_F\right)\parallel }_0\le K,n=0,1,2,\dots \dots F-1 \end{gathered}$$ (2.7)

Where H(t + nt_F) denotes the sparse support, varying over time corresponding to the motion of the clutter source, and t_F denotes the time interval between frames. For a given ultrasound imaging sequence, M is constant as the PSF at a given location does not vary between frames. Consequently, we consider variation in the clutter as solely due to the variation in H(t + nt_F) due to the motion of the clutter source.

Defining R_m as the matrix of clutter superimposed on the M-mode, where each column is an instance of R(t + nt_F) and H_m as a similar matrix where each column is an instance of H(t + nt_F), we can write

$${\mathit{R}}_{\mathit{m}}=\mathit{M}{\mathit{H}}_{\mathit{m}},subjectto{\parallel {\mathit{H}}_{\mathit{m}}\parallel }_0\le K$$ (2.8)

For quasi-static clutter, we impose an additional constraint

$$Rank\left({\mathit{R}}_{\mathit{m}}\right)\ll F$$ (2.9)

It may be observed, that the rank of R_m is one only if the support of R(t + nt_F) (locations of non-zero elements in H(t + nt_F) ) is the same at all time instants (static clutter), this is not true in general, as the motion of the clutter source can vary over time.

Formulating this as a basis learning problem,

$$\begin{array}{l}argmi{n}_{\mathit{D},\mathit{\alpha }}\parallel {\mathit{I}}_{\mathit{m}}-\mathit{D}\mathit{\alpha }{\parallel }_{\mathit{F}}^{\mathbf{2}},\hfill \\ \text{subject}\text{to}\parallel \mathit{\alpha }{\parallel }_{\mathit{0}}\le K_1,Rank(\mathit{D}\mathit{\alpha })\mathbf{\ll }F\hfill \end{array}$$ (2.10)

Here, I_m represents a single M-mode matrix. Now, since we know that the clutter is axially sparse, low rank, while the tissue is not required to satisfy either of these criteria, this problem can be simplified if we focus only on the regions of clutter we define a data determined weighting matrix P_c which exploits these characteristics of the clutter, such that,

$${\mathit{P}}_{\mathit{c}}{\mathit{I}}_{\mathit{m}}=\mathit{C}\mathbf{+}{\mathit{T}}_{\mathit{c}}$$ (2.11)

Where C denotes the clutter, and T_c denotes the underlying tissue in the region where clutter is present. Defining P_c as a sparse matrix containing one where the clutter is present, and zero elsewhere.

Consequently, we can modify (2.10) as.

$$\begin{array}{l}argmi{n}_{\mathit{D},\mathit{\alpha }}{\Vert {\mathit{P}}_{\mathit{c}}{\mathit{I}}_{\mathit{m}}-\mathit{D}\mathit{\alpha }\Vert }_{\mathit{F}}^{\mathbf{2}}\hfill \\ \text{subject}\text{to}Rank(\mathit{D}\mathit{\alpha })\ll F\hfill \end{array}$$ (2.12)

Hence, it suffices to use a suitable low rank estimator to remove the clutter, after detection. It may be appreciated, that the columns of D obtained as the solution to this problem, are linear combinations of the columns of M (2.6–2.10). In the next section, we describe an automatic method of detecting clutter (obtaining P_c) without using the clutter’s spectral properties.

III. DETECTION OF CLUTTER

In this section, we describe a procedure to obtain the matrix P_c. As we note in section II, the clutter is temporally more correlated when compared with the tissue. For quasi-static (low rank) clutter, it remains more highly correlated under a random permutation of the columns (frames in each A-line through time (M-mode)), and remains low rank compared to the tissue. Using this principle, we demonstrate an inexpensive method of detecting the clutter without using its spectral properties as illustrated in Fig. 2. We use a bootstrapped method [25] (randomly sampling with replacement) to detect the clutter.

Fig. 2.

(a) Schematic of clutter detection method. (b) Clutter map (c) Detected clutter

A bootstrapped subset of columns from the M-mode (depth x time) is randomly permuted, and the phase of these two subsets is correlated with a suitable regularizer (λ) to prevent spurious detection. The possibility of a spurious detection can be further reduced by repeating this procedure several times (E) and averaging the results.

It may be observed that, for a given subset of (K_a) A-lines (yellow lines in Fig. 2), the number of possible permutations grows combinatorially with the size of the subset. Thus, the larger the size of the subset, the greater the number of possible unique permutations. However, to obtain a robust estimate, it suffices that we obtain a sufficient contrast between clutter and tissue or noise, and hence, these permutations need not be evaluated exhaustively, as may be observed in Fig. 2(b).

Once the clutter is detected, all subsequent steps are performed only on the cluttered regions. Consequently, computational complexity is reduced and retention of useful signal is enhanced. In cases where there is a large motion of clutter, this detection can also be applied locally in time. We now analyse the clutter detection procedure. Consider the correlation of phase (C_r), for single sampling (X_r) and permutation (Y_r) of the clutter

$${\mathit{C}}_{\mathit{r}}=\frac{{\mathit{X}}_{\mathit{r}}\odot {\mathit{Y}}_{\mathit{r}}^{\prime }}{\left|{\mathit{X}}_{\mathit{r}}\odot {\mathit{Y}}_{\mathit{r}}^{\prime }\right|+\lambda }$$ (3.1)

Where ⊙ denotes the element wise product and Y′ denotes the complex conjugate of Y. Note here that the division is element wise. We note that

$${\mathit{C}}_{\mathit{r}}=\frac{\left|{\mathit{X}}_{\mathit{r}}\odot {\mathit{Y}}_{\mathit{r}}^{\prime }\right|e^{j\text{Δ}{\varphi }_r}}{\left|{\mathit{X}}_{\mathit{r}}\odot {\mathit{Y}}_{\mathit{r}}^{\prime }\right|+\lambda }$$ (3.2)

Where Δφ_r represents the difference in phase at the rth sample. To obtain the i^th estimate Cⁱ_E, we average the columns of C_r, and for each estimate, we have

$${\mathit{C}}_{\mathit{E}}^{\mathit{i}}=\left(\frac{1}{K_a}\right)\sum _{r=1}^{K_a}\frac{\left|{\mathit{X}}_{\mathit{r}}\odot {\mathit{Y}}_{\mathit{r}}^{\prime }\right|e^{j\text{Δ}{\varphi }_r}}{\left|X_r\odot Y_r^{\prime }\right|+\lambda }$$ (3.3)

To avoid cumbersome notation and obtain a more general case, we can write each column of X_r and Y_r as a diagonal matrix.

$${\mathit{X}}_{\mathit{r}}=\left[{\mathit{X}}_{\mathbf{1}}{\mathit{X}}_{\mathbf{2}}\dots {\mathit{X}}_{K_a}\right]$$ (3.4)

X_i denotes the ith column of X_r.

$${\mathit{D}}_{\mathit{X}}^{\mathit{i}}=\left[\text{diag}\left({\mathit{X}}_1\right)\text{diag}\left({\mathit{X}}_2\right)\dots .\text{diag}\left({\mathit{X}}_{K_{\text{a}}}\right)\right]$$ (3.5)

Thus, Dⁱ_X is matrix consisting of appended diagonal matrices of size M rows and K_ax M columns for the i^th estimate.

Similarly,

$${\mathit{D}}_{\mathit{Y}}^{\mathit{i}}=\left[\text{diag}\left({\mathit{Y}}_{\mathbf{1}}\right)\text{diag}\left({\mathit{Y}}_{\mathbf{2}}\right)\dots .\text{diag}\left(Y_{K_a}\right)\right]$$ (3.6)

And defining,

$${\mathit{D}}_{\mathit{X}}^{\mathit{i}}{\left({\mathit{D}}_{\mathit{Y}}^{\mathit{i}}\right)}^{\mathit{H}}={\mathit{R}}_{\mathit{x}\mathit{y}}^{\mathit{i}}$$ (3.7)

This allows us to express (3.3) as

$${\mathit{C}}_{\mathit{E}}^{\mathit{i}}=\left(\frac{1}{K_a}\right){\left(\left|{\mathit{R}}_{\mathit{x}\mathit{y}}^{\mathit{i}}\right|+\mathbf{\text{Λ}}\right)}^{-1}{\mathit{R}}_{\mathit{x}\mathit{y}}^{\mathit{i}}$$ (3.8)

Where Λ is a diagonal matrix of depthwise weighting and ${\left(D_Y^i\right)}^H$ represents the Hermitian operation (transpose conjugate) of $D_Y^i$ for the i^th estimate.

It may be observed that, as long as the matrix $\left(\left|R_{xy}^i\right|+\mathbf{\text{Λ}}\right)$ is strictly diagonally dominant $C_E^i$ exists. For each M-mode, several such estimates can be performed and averaged to reduce the possibility of spurious detection. The averaging step can be represented by multiplying (3.8) with $\left(\frac{1}{E}\right)1_{E\times M}$ (a vector of ones of length ExM), scaled by the number of permutations.

$${\tilde{C}}_E=\left(\frac{1}{K_{a}E}\right){\left(\left(\left|{\mathit{R}}_{\mathit{x}\mathit{y}}^{\mathit{i}}\right|+\mathbf{\text{Λ}}\right)\right)}^{-1}{\mathit{R}}_{\mathit{x}\mathit{y}}^{\mathit{i}}{\mathbf{1}}_{E\text{x}M}$$ (3.9)

Where ${\tilde{C}}_E$ denotes the component-wise average value of C_E, the estimate for the clutter.

IV. COMPLEXITY ANALYSIS OF DETECTION

Considering a dataset of M axial samples and N lateral samples (A-lines) with F frames. For a single estimate of the clutter, of K_a A-lines, we obtain a complexity of O(K_a (M)²) [26]. This scales linearly with the number of such estimates, so for E such estimates, we obtain E(K_a (M)²) operations per A-line. For example, using nominal values of M, N and K_a, comprising one thousand axial samples, one hundred A-lines, fifty frames, five lines per sample and ten estimates per sample, the number of operations required to detect the clutter using our method is approximately 5 × 10⁹ operations at the first iteration.

For subsequent iterations, the number of operations reduces as the procedure is performed only on regions of detected clutter, which is sparse relative to the axial dimension, and as the clutter is removed with each iteration. Using a nominal value of the sparsity being 20% of the A-lines and 20% of the axial samples, the cost of the detection on the second iteration becomes 4 × 10⁸ operations.

In contrast, existing methods [4,14] exhaustively search through patches and evaluate the presence of clutter using spectral properties. Consequently, they do not adapt their computational cost depending on the clutter. For the SVF method [4] the complexity is dominated by performing the SVD (O(mn²+n³)) [26] repeatedly for each patch, while for the MCA method [14], the computational complexity is dominated by obtaining the dictionary atoms. Once obtained, the cost of finding clutter atoms is greatly reduced as the spectral properties of only the dictionary atoms need to be evaluated. For example, using a nominal patch size of 10 × 10 samples, on the dataset described above, we obtain 1×10¹⁰ operations for the SVF.

In case of the MCA, a number of factors influence the computational cost such as the number of iterations of dictionary learning, the sparsity of each patch in the learned dictionary, and the number of dictionary atoms to be learned. For simplicity, we assume the sparsity to be constant, at 10% of the atom dimensionality, so for a 10 × 10 patch from the dataset described above, the sparsity is 10, and the dictionary size to be twice the atom dimensionality (200 atoms), executed for 20 iterations. In addition, we assume that 20% of the patches (10⁶ patches) are used for training the dictionary and the orthogonal matching pursuit (OMP) is used for the sparse coding stage.

Since the number of patches is far larger than the number of atoms, the sparse coding stage dominates the computational complexity of the algorithm. For each patch, the cost of OMP is O (pqk+k³), where p and q are the size of the dictionary, and k is the sparsity. The cost is dominated by the orthogonalization, which involves computing the pseudo-inverse of a matrix of size p × k (O (k³)). For the values described, we obtain 2.01 × 10⁵ operations per patch. Hence, we obtain 2.01 × 10¹¹ operations per iteration, for twenty iterations, we obtain 2.01 × 10¹² operations.

V. LOW RANK ESTIMATION

As described in section II, it suffices to estimate the low rank content of the detected cluttered regions in the M-mode, where C is the clutter in this region and T_c is the tissue signal. We can further simplify (2.12), by fixing D to be the principal components of S_c (U) multiplied by a matrix of corresponding coefficients ξ. Consequently, the low rank constraint can be replaced by a sparsity promoting penalty such as the L1 norm, which yields,

$${\mathit{S}}_{\mathit{c}}=\mathit{U}\mathit{\xi }$$ (5.1)

$${\mathit{C}}_{\mathit{c}}=\parallel \mathit{U}\mathit{x}-{\mathit{S}}_{\mathit{c}}{\parallel }_{\mathit{F}}^{\mathit{2}}+\mathit{\eta }\parallel \mathit{x}{\parallel }_1$$ (5.2)

(5.2) is solvable by sparse coding methods [27–29]. The parameter η creates a tradeoff between exact coding of cluttered region containing both moving tissue and clutter and sparsity in the principal component domain.

It has been established that the rank of clutter is lower than that of moving tissue and noise as in (5.2) due to its quasi-static nature [3, 4, 30]. Additionally, the use of principal components as the sparse coding basis implicitly assigns the highest coefficients, with large signal energy, to the lowest rank components of signal in the clutter signal. It may be observed that η can also be a vector of values which allows varying tradeoffs at different spatial locations of the clutter. Here, we describe the scalar case for simplicity.

The low rank signal can also be estimated using ‘Robust PCA’[32] approaches, or, nuclear norm minimization which is formulated as

$${\mathit{C}}_{\mathit{c}}=\min \parallel \mathit{x}{\parallel }_{\ast }\text{subject to}{\parallel {\mathit{C}}_{\mathit{c}}-{\mathit{S}}_{\mathit{c}}\parallel }_{\mathit{F}}^2\le \epsilon$$ (5.3)

Where $\parallel {\parallel }_{\ast }$ denotes the nuclear norm (sum of singular values)). We solve the nuclear norm minimization problem using the singular value thresholding method developed by Cai et al. [32, 33].

‘Robust PCA’ methods utilize similar approaches to separate a matrix into a combination of low rank and sparse matrices. In our approach, we use these methods to estimate the low rank component (clutter) in the aggregated clutter depths. Multiple methods to solve this minimization exist, such as the augmented Lagrangian method [34, 35] and singular value thresholding method [34]. In this paper, we use the singular value thresholding approach due to its simplicity of implementation. Here, the soft thresholding operator is applied to the singular value matrix [33] rather than thresholding the projections onto the principal components.

Our algorithm alternates between detection using the method described in the previous section, and estimation of the low rank content in the detected region. The low rank content is subsequently removed and the process is repeated until no clutter is detected or a fixed number of iterations has elapsed. We describe these methods in greater detail in Appendix I.

It may be appreciated that leveraging the sparsity of the clutter considerably reduces the computational complexity associated with this method due to the repeated use of the SVD. Since the low rank estimators are themselves iterative algorithms, to avoid ambiguity, we refer to the iterations of the low rank estimators as ‘inner’ iterations, and the number of iterations for which the alternating detection and estimation algorithm is executed as ‘outer’ iterations.

VI. COMPLEXITY ANALYSIS OF CLUTTER REMOVAL

In this section, we analyze the computational complexity of the clutter removal step. Here, we restrict ourselves to the spectral domain and examine the cost of reconstructing the dataset, once the bases and dictionary have been obtained for the SVF and MCA respectively. For CLEAR, we evaluate the computational complexity of the sparse coding and nuclear norm estimators operating on the detected clutter regions. We know that the singular value thresholding (SVT) algorithm is much more computationally complex compared to the sparse coding algorithm, owing to the computation of a fresh SVD at each inner iteration [32]. In contrast, the sparse coding is much less computationally complex since the SVD is computed only at each outer iteration.

Additionally, since we know that the left and right singular vectors are orthonormal, some of the products in the iterative shrinkage algorithm can be calculated only once for each outer iteration (see Appendix I), which further reduces the computational complexity. Let the number of outer iterations be I_o and the number of inner iterations be I_i. Since the SVD is calculated only once for each outer iteration, we obtain I_oO (mn²+n³) operations for a matrix of size (m × n). The size of the matrix is limited by the size of the clutter, and we treat the ensemble of all frames together. The complexity of each inner iteration is dominated by the matrix multiplication of the basis and coefficient matrices which requires O (mn²) operations. It may be appreciated that in CLEAR, these steps will be evaluated only on the A-lines where clutter is detected.

The total complexity for the sparse coding method is I_oO (mn²+n³)+ I_i O (mn²). For the nominal dataset considered in section IV, viz. one thousand axial samples, one hundred A-lines and fifty frames, with the clutter occupying 20% of the A-lines and axial samples, we assume a maximum number of 100 inner iterations and five outer iterations, and we assume that the convergence occurs when the maximum number of iterations has elapsed, and all the clutter is detected for all the iterations to obtain a worst case complexity. In this case, we obtain 5.313 × 10⁷ operations per A-line, or 1.062 × 10⁹ operations overall (Evaluated on only 20 A-lines because of clutter detection). In contrast, for the nuclear norm minimization, an SVD is required for every inner iteration. Hence, the computational complexity is I_oI_iO(mn²+n³). Using the same values as discussed earlier (100 inner iterations, five outer iterations, clutter occupies 20% of A-lines and axial samples), we obtain 3.13 × 10⁸ operations per A-line, or 6.26 × 10⁹ operations overall.

For the SVF, the clutter suppression is dominated by the multiplication of the SVD matrices together, albeit after suppressing the low rank part, since we assume a square patch of size 10 × 10 samples. The cost per patch is O(n³). Since it operates on every sample, for the nominal dataset, we obtain 5 × 10⁹ operations. Similarly, for the MCA, the reconstruction involves sparse coding each patch using OMP, as assumed earlier and subtracting the clutter components. The cost is dominated by OMP, which is O(pqk+k³) as discussed in section IV. Using the assumptions on a nominal dictionary size, and sparsity, and running on all patches to completely reconstruct the image, we have 1.005 × 10¹² operations.

VII. METHODS

A. Detection of clutter

First, we vary the different parameters of the clutter detection algorithm as described in section II so as to obtain optimal detection of clutter. This was performed on simulated data in Field II pro [36], with known tissue and clutter, simulated as closely spaced point scatterers superimposed on a phantom (6 mm × 3.5 mm) with an anechoic cyst of approximately 2 mm diameter as described by Mauldin et al. [4] and Turek et al.[14]. The parameters of these simulations are given in Table 1. The ratio of the intensity of the detected signal and the known clutter was used as a metric of accurate detection.

TABLE I.

FIELD II SIMULATION PARAMETERS

Parameter	Values
Center frequency	25 MHz
Sampling frequency	150 MHz
Bandwidth (−6Db)	66.6%
Tissue correlation	0.98
Artifact correlation	1.0
Tissue displacement	1,0.5,0.25 periods per frame
Artifact displacement	0.5,0.25, 0.125 periods per frame

$$d_r=\frac{\sum d_w\odot C_d}{\sum C_d}$$ (7.1)

Here, d_r is the ratio of the detected clutter to the ground truth clutter, d_w is the binary mask obtained from the method described in section III.

We prefer to have a slight overestimation of the cluttered region, rather than an underestimation, so that the clutter can be removed completely and the parameters were tuned accordingly. Additionally, we only consider cases where the tissue motion is strictly greater than that of the artifact motion to satisfy the clutter being quasi-static. The parameters of the field II simulation are as shown in Table 1.

Additionally, we verify the efficacy of the algorithm in instances where there are two quasi-static sources of clutter, but with different motion profiles (e.g. Tissue moving at 1 period per frame, artifact 1 moving at 0.5 periods per frame, artifact 2 moving at 0.25 periods per frame.) As stated earlier, we only consider cases where the artifact motion is strictly less than that of the tissue motion.

The parameters of the clutter detection algorithm, namely, the number of random samples chosen in an A-line over time (M-mode), the number of permutations, the regularization parameter (λ), and the size of the local window were varied, and the ratio of the detected signal’s magnitude to that of the known clutter is calculated. For each set of parameters, ten runs were performed for each combination of tissue and artifact displacements. It may be observed that the entire parameter space is multi-factorial, and exploring it exhaustively quickly becomes untenable. Hence, we sample the parameter space as follows.

As discussed in section III, linear increase in the size of the random sample (K) results in the combinatorial growth of the number of distinct possible random permutations i.e. capable of generating a far greater number of distinct permutations relative to it (approximately 10 times). The smallest such number is five, generating one hundred and twenty possible permutations. This suggests that the sample size must be at least five.

We know from (3.3) that obtaining the detected clutter depends on the regularization parameter λ and the number of permutations (E). We vary the number of permutations, initially in coarse increments of ten (Fig. 4(a)), while simultaneously varying λ from 0.5σ to 2.5σ in steps of 0.5σ. Using this, we obtain an approximately optimal setting for E, which we subsequently refine in finer increments (Fig. 4(c)). Once these parameters were obtained, they were fixed at their optimal values, and the variation in optional window size (Fig. 4(f)) and the sample size Fig. (4(e)) were examined holding E and λ at their optimal values. These experiments were performed for datasets with one clutter source for six datasets, and eight datasets containing two clutter sources with different motion profiles moving at different rates albeit still satisfying the quasi-static assumption.

Fig. 4.

Variation of detected clutter with parameters (a) Varying detection with regularization parameter along permutation axis (b) Varying detection with permutations along regularization parameter axis (c) Refining number of permutations (sample size (5), (regularization parameter (1.5 σ)) (d) Random sample size (regularization parameter (1.5 σ), permutations (10)). (N=10 for each bar)

B. Parameter sensitivity

The regularization parameters (η for the sparse code, and τ for nuclear norm minimization), control the tradeoff between sparsity and equivalence between the contaminated data and estimated clutter. These parameter values play an important role in clutter reduction and preservation of underlying tissue signal. We examine the impact of these parameters by varying their value over a range. For each value of the parameter, ten iterations were performed for each combination of tissue and clutter motion (60 iterations per value). The error with respect to the ground truth tissue data was calculated in the regions of interest (Fig. 3 (a) (white rectangles)). The error in regions containing only tissue (Fig. 3(a)) (red rectangle)) was also calculated to demonstrate the specificity of our approach.

Fig. 3.

(a) Cluttered data with regions of interest of clutter reduction (white), and tissue preservation (red). (b) Representative learned dictionary sorted by NSSA of atoms (c) Tuning cutoff for MCA (d) Tuning cutoff for SVF, where dotted red line in (c) and (d) indicates optimal cutoff.

Representative learned in silico atoms designated as contributing to clutter are indicated in the dotted white rectangle in Fig. 3(b).

C. In silico clutter reduction

The algorithm was validated in silico on simulated datasets in Field II pro [17, 36] as in Mauldin et al.[4] and Turek et al.[14]. Two point scatterers were used to simulate clutter. The tissue displacement and artifact displacement were varied between 0.125 to 1 period of the transducer center frequency per frame leading to six possible combinations of tissue and artifact motion as in Table 1 with one clutter source, and eight possible combinations with two sources of clutter as shown in Table 2. Of these, only combinations where the artifact motion was strictly less than the tissue motion between frames were considered.

TABLE II.

DICTIONARY LEARNING PARAMETERS (IN SILICO)

Parameter	Values
Patch size	24 × 9 (M × N)
Dictionary atoms	432
Maximum sparsity	20% of atom dimensionality
Error	2.3σ$\sqrt{2MN}$
Iterations	20

The algorithms were benchmarked against two known methods, the SVF of Mauldin et al. [4], and the MCA method of Turek et al.[14]. The parameters of the MCA and SVF methods were optimized by running on a balanced dataset consisting of 1.44 × 10⁶ patches of IQ data randomly chosen from all the datasets. With each loop of detection, low rank estimation and clutter removal, the penalty parameters in the low rank estimators η and τ are annealed as iterations progress, because the remaining signal is of lower magnitude (intensity) compared to the signal at the start.

For the MCA method, care was taken to balance the datasets so as to avoid bias in the training. The parameters for the dictionary training are given in Table 2. The patch size was kept constant between the methods to facilitate a fair comparison. The cutoff for the algorithms was chosen by comparing the error at various cut off values to the ground truth error for ‘perfect filtering’. (dotted red line (Fig. 3 (c)–(d)) The error value was chosen in accordance with [14]. Additionally, since the clutter is progressively reduced with each iteration, the parameters of the sparse coding and nuclear norm estimation can be tied to the maximum principal component projections and singular values, respectively, which reduces the need to empirically choose these parameters.

For CLEAR, the optimal values of η and τ obtained in section VII.B were used. For the SVF method, the cutoff value of 0.55 with a sigmoid slope parameter of 70 was found to be optimal (Fig. 3 (d)), while for the MCA method, a cutoff value of 0.7 (Fig. 3 (c)) was found to be optimal, consistent with [14].

Additionally, we evaluate the clutter reduction and tissue preserving characteristics of the proposed method with respect to the SVF and MCA methods applying the clutter reduction methods of the SVF [4] and MCA [14] only on the regions of clutter detected by the method described in section III.

D. In vivo clutter reduction

The performance of the algorithms was also compared in vivo on mouse heart cineloops as in Mauldin et al.[4]. The datasets were acquired using the VEVO 2100 scanner (Fujifilm VisualSonics, Toronto, Canada) with the MS-400 transducer operating at 30 MHz. ‘Cluttered’ datasets were simulated using a piece of solder wire (Kester 44, 0.02 inch diameter) placed between the mouse’s chest and the transducer [4]. Although this approach does not provide an entirely realistic form of clutter, it does produce clutter of approximately the correct extent and possessing the necessary quasi-static quality. Additionally, compared with more accurate methods (e.g. surgically induced scarring), this approach provides a near perfect matched “gold standard” by way of removing the wire when desired. The performance of the clutter reduction metrics was quantified using a measure of tracking error (7.2), the error between the initial (x1, y1) and final points (x2, y2) of the trajectory of motion normalized to the corresponding trajectory length.

$$\frac{\sqrt{{(x2-x1)}^2+{(y2-y1)}^2}}{\sqrt{{\left({\sum }_{i\in Roi}{x}_i\right)}^2+{\left({\sum }_{i\in Roi}{y}_i\right)}^2}}$$ (7.2)

The motion vectors were obtained using normalized cross correlation with a window of approx. (0.6 mm × 0.6 mm). Subsequently, the displacements were filtered by a median filter to remove outliers. Subsequently, a temporal fifth order Fourier fit was performed on the filtered vectors [38, 39].

The myocardium was manually masked in the first frame and the displacement vectors were used to interpolate the mask to subsequent frames. Traces of large displacements, due to vectors detected in the blood pool in the center of the ventricle remained after this procedure, which were removed by discarding vectors with magnitude greater than three standard deviations from the mean. The parameters used for CLEAR are given in Table 3. The same parameters were used for all datasets.

TABLE III.

PARAMETERS OF CLEAR (IN VIVO)

Parameter	LI minimization	Nuclear norm minimization
Maximum number of outer iterations	5	5
Initial $\eta (\tau )$	2000	20 times the maximum singular value of detected clutter
Inner iteration	100	100

For the dictionary learning and SVF methods, the patch size was chosen to be 15 × 15 in accordance with [4] and [14]. The cutoff value for both methods was found to be optimal at 0.9. The sparsity of the dictionary was 20% of the atom dimensionality as in the in silico experiments and the error value was used as the same function of the noise standard deviation and patch size.

Additionally, as in section VII. C, we evaluate the performance of the algorithms on the datasets using the same cutoff parameters both using the mask detected by the method described in section III and without using the mask.

VIII. RESULTS

A. Detection of clutter

The variation of the detected signal with respect to the ground truth is illustrated in Fig. 4. It may be observed in Fig 4 (a). that increasing the value of the regularization parameter λ, regardless of the number of permutations, generally results in better detection. A similar trend was observed while varying the regularization parameter in Fig. 4 (b).

We observe from the fine tuning of the number of permutations (Fig. 4(c)) that 10 permutations were sufficient for adequate signal detection and the number of permutations need not be exhaustively evaluated. Additionally, we observe in Fig.4 (d) that the detection is nearly independent of the sample size, once it is chosen to be ‘large enough’ as discussed in section VII.A.

Additionally, it may be observed that the regularization parameter Fig. 4 (b) is tuned with respect to the noise variance, and a value of one and a half times the noise standard deviation was observed to be optimal, for a sample size of five A-lines and ten permutations. Similar results were obtained for datasets of two sources of clutter as seen in Fig. 5 (a) and (b).

Fig. 5.

Variation of detected clutter with parameters for 2 (a) Varying detection with regularization parameter along permutation axis (b) Varying detection with permutations along regularization parameter axis. (N=10 for each bar)

B. Parameter sensitivity

We examined the effect of the regularization parameters η and τ while using either the L1 minimization by iterative shrinkage, or the nuclear norm minimization by singular value thresholding, as discussed in sections IV A and B. The error was measured as the mean squared error of the images obtained within the ROI of the white boxes as seen in Fig. 3. (a).

$$\begin{array}{c}\frac{{\sum }_{roi1}10{log}_{10}{\left(I_{noclutter}-I_{decluttered}\right)}^2}{{\sum }_{roi1}roi1}\\ +\frac{{\sum }_{roi2}10{log}_{10}{\left(I_{noclutter}-I_{decluttered}\right)}^2}{{\sum }_{roi2}roi2}\end{array}$$ (8.1)

Here, ${\sum }_{roi2}roi2$ denotes the summation of the number of pixels in the respective regions of interest, and I_noclutter and I_decluttered denote the intensity with no clutter and after applying each of the decluttering algorithms.

We quantified the performance of these methods by measuring the error with respect to the ground truth for ten runs of each of the six combinations of tissue and clutter motion as indicated in Table 4 for ten runs at each value. We observe a minimum error when η takes the value 10⁻² and τ takes the value 100 as can be observed in Fig. 6 (a, b). This is consistent with our signal model in section III.

TABLE IV.

CLUTTER REDUCTION PERFORMANCE (ERROR WITH RESPECT TO GROUND TRUTH) OF CLEAR (L1 AND NUCLEAR NORM) COMPARED TO MCA AND SVF (1 SOURCE OF CLUTTER)

Tissue motion (Periods per frame)	Artifact motion (Periods per frame)	SVF error (dB) (Mean ± S.D.) N=10 (Without mask)	SVF error (dB) (Mean ± S.D.) N=10 (With mask)	MCA error (dB) (Mean ± S.D.) N=10 (Without mask)	MCA error (dB) (Mean ± S.D.) N=10 (With mask)	L1 minimization error (dB) (Mean ± S.D.) N=10	Nuclear norm minimization error (dB) (Mean ± S.D.) N=10
1	0.5	8.49 ±0.18	6.63 ± 0.07	11.97 ±0.28	12.22 ± 0.27	5.52± 0.62	9.2 ±0.14
1	0.25	4.98 ±0.13	4.79 ± 0.08	7.57 ±0.23	7.18 ±0.38	3.38 ± 0.05	6.96 ±0.16
1	0.125	1.88 ± 0.17	2.65 ± 0.05	6.09 ± 0.20	2.30 ±0.31	2.29 ± 0.02	4.82 ±0.35
0.5	0.25	4.28 ± 0.16	5.17 ±0.06	7.65 ± 0.38	4.65 ± 0.41	4.28 ± 0.12	8.54 ±0.14
0.5	0.125	2.63 ±0.13	2.54 ± 0.05	9.25 ± 0.27	5.44 ±0.51	3.72± 0.72	6.83 ±0.19
0.25	0.125	5.94 ± 0.23	2.61 ± 0.07	12.9 ±0.38	4.53 ± 0.32	2.25 ± 0.04	4.72 ±0.68

Fig. 6.

Variation of Error with respect to ground truth with regularization parameters (a) Variation of error with respect to η (b) Variation of Error with respect to τ. Error bars indicate standard deviation. (N=10 for each bar)

The parameters control the trade off the error with respect to the true corrupted data. It follows that at these values, most of the estimated low rank from the detected clutter data is clutter and very little underlying tissue signal is removed, if any. All subsequent in silico testing was carried out using these values.

Additionally, the sensitivity of the CLEAR algorithm was also evaluated for eight datasets with varying motion profiles for both L1 minimization and nuclear norm minimization. Similar results were observed in both cases as seen in Fig. 6 (c,d).

C. In silico clutter reduction

Prior works by Mauldin et al. [4] and Turek et al. [14] have reported the efficacy of the SVF and MCA methods respectively on in silico simulated data using Field II.

However, the in silico analysis in these works were limited to one set of tissue and artifact motion (Tissue motion 1 period per frame, and artifact motion 0.125 period per frame). Here, we compare the performance of CLEAR with these methods as the artifact and tissue motion varies. We summarize our results in Table 4 and Table 5, the best results are highlighted in bold. In Table 4, the performance was quantified in two ways; first, by measuring the error with respect to the ground truth tissue signal in the regions of interest indicated by the white box in Fig. 3 (a), as a measure of clutter reduction.

TABLE V.

TISSUEPRESERVING PERFORMANCE OF PROPOSED METHOD COMPARED TO MCA AND SVF (NO ERROR OBSERVED USING PROPOSED METHOD)

Tissue motion (Periods per frame)	Artifact motion (Periods per frame)	SVF Error (dB) (Mean ± S.D.) N=10 (Without mask)	MCA Error (dB) (Mean ± S.D.) N=10 (Without mask)
1	0.5	0.24 ±0.1	3.45 ±0 .41
1	0.25	0.25 ± 0.09	3.38 ±0.44
1	0.125	0.28 ± 0.09	3.38 ±0.43
0.5	0.25	0.72 ±0.12	2.89 ± 0.72
0.5	0.125	0.8 ±0.12	2.83 ± 0.75
0.25	0.125	5.68 ±0.18	7.61 ±0.78

It may be observed, from Table 4, that the performance of CLEAR is comparable to that of the SVF and MCA methods when the tissue motion is high (1 period per frame and 0.5 period per frame), but as the tissue motion drops and approaches that of the clutter, the proposed algorithm shows a clear advantage with better preservation of the underlying signal, illustrated by the smaller error (highlighted in bold). Both these methods, being patch-wise methods lack the ability to discriminate between signals of similar local rank.

The MCA results were calculated with respect to the reconstructed ground truth data using the same dictionary for a fair comparison, to avoid bias due to loss of signal from the error constraint. In accordance with our signal model, CLEAR does not suffer from these limitations as it does not operate in a patch-wise manner while removing the clutter.

Representative figures from the clutter suppression tests for artifact motion 0.25 periods per frame, and tissue motion 0.5 periods per frame on data simulated in Field II are shown in Fig.6. Additionally, since both these methods act on the entire dataset, we consider the performance of both methods with respect to the preservation of tissue signal that is not obscured by clutter in Table 5.

The region of interest for this measurement is denoted by the red box as shown in Fig.3 (a) and Fig. 6(a). Since CLEAR does not act on the region outside the detected clutter, no error was observed using either nuclear norm or L1 minimization. Additionally, it may be observed that the error increases as the tissue motion approaches that of the artifact.

The results in Table 5 reinforce our results in Table 4 that CLEAR preserves more tissue signal, by only acting in regions where the clutter is detected. However, when operating only on the regions where the mask is applied, all clutter estimators have similar performance as seen in Table 4. The error for the MCA method was calculated with respect to the reconstructed dataset so as to not bias the result toward potential signal loss due to sparsity or error constraints. When the SVF and MCA are estimators are applied only on regions of clutter detected using the method described in section III, no error was observed and the tissue was preserved outside the detected region.

Additionally, the performance of the SVF, MCA and CLEAR algorithms were compared on eight datasets with two clutter sources with different motion profiles. Both motion profiles were kept strictly smaller than that of the tissue to keep the clutter quasi-static. The results of the clutter reduction and tissue preservation experiments are shown in Tables 6 and 7 respectively, the best performances are indicated in bold. As seen when one clutter source was considered, there was no error in the tissue ROI when the MCA and SVF methods were applied on the clutter detected using the mask obtained from our method.

TABLE VI.

CLUTTER REDUCTION PERFORMANCE OF PROPOSED METHOD COMPARED TO MCA AND SVF (2 CLUTTER SOURCES)

Tissue motion (periods per frame)	Artifact 1 motion (periods per frame)	Artifact 2 motion (periods per frame)	SVF Error wrt. ground truth (no mask) (dB) (N=10)	SVF Error wrt. ground truth (mask) (dB) (N=10)	MCA Error wrt. ground truth (no mask) (dB) (N=10)	MCA Error wrt. ground truth (mask) (dB) (N=10)	L1 minimization error wrt. ground truth (dB) (N=10)	Nuclear norm minimization error wrt. ground truth (dB) (N=10)
1	0.5	0.25	6.39 ± 0.08	5.56 ± 0.18	9.19 ± 0.27	6.54 ± 0.11	4.27 ± 0.14	2.32 ± 0.31
1	0.5	0.125	5.42 ± 0.09	4.62 ± 0.17	9.05 ± 0.28	5.38 ± 0.09	3.97 ±0.10	2.38 ± 0.34
1	0.25	0.5	6.97 ± 0.10	5.46 ± 0.13	14.88 ±0.30	9.18 ± 0.20	4.09 ± 0.04	6.73 ±1.14
1	0.25	0.125	5.49 ± 0.10	4.10 ± 0.11	15.30 ±0.30	7.90 ± 0.22	3.65 ± 0.002	1.83 ±0.37
1	0.125	0.5	6.89 ± 0.12	5.68 ± 0.16	16.42 ± 0.29	9.66 ± 0.47	3.99 ± 0.006	7.40 ±1.72
1	0.125	0.25	6.39 ±0.13	5.21 ±0.18	16.95 ± 0.28	9.68 ±0.15	3.76 ±0.14	2.78 ±0.34
0.5	0.25	0.125	5.79 ± 0.09	4.41 ± 0.09	17.52 ± 0.29	7.65 ±0.12	3.27 ±0.002	1.90 ±0.12
0.5	0.125	0.25	6.79 ± 0.12	5.66 ±0.16	15.45 ± 0.19	10.88 ±0.21	4.00 ±0.14	2.39 ±0.41

TABLE VII.

TISSUE PRESERVING PERFORMANCE OF PROPOSED METHOD COMPARED TO MCA AND SVF (2 CLUTTER SOURCES)

Tissue motion (periods per frame)	Artifact 1 motion (periods per frame)	Artifact 2 motion (periods per frame)	MCA Error w.r.t ground truth (dB) (Mean ± s.d.) 2 clutter sources (without mask) (N=10)	SVF Error w.r.t ground truth (dB) (Mean ± s.d.) 2 clutter sources (without mask) (N=10)
1	0.5	0.25	0.09 ± 0.04	0.061 ± 0.04
1	0.5	0.125	0.09 ± 0.04	0.062 ± 0.04
1	0.25	0.5	0.10 ± 0.06	0.11 ± 0.05
1	0.25	0.125	0.10 ± 0.05	0.11 ± 0.06
1	0.125	0.25	0.07 ± 0.04	0.09 ± 0.03
1	0.125	0.5	0.070 ± 0.04	0.09 ± 0.04
0.5	0.25	0.125	3.58 ± 0.12	0.10 ± 0.12
0.5	0.125	0.25	3.66 ± 0.17	0.07 ± 0.17

D. In vivo clutter reduction

We validate the proposed algorithm in vivo on five short axis mouse heart datasets acquired using the VisualSonics VEVO 2100 scanner at 30 MHz [4] using both the nuclear norm minimization and L1 minimization as low rank estimators. No local refinement of the detected clutter was found to be necessary for these datasets. A representative in vivo result using CLEAR is shown in Fig. 7. Fig. 7 (a-c) depict the same cluttered B-mode after each outer iteration. Fig.7 (d,e) depict the detected clutter at each iteration. Fig.7 (f-h) depict the M-mode (a single A-line through time), indicated in Fig. 7 (a-e) by the yellow line.

Fig. 7.

Representative in silico result (Tissue motion 0.5 periods per frame, artifact motion 0.25 periods per frame) (a) Cluttered data (b) decluttered using SVF (c) decluttered using MCA (d) decluttered using CLEAR (L1 minimization) (e) decluttered using CLEAR (nuclear norm minimization) (f) ground truth data (no clutter) (g) ground truth clutter artifacts (h) detected clutter for CLEAR.

A representative result of the reduction in clutter using the proposed method compared to the SVF and MCA methods is shown in Fig. 8, with the white arrows indicating the region of suppressed clutter. All the datasets were normalized to the same intensity baseline. It may be observed that the SVF and MCA methods Fig. 8 (d) and (f) slightly reduce the contrast as they act on the entire dataset when compared to the CLEAR using either low rank estimate Fig.8(b) and (c). The performance of the method is quantified using tracking error normalized to the length of the trajectory over a cardiac cycle. The vectors from end systole and end diastole were used to obtain the error as in Mauldin et al. [4]. Representative results are shown in Fig. 9. The region of interest (ROI) chosen is indicated in white, with white arrows indicating the region of clutter removed with improved tracking in the corresponding ROI.

Fig. 8.

Representative in vivo result of CLEAR (a) Cluttered data (b) Decluttered B-mode using CLEAR (Iteration 1) (c) Decluttered B-mode using CLEAR (Iteration 2) (d) Detected clutter iteration 1 (e) Detected clutter iteration 2 (f) Cluttered M-mode (Yellow line in (a)) (g) Decluttered M-mode (Yellow line in (b)) (h) Decluttered M-mode (Yellow line in (c)).

Fig. 9.

Representative in vivo result (a) Cluttered data (b) decluttered using CLEAR (L1) (c) decluttered using CLEAR (nuclear norm) (d) decluttered using MCA (with mask) (e) decluttered using MCA (without mask), (f) decluttered using SVF (with mask) (g) decluttered using (SVF (without mask)

The results are indicative of underlying tissue signal being preserved after the clutter reduction algorithms are applied. This is further strengthened by the reduction of the tracking error using the SVF and MCA when the mask is applied, as compared to the absence of the mask. In Fig.10, we show the results of the tracking error metric described. CLEAR performs comparably with the MCA, reducing the tracking error by slightly more than half, and better than that of the SVF which reduces the tracking by approximately 25%. The tracking failed to produce usable tracking vectors as outlined in section V c in the ROI of one of the cluttered sets and the corresponding MCA set as the clutter was not adequately removed. Hence, we indicate the corresponding number of sets next to each bar in Fig.10. The error bars indicate standard deviation.

Fig. 10.

Representative in vivo tracking result (a) Cluttered data (b) decluttered using SVF (c) decluttered using MCA (d) decluttered using CLEAR(L1 minimization) (e) decluttered using CLEAR(nuclear norm minimization), (f-j) Corresponding (a)-(e) with tracking vectors super imposed. ROI indicated in white in (a)-(e) white arrows show removed clutter and improved tracking in ROI.

E. Runtime

Finally, we also compare the runtimes of the algorithm on an A-lines with clutter from the in silico datasets. The runtimes of the different algorithms were compared firstly, for clutter removal for the SVF, MCA, and CLEAR. We omit the time required to train the dictionary for MCA and assume we already have a trained dictionary in place. Additionally, both the SVF and MCA detection methods were allowed to run in parallel on the computer’s cores while the CLEAR method was not.

The average of ten runs was calculated and the results are as shown below in Fig. 11, the error bars indicate standard deviation. All benchmarking was performed on an Intel i7 CPU (4 cores), with 16 GB RAM, running Windows 10 (64 bit) and MATLAB R2018a. The number of ‘outer’ iterations of CLEAR was set to five, while the maximum number of inner iterations was set to one hundred.

Fig. 11.

Tracking loop error normalized to the corresponding trajectory length in the chosen ROI.

The patch-size used was of size 24×9 (axial × temporal) samples. Without masking, the procedure was applied on approximately sixteen thousand patches, while being applied on approximately two thousand six hundred patches after masking.

It may be observed from Fig. 11, that applying the clutter estimation only in regions of detected clutter significantly lowers the run time per A-line. While CLEAR operates on a smaller matrix, it converges to an estimate within the specified tolerance (10⁻⁶) rapidly, in approximately 2–3 outer iterations and approximately twenty internal iterations per outer iteration. The runtime for the SVF and MCA may be higher than expected due to overhead during parallel processing, may be further brought down by using incremental refinement methods, as developed by Brand [39, 40], or approximation methods as developed by Rubinstein et al. [41].

IX. DISCUSSION

In this paper, we present a generalized ‘closed loop’ method and model to estimate and remove detected quasi-static clutter in echocardiography. The method uses the axial sparsity and high temporal coherence of quasi-static clutter to detect and remove the clutter iteratively. In addition, detection of the clutter is not performed by using spectral characteristics, decoupling it from the signal separation problem. The decoupling enables us to deal with each part separately using computationally inexpensive operations, thus enabling algorithmic acceleration.

Additionally, the method accounts for spatial variation in the rank of the clutter without resorting to the computationally cumbersome option of utilizing local signal characteristics. Through plane motion, multipath artifacts and other possible mechanisms may increase the local estimate of the rank, making it comparable to the size of the local neighborhood used, and consequently impairing patch-based methods. In general, the number of frames is much larger than the size of the block used, and consequently, utilizing the entire dataset overcomes this limitation.

Our method is also faster than existing patch-based methods as it operates only on the regions of clutter, detected based on the coherence of the signal. Additionally, we do not rely on the spectral properties of the clutter for detection. Instead, it is estimated from bootstrapped samples from the M-mode. In the presence of exaggerated motion, such as due to respiration, the window may be optionally refined locally to detect moving clutter.

CLEAR also preserves more usable tissue signal, comparing favourably with the SVF (4.7 ± 0.17 dB, without the mask, 4.07 ± 0.07 dB with the mask) and MCA methods (9.24 ± 0.30 dB, without the mask, 6.05 ± 0.38 dB with the mask) with a single source of clutter as seen in Table 4 (7.16 ± 0.39 dB using the L1 norm, 6.85 ± 0.34 dB using the nuclear norm) (averaged over all motion profiles) and outperforms the SVF and MCA methods for two sources of clutter may be observed in Tables 6 and 7 (3.88 ± 0.093 dB (sparse coding) and 3.47 ±0.78 dB using the nuclear norm), even when the mask determined from our clutter detection is applied. This is lower than that of the SVF (8.5 ± 0.7 dB) and the MCA (9.3 ± 0.5 dB). Additionally, since CLEAR acts only on regions of clutter, it preserves more tissue signal. For either low rank estimator, no error was seen in when CLEAR was used in data outside the cluttered region regardless of artifact motion or tissue motion. In both the exhaustively applied patch-wise methods, the error increased as the disparity of motion between the artifact and tissue reduced as seen in Table 4.

Despite the closed loop nature of our method, it could also be implemented online (as images are acquired) by progressively refining the detection from the first few frames, with each frame acquired. In addition, The principal components may be updated using incremental refinement as was proposed by Brand [39, 40]. In this implementation too, our method is potentially less expensive compared to patch-based methods, such as the SVF, because fewer refinements would be required.

Additionally, our study demonstrates that detection of quasi-static reverberant clutter can be achieved without using spectral properties in a local neighborhood of the signal. While we use this method to detect clutter here, it may not be limited to just reverberant clutter. In this work, (as described in section I) we exploit prior knowledge of acoustic propagation to sparsely model reverberant clutter. Other forms of clutter in echocardiography, such as grating lobe artifacts from off axis reflections, may be sparsely represented in the Fourier domain. The CLEAR algorithm may also be applied in the transform domain in future studies.

Our study is limited in the simulating the motion of the clutter in the axial direction, other motion profiles, such as lateral or through plane motion may be examined in future studies. Lateral resolution is coarser than axial resolution, and significant lateral motion may necessitate the use of multidimensional (3 D, or 2D+T) dictionary atoms as suggested by Turek et al. [14]. Additionally, we artificially create artifacts in vivo by placing solder wire between the transducer and the mouse chest, as was performed by Mauldin et al. [4]. While this method may not provide entirely realistic artifacts, using this method allows us to obtain reproducible quasi-static clutter.

Furthermore, in this study, the initial dictionary learning is entirely unsupervised, and a suitable discriminating function (Rank 1 content of the atoms) [14] is used to provide discrimination between tissue and clutter. The dictionaries were trained on patches of tissue and clutter, atoms that contribute to tissue or clutter are not constrained to solely contribute to regions containing only tissue or clutter. It remains to be seen if the MCA method can potentially be improved by incorporating this prior information into the dictionary training.

Finally, our method reduces the tissue tracking error compared to the cluttered signal on mouse heart datasets. The performance compares favourably with the MCA method (approximately 50% reduction), while the SVF reduces the error by approximately 25%. This may be due to the SVF removing more tissue signal when compared to the CLEAR, as is seen in the work of Turek et al. [14].

Fig. 12.

Runtime per A-line comparing SVF, MCA and CLEAR. (N=10) for each bar

APPENDIX

A. Sparse coding

In this paper, we utilize iterative shrinkage methods for sparse coding due to their ease of implementation using gradient descent type approaches. These approaches seek a tradeoff between the sparsity of the signal and the reconstruction error. The parameter η governs this tradeoff. In our approach, this denotes the difference of the clutter energy and the noise floor or underlying tissue.

$${\mathit{C}}_{\mathit{c}}=\parallel \mathit{U}\mathit{x}-{\mathit{S}}_{\mathit{c}}{\parallel }_{\mathit{F}}^{\mathbf{2}}+\mathit{\eta }\parallel \mathit{x}{\parallel }_{\mathbf{1}}$$ (1)

The algorithm proceeds iteratively as follows

$${\mathit{x}}_{\mathit{n}\mathbf{+}1}=S_{\eta }(\mathit{x}-\mathit{l}\left({\mathit{U}}^{\mathit{H}}\left(\mathit{U}{\mathit{x}}_{\mathit{n}}-{\mathit{S}}_{\mathit{c}}\right)\right)$$ (2)

Where l represents the learning rate, S_η represents the soft thresholding operator (3), and U denotes the singular vectors of the aggregated clutter matrix S_c and U^H represents the transpose conjugate (Hermitian operator) of U.

$$S_{\eta }(\mathit{x})=(sgn(\mathit{x})(max(|\mathit{x}|-\eta ,0))$$ (3)

In (3) sgn(x) is the signum function.

(2) is repeated until a specified number of iterations elapses or the error between the estimated clutter and the detected signal $\parallel Ux-S_c{\parallel }_F^2$ is less than a specified tolerance. Since U is orthonormal, UU ^H = I where I is the identity matrix. Additionally U^H S_c does not vary with each iteration and can be calculated only once, speeding up computation.

B. Nuclear norm minimization

In this paper, we use the singular value thresholding approach due to its simplicity of implementation. Here, the soft thresholding operator is applied to the singular value matrix [34] rather than thresholding the projections onto the principal components.

The algorithm proceeds as follows:

Let the initial estimate of C_c be a 0 matrix.

$${\mathit{C}}_{\mathit{c}}=\mathit{U}\mathbf{\text{Σ}}{\mathit{V}}^{\mathbf{\prime }}$$ (4)

$${\mathbf{\text{Σ}}}^{\ast }=S_{\tau }(\mathit{\text{Σ}})$$ (5)

$${\mathit{C}}_{\mathit{c}}^{\mathit{n}\mathbf{+}\mathbf{1}}={\mathit{C}}_{\mathit{c}}^n\mathbf{+}\mathit{l}\left({\mathit{S}}_{\mathit{c}}-\mathit{U}{\mathbf{\text{Σ}}}^{\ast }{\mathit{V}}^{\mathbf{\prime }}\right)$$ (6)

Where S_τ denotes the soft thresholding operator

$$S_{\tau }(\mathit{x})=(sgn(\mathit{x})(max(|\mathit{x}|-\tau ,0))$$ (7)

(4) – (7) are repeated until convergence criteria are met, as described in Appendix I A.