Dynamic Morphological Component Analysis (DMCA)

• Dynamic Morphological Component Analysis (DMCA) is a data-adaptive extension of MCA that builds dictionaries from video data using dynamic mode decomposition.
• It combines sliding-window DMD, eigenvalue clustering, and ℓ1-regularized sparse coding to decompose videos into distinct component layers based on temporal dynamics.
• DMCA demonstrates improved video denoising and target enhancement by leveraging dynamic behavior to separate overlapping spatial components.

Dynamic Morphological Component Analysis (DMCA) is an unsupervised, data-adaptive extension of Morphological Component Analysis (MCA) for video and time-ordered image data that uses Dynamic Mode Decomposition (DMD) to build data-driven dictionaries directly from the input sequence. In the formulation introduced in "A Dynamic Mode Decomposition Approach to Morphological Component Analysis" (Huber et al., 7 Oct 2025), a video is arranged as a matrix $X \in \mathbb{C}^{m \times n}$ whose columns are vectorized frames; DMD is then applied on sliding temporal windows, the resulting DMD eigenvalues are clustered, and the associated modes are assigned to cluster-specific dictionaries for sparse frame-wise reconstruction. The output is a decomposition into $k$ component videos, each intended to correspond to a distinct morphology or dynamic class. The method is positioned as an alternative to predefined MCA dictionaries and to externally trained learned dictionaries, particularly when components overlap spatially but differ in temporal behavior.

1. Definition and conceptual scope

DMCA extends the classical MCA paradigm from static dictionary selection to dictionary generation driven by temporal dynamics. In the paper’s formulation, standard MCA assumes that a signal is a sum of morphologically distinct sources and that each source is sparse in its own dictionary but not in the others. DMCA preserves that sparse-separation logic, but replaces fixed or externally trained dictionaries with dictionaries derived from the same video being decomposed through sliding-window DMD and eigenspace clustering (Huber et al., 7 Oct 2025).

The central idea is that morphologically distinct components in video often differ not only spatially but dynamically. DMD supplies, for each local temporal window, a set of spatial modes $\varphi_{(i,j)}$, eigenvalues $\lambda_{(i,j)}$, and amplitudes $b_{(i,j)}$. In DMCA, the DMD modes become candidate atoms, and the DMD eigenvalues provide the dynamic descriptors used to partition those atoms into multiple dictionaries. The method therefore treats clustered dynamic behavior as a proxy for morphology.

The intended decomposition is into component videos rather than into a single low-dimensional subspace. This is a major distinction from "Unsupervised Discovery of Temporal Structure in Noisy Data with Dynamical Components Analysis" (Clark et al., 2019), which studies Dynamical Components Analysis (DCA), an unsupervised linear dynamical dimensionality reduction method that seeks a single $d$-dimensional subspace maximizing predictive information between past and future. That method is explicitly not a morphological sparse decomposition method. The acronymic similarity is a likely source of confusion, but the objectives, assumptions, and outputs are different.

2. Relation to MCA and the role of DMD

In the paper’s summary of MCA, a signal $s$ is modeled as a sum of $k$ source components,

$s = \sum_{p \le k} s_{(p)},$

with each component represented as

$s_{(p)} = \Phi_{(p)} \alpha^{opt}_{(p)},$

where $k$0 is sparse. The ideal sparse decomposition problem is

$k$1

The underlying separability assumption is morphological diversity: each component should be sparse in its own dictionary and not sparse in the others (Huber et al., 7 Oct 2025).

DMCA keeps this MCA objective structure but changes how the dictionaries are obtained. Instead of using predefined mathematical dictionaries such as Fourier, wavelets, Gabor, STFT, or ridgelets, or externally learned dictionaries such as generalized PCA, K-SVD, or union of orthobases, it constructs dictionaries directly from the observed video through DMD. In this sense, the dictionaries are data-driven and video-specific, but not trained over an external dataset.

DMD contributes both atoms and a dynamic labeling mechanism. Given sequential measurements $k$2, $k$3, collected into

$k$4

DMD seeks a rank-$k$5 best-fit linear operator

$k$6

and in the full-rank case,

$k$7

With the truncated SVD

$k$8

one forms

$k$9

computes

$\varphi_{(i,j)}$0

and recovers the high-dimensional DMD modes

$\varphi_{(i,j)}$1

The DMD modal expansion is

$\varphi_{(i,j)}$2

with mode-wise temporal coefficient

$\varphi_{(i,j)}$3

Writing $\varphi_{(i,j)}$4, the paper gives

$\varphi_{(i,j)}$5

Accordingly, $\varphi_{(i,j)}$6 controls growth or decay and $\varphi_{(i,j)}$7 controls oscillatory behavior. DMCA uses this interpretation to argue that eigenvalues that are close in the complex plane correspond to modes with similar temporal behavior, and that such dynamic similarity can be used to form morphology-specific dictionaries.

A nearby but distinct reference point is "Approximate Extraction of Late-Time Returns via Morphological Component Analysis" (Goehle et al., 2022). That paper is highly relevant as MCA background because it uses additive sparse decomposition, morphology-specific dictionaries, and $\varphi_{(i,j)}$8-regularized separation, but it does not introduce dynamic dictionary generation, temporal state evolution, or sequence-level coupling. It is therefore best understood as static MCA applied to temporal acoustic data rather than DMCA.

3. Mathematical formulation of DMCA

For a video matrix $\varphi_{(i,j)}$9, DMCA defines the $\lambda_{(i,j)}$0-th sliding DMD window of length $\lambda_{(i,j)}$1 by

$\lambda_{(i,j)}$2

For each $\lambda_{(i,j)}$3, full-rank DMD is applied to the window, fitting

$\lambda_{(i,j)}$4

subject to

$\lambda_{(i,j)}$5

This yields the triplets

$\lambda_{(i,j)}$6

Within each window, the reconstruction is

$\lambda_{(i,j)}$7

DMCA then assigns labels to eigenvalues through a clustering function $\lambda_{(i,j)}$8. The paper gives three clustering rules. The first is magnitude-threshold clustering: $\lambda_{(i,j)}$9 for decreasing thresholds $b_{(i,j)}$0. The second is radial clustering: $b_{(i,j)}$1 The third is K-means or K-medians on $b_{(i,j)}$2. Each DMD mode inherits the label of its eigenvalue.

For each frame $b_{(i,j)}$3, DMCA constructs local cluster-specific dictionaries from modes whose windows lie within a neighborhood $b_{(i,j)}$4. The cluster-$b_{(i,j)}$5 dictionary is formed from all modes $b_{(i,j)}$6 satisfying

$b_{(i,j)}$7

The combined dictionary is

$b_{(i,j)}$8

The per-frame reconstruction model is

$b_{(i,j)}$9

Because full-rank DMD is used, the paper states

$d$0

and

$d$1

The paper uses this to argue that each $d$2 lies in the union of the relevant local dictionary column spaces and can be reconstructed exactly.

Since the local combined dictionary is overcomplete, the least-squares problem

$d$3

may have multiple solutions with unbounded norm. DMCA therefore introduces $d$4-regularized sparse coding: $d$5 Equivalently,

$d$6

After coefficient estimation, the output layers are reconstructed as

$d$7

4. Algorithmic pipeline and implementation parameters

The algorithm summarized in the paper, "Algorithm 1: Dynamic Morphological Component Analysis," takes as input the data matrix $d$8, the DMD window length $d$9, a labeling function $s$0, and the reconstruction neighborhood $s$1. Its dictionary-generation stage runs DMD on each sliding window; its component-separation stage determines $s$2 from the distinct labels returned by $s$3, solves the sparse reconstruction problem frame by frame, and appends each cluster-specific reconstruction to the corresponding output matrix (Huber et al., 7 Oct 2025).

The principal parameters are:

• $s$4: DMD window length;
• $s$5: reconstruction neighborhood size;
• $s$6: eigenvalue labeling or clustering function;
• $s$7: $s$8 regularization weight;
• $s$9: number of output components or clusters.

Several concrete parameter choices appear in the experiments. For the still-image example, the paper uses $k$0, $k$1, and a two-cluster rule based on $k$2. For Adobe 240fps video denoising, it uses $k$3, $k$4, and a threshold at $k$5. For the sea-state experiment, it uses $k$6, $k$7, and K-medians on $k$8 with $k$9. For ISAR bicycle-versus-wind separation, it uses $s = \sum_{p \le k} s_{(p)},$0, $s = \sum_{p \le k} s_{(p)},$1, and a two-cluster threshold at $s = \sum_{p \le k} s_{(p)},$2.

The paper states that the sliding-window DMD uses full rank,

$s = \sum_{p \le k} s_{(p)},$3

as a practical design choice to ensure local spanning and reconstruction properties. Frames are vectorized into columns; in several experiments, the combined video or noise-corrupted data are rescaled to the pixel range $s = \sum_{p \le k} s_{(p)},$4; and ISAR data are used directly in complex matrix form.

A notable implementation omission is the sparse solver. The paper mentions several efficient algorithms in passing—“joint-basis pursuit, SALSA, gradient-descent, etc.”—but does not specify which solver was used in the experiments. Optional postprocessing is also mentioned: a Savitzky–Golay filter may be applied to target modes after clustering to suppress residual noise and improve signal-to-clutter ratio.

5. Empirical demonstrations

The paper presents four classes of demonstrations: a still-image example, non-Gaussian video denoising on Adobe 240fps videos, faint-target enhancement in sea clutter, and complex-valued ISAR target-versus-clutter separation (Huber et al., 7 Oct 2025).

In the motivational still-image experiment, the columns of a still image are treated as sequential snapshots. A mandrill image is summed with a checkerboard image whose pixel values alternate between $s = \sum_{p \le k} s_{(p)},$5 and $s = \sum_{p \le k} s_{(p)},$6, then rescaled to $s = \sum_{p \le k} s_{(p)},$7. The corrupted image has PSNR = 5.69 dB and SSIM = 0.0061. With $s = \sum_{p \le k} s_{(p)},$8, $s = \sum_{p \le k} s_{(p)},$9, and the rule

$s_{(p)} = \Phi_{(p)} \alpha^{opt}_{(p)},$0

DMCA reconstructs a target image and a texture layer, and the target reconstruction attains PSNR = 31.7 dB and SSIM = 0.96.

For video denoising, the task is to remove time-varying, non-Gaussian additive noise from Adobe 240fps videos. Each pixel is modeled as

$s_{(p)} = \Phi_{(p)} \alpha^{opt}_{(p)},$1

where $s_{(p)} = \Phi_{(p)} \alpha^{opt}_{(p)},$2 are drawn from changing Gaussian, uniform, and Laplace distributions whose parameters are modulated over time. For the shipyard video at noise level $s_{(p)} = \Phi_{(p)} \alpha^{opt}_{(p)},$3, the configuration is $s_{(p)} = \Phi_{(p)} \alpha^{opt}_{(p)},$4, $s_{(p)} = \Phi_{(p)} \alpha^{opt}_{(p)},$5, and

$s_{(p)} = \Phi_{(p)} \alpha^{opt}_{(p)},$6

The reported findings are qualitative and comparative: DMCA separates a denoised component from evolving noise; on the displayed 30th frames, DMCA has higher PSNR than V-BM4D for each of the three videos shown; over sequences, V-BM4D sometimes attains a much higher maximum PSNR; and DMCA exhibits more compact 85% confidence intervals of PSNR across tested frames and noise strengths. This suggests that the paper’s main denoising claim is about stability under changing non-Gaussian noise dynamics rather than uniform superiority in average denoising quality.

In the sea-state experiment, the method separates a weak moving target from a simulated sea-state height map plus additive Gaussian noise. The simulation uses a facet size of 1 m, scene size of 1000 m, wind of 10 m/s at 10 m above sea level, fetch of 50 km, wind angle of 35 degrees, frames every $s_{(p)} = \Phi_{(p)} \alpha^{opt}_{(p)},$7 second, and a crop to $s_{(p)} = \Phi_{(p)} \alpha^{opt}_{(p)},$8 pixels. The target is an “X,” 20 pixels wide and long, with height 0.8 m, undergoing a symmetric random walk with five equiprobable transitions, including no motion. After adding Gaussian noise with standard deviation $s_{(p)} = \Phi_{(p)} \alpha^{opt}_{(p)},$9 m and rescaling to $k$00, the input target visibility is SNR = -1.08 dB. Using $k$01, $k$02, K-medians clustering on $k$03, and $k$04, DMCA returns five component videos. The target-containing cluster attains SNR = 5.86 dB, compared with input SNR = -1.08 dB.

In the ISAR experiment, DMCA separates a moving bicycle target from wind clutter in complex-valued inverse synthetic aperture radar image sequences. The wind speeds are 2.5, 5, 7.5, and 10 m/s. For the sequence with the bicycle turning left at wind speed 10 m/s, the data matrix is formed by vectorized complex-valued ISAR images, and the configuration is $k$05, $k$06, with

$k$07

The output is a target sequence and a wind-clutter sequence. For the 8th frame, the input SCR = 24.8 dB and the output SCR = 28.0 dB. Against baselines, the paper reports a consistent average SCR increase over both the input frames and V-BM4D outputs for all wind speeds; and for the two highest wind speeds, DMCA’s confidence interval is separated from those of the input and V-BM4D. A classical MCA baseline with predefined STFT dictionaries fails after exhaustive search over window lengths and regularization, returning components identical to the input up to a scalar factor.

6. Assumptions, limitations, and disambiguation

The paper states several explicit assumptions. DMCA assumes that the video is a linear superposition of components, that different components have distinct enough dynamics to produce separable DMD eigenvalue clusters, that local sliding-window DMD captures transient structures effectively, and that sparse reconstruction in the resulting dictionaries is meaningful (Huber et al., 7 Oct 2025).

Its limitations are equally explicit. The paper states that there is no theory yet for when DMCA succeeds and no numerical bounds. Clustering is heuristic, and manual clustering may be needed. K-means or K-medians may place centers too close together or may mismatch the true number of clusters. The relation between clustered eigenspaces and dictionary incoherence is empirical rather than formally established. Solver details and some implementation specifics are under-described.

These caveats matter because DMCA relies on an empirical chain of reasoning: distinct morphological components are assumed to induce distinct temporal dynamics; those dynamics are assumed to produce distinct DMD eigenvalue clusters; and those clusters are assumed to define effective, sufficiently incoherent MCA dictionaries. The paper’s conclusion is that DMCA works effectively when components have dissimilar dynamics, but it does not provide a formal coherence theorem.

The strongest adjacent comparison is with static MCA. "Approximate Extraction of Late-Time Returns via Morphological Component Analysis" (Goehle et al., 2022) provides a clear MCA template: additive mixture modeling,

$k$08

$k$09-regularized sparse decomposition,

$k$10

and morphology-specific dictionary design, including FFT dictionaries and physics-informed Enveloped Sinusoid Parseval frames. That paper is highly relevant as methodological background because it shows how dictionary choice governs physical interpretability and robustness. However, it processes each time series separately and introduces neither dynamic coefficient evolution nor data-driven dictionary generation from sequence dynamics.

A second essential disambiguation concerns Dynamical Components Analysis. "Unsupervised Discovery of Temporal Structure in Noisy Data with Dynamical Components Analysis" (Clark et al., 2019) optimizes predictive information between past and future windows to identify a low-dimensional dynamical subspace. It is linear, unsupervised, and temporal, but it is not a sparse coding method, not a dictionary learning method, and not a morphological decomposition method. DMCA, by contrast, is a sparse multi-dictionary separation framework whose dictionaries are induced from clustered DMD dynamics.

A concise comparison is therefore:

Method	Core object learned	Primary mechanism
MCA	Multiple morphology-specific components	Sparse decomposition in predefined or otherwise chosen dictionaries
DMCA	Multiple component videos tied to dynamic classes	Sliding-window DMD, eigenvalue clustering, and sparse reconstruction
DCA	A single low-dimensional dynamical subspace	Maximization of past-future predictive information

This suggests that DMCA occupies a distinct position at the intersection of MCA and DMD: it is not merely MCA on temporal data, and it is not merely DMD-based foreground/background separation. Its defining feature is the use of clustered DMD modes as a dictionary system for sparse morphological component separation.