U2-rPCA for Ultrasound Clutter Filtering

• The paper presents a deep unfolded robust PCA method that effectively separates low‐rank tissue clutter from sparse blood flow in ultrasound imaging.
• It unfolds an IRLS-based rPCA algorithm and integrates a sparse enhancement unit (SEU) to refine vessel recovery while retaining interpretability.
• Evaluations demonstrate that U2-rPCA significantly improves power-Doppler contrast and maintains Doppler velocity accuracy over conventional methods.

Unsupervised Unfolded rPCA (U2-rPCA) is a deep-unfolded robust principal component analysis method for clutter filtering in ultrasound microvascular imaging. It is formulated from an iteratively reweighted least squares (IRLS) rPCA baseline, preserves an explicit low-rank-plus-sparse decomposition, and is trained without ground-truth tissue or blood-flow labels by using a reconstruction-based self-consistency loss. In the ultrasound setting, U2-rPCA treats tissue clutter as low-rank, blood flow as sparse, and adds a sparse-enhancement unit (SEU) to improve micro-flow recovery while retaining the mathematical structure of the underlying optimizer (Li et al., 1 Oct 2025).

1. Problem formulation and signal model

U2-rPCA is designed for ultrafast ultrasound microvascular imaging, which begins from a raw beamformed IQ sequence

$$\mathbf{D}\in \mathbb{C}^{N_z\times N_x\times N_t}.$$

For clutter filtering, this three-dimensional sequence is reshaped into a Casorati matrix

$$\mathbf{D}\in \mathbb{C}^{N_s\times N_t}, \quad N_s=N_zN_x,$$

with each column representing one vectorized frame. The decomposition model is

$\mathbf{D}=\mathbf{T}+\mathbf{B}+\mathbf{N},$

where $\mathbf{T}$ is tissue/clutter, $\mathbf{B}$ is blood-flow or microvascular signal, and $\mathbf{N}$ is noise (Li et al., 1 Oct 2025).

The structural prior is standard but operationally important: tissue is modeled as low-rank because it moves slowly and remains highly correlated across frames, whereas blood flow is modeled as sparse because vessels occupy a small spatial fraction in each frame. The purpose of the decomposition is to suppress tissue clutter while preserving sparse micro-flow signals for subsequent power Doppler imaging and, in the simulated setting, Doppler velocity estimation (Li et al., 1 Oct 2025).

The method is motivated by limitations in both SVD-based clutter filtering and classical rPCA. SVD filtering depends on finely chosen singular-value cutoffs, does not guarantee clean subspace separation because tissue, flow, and noise are not perfectly orthogonal, and may miss flow in some branches, producing “hollows” in power Doppler images. Standard rPCA, although better aligned with low-rank-plus-sparse modeling, remains sensitive to hyperparameters such as $\lambda$, inner rank $d$, and thresholds, and requires repeated expensive updates, especially SVD-like operations (Li et al., 1 Oct 2025).

2. IRLS-rPCA baseline and unfolded derivation

Rather than unfolding the nuclear-norm/$\ell_1$-norm relaxation directly, U2-rPCA is derived from an IRLS-based rPCA baseline. Tissue is factorized as

$\mathbf{T}=\mathbf{U}\mathbf{V}^*$

with $$\mathbf{D}\in \mathbb{C}^{N_s\times N_t}, \quad N_s=N_zN_x,$$0, $$\mathbf{D}\in \mathbb{C}^{N_s\times N_t}, \quad N_s=N_zN_x,$$1, and $$\mathbf{D}\in \mathbb{C}^{N_s\times N_t}, \quad N_s=N_zN_x,$$2 an inner dimension satisfying $$\mathbf{D}\in \mathbb{C}^{N_s\times N_t}, \quad N_s=N_zN_x,$$3. The observation model becomes

$$\mathbf{D}\in \mathbb{C}^{N_s\times N_t}, \quad N_s=N_zN_x,$$4

This factorization avoids direct repeated nuclear-norm proximal steps and converts the problem into smoother Frobenius-norm terms (Li et al., 1 Oct 2025).

The unfolded baseline is built from the IRLS objective

$$\mathbf{D}\in \mathbb{C}^{N_s\times N_t}, \quad N_s=N_zN_x,$$5

Here $$\mathbf{D}\in \mathbb{C}^{N_s\times N_t}, \quad N_s=N_zN_x,$$6 is the sparse reweighting matrix and $$\mathbf{D}\in \mathbb{C}^{N_s\times N_t}, \quad N_s=N_zN_x,$$7 is the low-rank reweighting matrix. Their IRLS definitions are

$$\mathbf{D}\in \mathbb{C}^{N_s\times N_t}, \quad N_s=N_zN_x,$$8

and

$$\mathbf{D}\in \mathbb{C}^{N_s\times N_t}, \quad N_s=N_zN_x,$$9

Small sparse entries therefore receive larger weights, and small latent rank components are penalized more strongly (Li et al., 1 Oct 2025).

The alternating IRLS updates are

$\mathbf{D}=\mathbf{T}+\mathbf{B}+\mathbf{N},$0

$\mathbf{D}=\mathbf{T}+\mathbf{B}+\mathbf{N},$1

$\mathbf{D}=\mathbf{T}+\mathbf{B}+\mathbf{N},$2

U2-rPCA unfolds these iterations into a network with $\mathbf{D}=\mathbf{T}+\mathbf{B}+\mathbf{N},$3 layers, where each layer corresponds to one algorithmic iteration. The resulting architecture remains interpretable because each module still has the role of sparse blood extraction, sparse refinement, or low-rank tissue-factor estimation (Li et al., 1 Oct 2025).

3. Network architecture, sparse enhancement, and interpretability

Each U2-rPCA layer performs four operations in sequence: update the sparse blood-flow estimate $\mathbf{D}=\mathbf{T}+\mathbf{B}+\mathbf{N},$4, refine it through the SEU to obtain $\mathbf{D}=\mathbf{T}+\mathbf{B}+\mathbf{N},$5, update the tissue coefficients $\mathbf{D}=\mathbf{T}+\mathbf{B}+\mathbf{N},$6, and update the tissue basis $\mathbf{D}=\mathbf{T}+\mathbf{B}+\mathbf{N},$7 (Li et al., 1 Oct 2025).

The blood-flow update is

$\mathbf{D}=\mathbf{T}+\mathbf{B}+\mathbf{N},$8

where $\mathbf{D}=\mathbf{T}+\mathbf{B}+\mathbf{N},$9 is deterministic and $\mathbf{T}$0 is learnable. The paper states that keeping $\mathbf{T}$1 tied to $\mathbf{T}$2 is essential for maintaining intrinsic sparse regularization. This point is not cosmetic: in ablation, making $\mathbf{T}$3 learnable causes the method to fail to extract effective blood flow and degrades power Doppler quality (Li et al., 1 Oct 2025).

The SEU is the main architectural addition beyond the raw IRLS unfolding. It is a dual-frame U-Net implemented in the complex domain, with a symmetric encoder-decoder structure and skip-like feature interactions in which features from decoder upsampling are subtracted from encoder features at corresponding scales. Its output is

$\mathbf{T}$4

The stated role of the SEU is to enhance vessel-like sparse structures, suppress residual tissue interference, improve micro-flow recovery, and increase power-Doppler contrast, especially PSL and CNR (Li et al., 1 Oct 2025).

The low-rank updates after SEU are

$\mathbf{T}$5

$\mathbf{T}$6

Here $\mathbf{T}$7 is learnable, and $\mathbf{T}$8 is effectively absorbed into $\mathbf{T}$9 (Li et al., 1 Oct 2025).

Component	Status	Role
$\mathbf{B}$0	Deterministic	Sparse regularization
$\mathbf{B}$1	Learnable	Blood-flow update control
$\mathbf{B}$2	Learnable	Low-rank adaptation
SEU parameters $\mathbf{B}$3	Learnable	Sparse feature refinement
$\mathbf{B}$4, $\mathbf{B}$5	Predefined	Structural hyperparameters

This division between deterministic and learned elements is central to the method’s interpretability claim. Sparsity is not replaced by an unconstrained learned block; it is retained through the deterministic IRLS weighting, while data adaptation is concentrated in a limited set of tunable quantities (Li et al., 1 Oct 2025).

4. Unsupervised training and the meaning of “unsupervised”

The training objective in U2-rPCA is a pure self-consistency loss:

$\mathbf{B}$6

No ground-truth blood-flow or tissue labels are required. Training is performed on the early frames of a sequence, and the trained model is then applied to subsequent frames of the same acquisition; the paper therefore characterizes the method as behaving “like an adaptive filter” (Li et al., 1 Oct 2025).

This label-free training protocol distinguishes the ultrasound U2-rPCA formulation from several related unfolded RPCA variants. The tensor method in "Deep Unfolded Tensor Robust PCA with Self-supervised Learning" also removes the need for ground-truth low-rank tensors, but it does so through a residual-sparsity objective

$\mathbf{B}$7

and it learns only four hyperparameters for Tucker-structured tensor RPCA (Dong et al., 2022). By contrast, the nonconvex matrix solver in "Deep Unrolling for Nonconvex Robust Principal Component Analysis" is described as “unsupervised” in the RPCA sense, but its loss explicitly uses low-rank and sparse targets obtained either from synthetic ground truth or from a classical RPCA solver such as IRCUR (Tan et al., 2023).

Other related unfolded RPCA systems are not unsupervised in this stricter label-free sense. RPCANet for infrared small target detection uses mask supervision and reconstruction loss, and is therefore better described as supervised or weakly supervised with reconstruction regularization (Wu et al., 2023). Learned Robust PCA (LRPCA) is trained with ground-truth $\mathbf{B}$8 or pseudo-labels produced by a classic RPCA method (Cai et al., 2021). The automotive radar unfolded RPCA with ROC-AE blocks is trained on paired clean and corrupted signals from the ARIM dataset, using supervised decomposition targets (Ristea et al., 2020).

A plausible implication is that the term “unsupervised” is not uniform across unfolded RPCA literature. In the specific U2-rPCA ultrasound formulation, it denotes training directly from observed corrupted sequences via data consistency, rather than from manual labels or proxy decompositions (Li et al., 1 Oct 2025).

5. Empirical performance, datasets, and ablation findings

The ultrasound study evaluates U2-rPCA on one in-silico dataset and public in-vivo PALA datasets. The in-silico kidney-mimicking phantom includes multi-hierarchy vascular flow units, tissue deformation and respiration-like motion, randomized vessel geometry and velocities, scatterers distributed at 10 per resolution cell, and tissue scatterer amplitude about 20 times higher than flow scatterers. Field II is used for plane-wave compounding with five plane waves at steering angles $\mathbf{B}$9, PRF 5 kHz, frame rate 1 kHz, total acquisition 6 s or 6000 frames, and Gaussian white noise added at 25 dB SNR. The first 5400 frames are used for training and the last 600 for testing (Li et al., 1 Oct 2025).

The public in-vivo PALA datasets are rat kidney and mouse tumor acquisitions. They use five-angle plane-wave imaging with a 15 MHz probe. Image sizes are $\mathbf{N}$0 for kidney and $\mathbf{N}$1 for tumor. Total frame counts are 171,360 frames at 1 kHz for kidney and 30,000 frames at 500 Hz for tumor. The split is 14,400 frames for training, 5,000 for validation, and the remainder for testing (Li et al., 1 Oct 2025).

The compared methods are SVD-based adaptive thresholding using a spatial similarity matrix, the IRLS-rPCA baseline, 3D-Res-UNet, and U2-rPCA. Evaluation uses CNR, SNR, PSL, $\mathbf{N}$2 between estimated and ground-truth Doppler velocities in simulation, and frame rate (Li et al., 1 Oct 2025).

On the in-silico kidney phantom, U2-rPCA achieves CNR values of 27.95, 27.79, and 26.68 dB, compared with 16.07, 23.36, and 19.48 for IRLS-rPCA; 18.78, 16.98, and 16.45 for SVD; and 8.06, 10.56, and 9.90 for 3D-Res-UNet. For Doppler velocity correlation $\mathbf{N}$3, U2-rPCA reports 0.9530, 0.9002, and 0.8458, while IRLS-rPCA reports 0.9476, 0.8975, and 0.8538, and SVD reports 0.9471, 0.8722, and 0.8536. The stated conclusion is that U2-rPCA improves power-Doppler contrast substantially while keeping Doppler velocity estimation comparable across methods (Li et al., 1 Oct 2025).

On rat kidney data, reported CNR values are $\mathbf{N}$4 and $\mathbf{N}$5 for two ROIs, exceeding SVD by more than 8.48 dB and IRLS-rPCA by more than 2.22 dB. On mouse tumor data, reported CNR values are $\mathbf{N}$6 and $\mathbf{N}$7, exceeding SVD and 3D-Res-UNet by more than 2.87 dB and IRLS-rPCA by more than 1.91 dB. The paper summarizes the practical gain as an improvement of the power Doppler CNR by 2 dB to 10 dB relative to the other methods (Li et al., 1 Oct 2025).

The ablation studies are structurally informative. Performance improves as the number of layers $\mathbf{N}$8 increases from 1 to about 5–6 and then saturates; the chosen values are $\mathbf{N}$9 for kidney and $\lambda$0 for tumor. The SEU improves vessel recovery and PSL by about 5 dB on kidney when $\lambda$1 and by about 2 dB on average on tumor when $\lambda$2. Varying the inner dimension $\lambda$3 changes performance only slightly; the chosen values are $\lambda$4 for kidney and $\lambda$5 for tumor. The deterministic treatment of $\lambda$6 is also validated directly, since a learnable $\lambda$7 degrades extraction quality (Li et al., 1 Oct 2025).

6. Relation to broader unfolded RPCA literature

U2-rPCA belongs to a broader class of model-driven networks that map RPCA iterations into trainable layers. The automotive radar method in "Automotive Radar Interference Mitigation with Unfolded Robust PCA based on Residual Overcomplete Auto-Encoder Blocks" uses the decomposition $\lambda$8 in the Fourier domain, retains singular value thresholding for the low-rank component and thresholding for the sparse component, and replaces the six learned transforms $\lambda$9 with residual overcomplete auto-encoder blocks. It estimates both amplitude and phase from real and imaginary FFT channels and, on the reported test set, achieves the best AUC among non-oracle methods together with amplitude MAE 0.55 dB and phase MAE 2.55 degrees (Ristea et al., 2020).

The nonconvex unrolling paper based on AccAltProj adopts a rank-constrained formulation rather than a convex surrogate, learns only two shared scalar hyperparameters $d$0, and replaces hard thresholding by the MCP proximal map with $d$1. Its architecture is tightly tied to tangent-space projection, rank-$d$2 truncation, and firm thresholding, which preserves strong interpretability but requires access to decomposition targets during training (Tan et al., 2023).

The tensor counterpart generalizes the same design philosophy from matrices to Tucker-structured tensors. It unfolds the ScaledGD tensor RPCA solver, learns only four hyperparameters $d$3, and can be trained with a self-supervised residual loss that promotes sparse corruptions without ground-truth low-rank tensors (Dong et al., 2022). LRPCA, in turn, emphasizes scalability by using factorized low-rank variables $d$4 and $d$5, per-layer thresholds and step sizes, and a feedforward-recurrent-mixed neural network that extends finite-depth unfolding to indefinitely many iterations, but its primary training formulation remains supervised or pseudo-supervised (Cai et al., 2021). RPCANet preserves RPCA-style stagewise structure for infrared small target detection through background extraction, target extraction, and image reconstruction modules, yet it explicitly relies on segmentation supervision and reconstruction loss rather than label-free RPCA training (Wu et al., 2023).

This suggests that U2-rPCA is best understood in two senses. In the strict sense, it denotes the ultrasound clutter-filtering method unfolded from IRLS-rPCA, equipped with a complex-domain SEU, and trained by layer-averaged reconstruction consistency without labels (Li et al., 1 Oct 2025). In a broader methodological sense, it denotes a class of interpretable unfolded RPCA systems that preserve low-rank-plus-sparse structure while replacing manual tuning or expensive proximal updates with learned components adapted to a specific sensing modality.