CrossDenoise: Phase-Amplitude Denoising for SAR

• The paper introduces a novel reciprocal denoising strategy where phase and amplitude mutually guide each other in the Fourier domain for enhanced SAR object detection.
• It applies a 2D DFT to decompose features into amplitude and phase, using band-wise self-attention and token exchange to effectively suppress speckle noise.
• Empirical results on benchmarks like SARDet-100K demonstrate improved mAP and efficiency, especially in detecting small objects while reducing model complexity.

Searching arXiv for the specified paper to ground the article in the current record. CrossDenoise denotes the phase-amplitude cross denoising strategy introduced as the central idea of DenoDet V2 for SAR object detection. It operates in the Fourier or transform domain rather than only in the image space, decomposing each feature map into amplitude and phase components and then using one component to guide denoising of the other through a mutual modulation or token exchange mechanism. The method is motivated by the observation that SAR imagery is dominated by coherent speckle noise, that amplitude is more vulnerable to noise and perturbation, and that the phase spectrum is empirically more robust and better preserves structure and contours. Within this formulation, CrossDenoise treats phase and amplitude as complementary signals that denoise each other reciprocally (Ni et al., 12 Aug 2025).

1. Problem formulation in SAR object detection

Synthetic Aperture Radar object detection is difficult because SAR is a coherent imaging system whose signal formation process produces speckle noise that is intrinsic and unavoidable. The noise is not merely random background clutter; it is superimposed on objects and can distort appearance, boundaries, and small-object cues, causing missed detections and false alarms. In this setting, denoising is not a peripheral preprocessing step but a central part of robust detection (Ni et al., 12 Aug 2025).

The paper situates CrossDenoise against a background in which many prior methods, including prior denoising-based detectors, mainly operate in the spatial domain by enhancing local texture or objectness. The stated limitation is that speckle in SAR has a strong interaction with global signal structure, so purely spatial processing may not separate object information from noise effectively. This motivates a transform-domain treatment in which feature maps are analyzed after a 2D DFT, where they can be separated into amplitude and phase representations (Ni et al., 12 Aug 2025).

A central empirical premise is that amplitude is more vulnerable to noise and perturbation, whereas phase is more stable and better preserves structure and spatial relationships. CrossDenoise therefore emphasizes using phase as a reference to clean amplitude, and then using refined amplitude to improve phase in return. This suggests that denoising in SAR should exploit the differing functional roles of spectral components rather than applying generic spatial smoothing.

2. Conceptual basis of phase-amplitude cross denoising

CrossDenoise is defined by explicit amplitude-phase interaction in the transform domain. Earlier approaches are described as typically denoising in the image space, using frequency information only as an auxiliary cue, or processing amplitude and phase separately. By contrast, DenoDet V2 performs denoising in the transform domain, explicitly models amplitude-phase interaction, and uses a cross-guidance mechanism rather than independent refinement (Ni et al., 12 Aug 2025).

The method’s conceptual distinction from DenoDet V1 is equally explicit. DenoDet V1 is described as “Attention as Deformable Multi-Subspace Feature Denoising”, more generic, with higher complexity, and not centered on amplitude-phase reciprocity. DenoDet V2 instead introduces Phase-Amplitude Cross Denoising, Phase-Amplitude Token Exchange (PATE), and band-wise partition self-attention. The key conceptual difference is that DenoDet V2 does not just denoise features; it makes phase and amplitude act as denoising references for each other (Ni et al., 12 Aug 2025).

The reciprocal structure is central to the notion of “cross” in CrossDenoise. Amplitude is guided by phase, and phase is guided by amplitude. However, the paper particularly highlights phase-guided soft-thresholding of amplitude, reflecting the claim that phase is more stable under speckle corruption. This yields a denoising process that is not unidirectional. Instead, each spectrum functions both as a signal to be denoised and as a reference that denoises its counterpart. The paper explicitly characterizes this as the first reciprocal, reference-based feature denoising strategy of this kind in SAR detection (Ni et al., 12 Aug 2025).

A common misconception would be to interpret CrossDenoise as a conventional frequency-domain filtering step. The described mechanism is not a fixed transform-domain thresholding rule. It is an attention-based reciprocal modulation design in which amplitude and phase exchange information through learned token roles, band by band. A plausible implication is that the method should be understood less as handcrafted spectral suppression and more as transform-domain representation learning specialized for coherent SAR noise.

3. DFTDeno architecture and transform-domain processing

DenoDet V2 implements CrossDenoise through a plug-and-play module called DFTDeno, inserted into a standard detector backbone. The module contains three main stages: 2D DFT transformation, band-wise partition self-attention + phase-amplitude token exchange, and inverse DFT reconstruction (Ni et al., 12 Aug 2025).

For a feature map $\mathbf{M}\in\mathbb{R}^{C\times H\times W}$, the forward 2D DFT is given as

$$\mathbf{m}_{c,u,v} = \sum_{h=0}^{H-1}\sum_{w=0}^{W-1}\mathbf{M}_{c,h,w}e^{ - 2{\pi}i(\frac{uh}{H} + \frac{vw}{W})}.$$

Using Euler’s formula, the transformed representation is decomposed into real and imaginary parts:

$$\mathcal{R}_{c,u,v} = \sum_{h=0}^{H-1}\sum_{w=0}^{W-1}\mathbf{M}_{c,h,w}\cos 2{\pi}\left(\frac{uh}{H} + \frac{vw}{W}\right),$$

$$\mathcal{I}_{c,u,v} = -\sum_{h=0}^{H-1}\sum_{w=0}^{W-1}\mathbf{M}_{c,h,w}\sin 2{\pi}\left(\frac{uh}{H} + \frac{vw}{W}\right).$$

Amplitude and phase are then computed as

$$\mathcal{A}_{c,u,v} = \sqrt{\mathcal{R}_{c,u,v}^2 + \mathcal{I}_{c,u,v}^2},$$

$\mathcal{P}_{c,u,v} = \arctan2\left(\frac{\mathcal{I}_{c,u,v}{\mathcal{R}_{c,u,v}\right).$

The reconstructed modulated complex spectrum is written as

$\hat{\mathbf{m}_{c,u,v} = \hat{\mathcal{A}_{c,u,v}\cdot \left(\cos \hat{\mathcal{P}_{c,u,v} + i\sin \hat{\mathcal{P}_{c,u,v}\right),$

and the inverse DFT returns the representation to the spatial domain:

$\hat{\mathbf{M}_{c,h,w} = \sum_{u=0}^{H-1}\sum_{v=0}^{W-1}\hat{\mathbf{m}_{c,u,v}e^{ 2{\pi}i(\frac{uh}{H} + \frac{vw}{W})}.$

In implementation, the paper states that $\cos(\hat{\mathcal{P}})$ and $\sin(\hat{\mathcal{P}})$ are used instead of raw phase angles in order to avoid angular boundary discontinuity (Ni et al., 12 Aug 2025).

This architectural formulation makes the transform-domain decomposition operational rather than purely analytic. The forward DFT exposes amplitude and phase as separate objects of computation; the inverse DFT ensures that the transformed-domain modulation remains compatible with downstream spatial-domain detection. This suggests that CrossDenoise is best viewed as a spectral intervention embedded inside end-to-end detector training rather than as an external denoising front-end.

4. Band-wise partition self-attention and token exchange

A key design choice is the use of band-wise partition self-attention (BPSA). The frequency spectrum is partitioned into local bands, motivated by the argument that frequency components do not necessarily obey local image-like continuity, while global all-to-all interaction would be expensive and may over-process the spectrum. The paper defines a frequency attention map using pooling over channels:

$$\mathbf{m}_{c,u,v} = \sum_{h=0}^{H-1}\sum_{w=0}^{W-1}\mathbf{M}_{c,h,w}e^{ - 2{\pi}i(\frac{uh}{H} + \frac{vw}{W})}.$$0

followed by

$$\mathbf{m}_{c,u,v} = \sum_{h=0}^{H-1}\sum_{w=0}^{W-1}\mathbf{M}_{c,h,w}e^{ - 2{\pi}i(\frac{uh}{H} + \frac{vw}{W})}.$$1

In the basic band-wise self-attention formulation, the spectrum is split into groups $$\mathbf{m}_{c,u,v} = \sum_{h=0}^{H-1}\sum_{w=0}^{W-1}\mathbf{M}_{c,h,w}e^{ - 2{\pi}i(\frac{uh}{H} + \frac{vw}{W})}.$$2. For each band,

$$\mathbf{m}_{c,u,v} = \sum_{h=0}^{H-1}\sum_{w=0}^{W-1}\mathbf{M}_{c,h,w}e^{ - 2{\pi}i(\frac{uh}{H} + \frac{vw}{W})}.$$3

$$\mathbf{m}_{c,u,v} = \sum_{h=0}^{H-1}\sum_{w=0}^{W-1}\mathbf{M}_{c,h,w}e^{ - 2{\pi}i(\frac{uh}{H} + \frac{vw}{W})}.$$4

$$\mathbf{m}_{c,u,v} = \sum_{h=0}^{H-1}\sum_{w=0}^{W-1}\mathbf{M}_{c,h,w}e^{ - 2{\pi}i(\frac{uh}{H} + \frac{vw}{W})}.$$5

The reported interpretation is that partitioning reduces dimensionality and allows regional attention while preserving frequency-domain structure. In the ablation study, the best stride is 8 (Ni et al., 12 Aug 2025).

The core CrossDenoise mechanism is Phase-Amplitude Token Exchange (PATE). Rather than processing amplitude and phase through separate pipelines, DenoDet V2 partitions them into corresponding bands and exchanges token roles so that one spectrum provides query while the other provides key/value. For band $$\mathbf{m}_{c,u,v} = \sum_{h=0}^{H-1}\sum_{w=0}^{W-1}\mathbf{M}_{c,h,w}e^{ - 2{\pi}i(\frac{uh}{H} + \frac{vw}{W})}.$$6, the exchange is defined as

$$\mathbf{m}_{c,u,v} = \sum_{h=0}^{H-1}\sum_{w=0}^{W-1}\mathbf{M}_{c,h,w}e^{ - 2{\pi}i(\frac{uh}{H} + \frac{vw}{W})}.$$7

$$\mathbf{m}_{c,u,v} = \sum_{h=0}^{H-1}\sum_{w=0}^{W-1}\mathbf{M}_{c,h,w}e^{ - 2{\pi}i(\frac{uh}{H} + \frac{vw}{W})}.$$8

This token exchange constitutes the formal definition of the “cross” operation. Amplitude is denoised under phase-derived attention, and phase is correspondingly refined using amplitude-derived information. The paper describes this especially as phase-guided soft-thresholding of amplitude, followed by reciprocal enhancement of phase through the denoised amplitude (Ni et al., 12 Aug 2025).

5. Reciprocal enhancement and mathematical interpretation

The paper interprets the CrossDenoise process as mutual modulation. The first direction is phase-guided amplitude denoising: phase is more robust to speckle and retains shape and structure, so attention generated from phase helps suppress noisy amplitude responses. This is described as analogous to a learned, data-adaptive soft-thresholding in the frequency domain (Ni et al., 12 Aug 2025).

The second direction is amplitude-assisted phase refinement. Once amplitude has been cleaned, it provides a better spectral reference, which improves the phase representation as well rather than leaving it untouched. The result is a reciprocal denoising system in which neither spectrum is treated as a fixed supervisory anchor; instead, each spectrum alternates between being denoised and serving as a denoising reference (Ni et al., 12 Aug 2025).

This framing has methodological significance. CrossDenoise does not assume that phase alone is sufficient and amplitude dispensable. The ablation results reported in the paper indicate that refining amplitude helps, refining phase helps more, and refining both together is best. On SARDet-100K, the reported values are 55.6 for amplitude-only refinement, 56.2 for phase-only refinement, and 56.4 for refining both together. This supports both parts of the argument: phase is more robust to noise, and dual refinement is complementary (Ni et al., 12 Aug 2025).

A plausible implication is that the method operationalizes a division of labor between spectral variables: phase contributes structural stability, while amplitude, once denoised, supplies complementary spectral evidence. In this sense, CrossDenoise is neither phase-only nor amplitude-only; it is a reciprocal denoising framework whose efficacy depends on controlled exchange between asymmetrically robust signals.

6. Empirical evaluation, ablations, and performance profile

The paper evaluates DenoDet V2 on three SAR benchmarks: SARDet-100K, SAR-Aircraft-1.0, and AIR-SARShip-1.0. SARDet-100K contains 116,598 images, 10 sub-collections, 6 object categories, and a train/val/test split of 8:1:1. SAR-Aircraft-1.0 contains 7 aircraft categories, 3,489 training / 879 test images, and chips of size $$\mathbf{m}_{c,u,v} = \sum_{h=0}^{H-1}\sum_{w=0}^{W-1}\mathbf{M}_{c,h,w}e^{ - 2{\pi}i(\frac{uh}{H} + \frac{vw}{W})}.$$9. AIR-SARShip-1.0 contains 31 Gaofen-3 scenes, 461 annotated vessels, and $$\mathcal{R}_{c,u,v} = \sum_{h=0}^{H-1}\sum_{w=0}^{W-1}\mathbf{M}_{c,h,w}\cos 2{\pi}\left(\frac{uh}{H} + \frac{vw}{W}\right),$$0 chips (Ni et al., 12 Aug 2025).

Implementation is reported in MMDetection using 4 RTX 4090 GPUs, the DAdaptAdam optimizer, 12 epochs, input resized to $$\mathcal{R}_{c,u,v} = \sum_{h=0}^{H-1}\sum_{w=0}^{W-1}\mathbf{M}_{c,h,w}\cos 2{\pi}\left(\frac{uh}{H} + \frac{vw}{W}\right),$$1, horizontal flip augmentation for SARDet-100K, gradient clipping, and synchronized batch normalization. Metrics include COCO mAP on SARDet-100K with AP@50, AP@75, AP_S, AP_M, AP_L, AP’07 and AP’12 on SAR-Aircraft-1.0, and AP on AIR-SARShip-1.0 (Ni et al., 12 Aug 2025).

The principal quantitative results can be summarized as follows:

Dataset	Reported result	Model complexity note
SARDet-100K	56.71 mAP	52.47G FLOPs, 32.60M parameters
SAR-Aircraft-1.0	69.93 mAP (AP’07) / 70.73 mAP (AP’12)	37.15M parameters
AIR-SARShip-1.0	73.98 AP	Reported as SOTA

On SARDet-100K, DenoDet V2 is reported as the best among 25 compared methods, improving over DenoDet V1 by 0.8% mAP while keeping FLOPs nearly unchanged (52.47G vs 52.69G) and roughly halving parameter count (32.60M vs 65.78M). On SAR-Aircraft-1.0, the paper reports 69.93 mAP (AP’07) and 70.73 mAP (AP’12), with parameter count reduced from 70.33M to 37.15M relative to DenoDet V1. On AIR-SARShip-1.0, the reported score is 73.98 AP (Ni et al., 12 Aug 2025).

The paper further emphasizes strong gains on small and medium objects, which are more sensitive to speckle, reporting AP_S = 51.45 and AP_M = 68.75 on SARDet-100K. This is consistent with the stated motivation that speckle particularly degrades weak or small-object cues (Ni et al., 12 Aug 2025).

Ablation studies are used to support specific design decisions. For phase decomposition, the reported SARDet-100K mAP values are 55.6 for direct angle regression, 56.1 for orthogonal split, and 56.2 with trigonometric alignment, which the paper interprets as evidence that phase representation requires careful handling because of angular discontinuity. For band partitioning, stride 8 gives the best result: stride 1 is worst, larger strides improve until 8, and 16 slightly drops. For token exchange, performance improves from 56.4 under No-Exchange to 56.7 with token exchange, indicating that explicitly swapping amplitude/phase roles is beneficial rather than merely processing them in parallel (Ni et al., 12 Aug 2025).

The visual analyses using Eigen-CAM and detection visualizations are described as showing that attention concentrates more on object regions, background noise is suppressed, and there are fewer false alarms and missed detections, especially under noisy or crowded SAR scenes. This visual evidence is presented as qualitative support for the reciprocal denoising hypothesis (Ni et al., 12 Aug 2025).

7. Significance, scope, and interpretive context

The paper’s principal takeaway is that SAR denoising should not be treated as a generic spatial smoothing problem. Because SAR data are coherent and speckle-corrupted, denoising should exploit the different roles of phase and amplitude in the transform domain. CrossDenoise realizes this through DFT-based decomposition, band-wise self-attention, and phase-amplitude token exchange, producing a detector that is reported as more accurate, more robust to SAR speckle, more efficient than DenoDet V1, and state of the art across multiple SAR benchmarks (Ni et al., 12 Aug 2025).

In methodological terms, CrossDenoise occupies the intersection of SAR detection, transform-domain modeling, and attention-based denoising. Its novelty lies not merely in moving denoising to the Fourier domain, but in making amplitude and phase participate in a reciprocal denoising loop. The paper’s one-sentence summary captures this precisely: CrossDenoise is the reciprocal transform-domain denoising mechanism in DenoDet V2 that uses phase and amplitude to cross-guide each other, band by band, for robust SAR object detection (Ni et al., 12 Aug 2025).

Two interpretive cautions follow from the reported evidence. First, the claims are established in the context of SAR object detection rather than generic image denoising. Second, the argument for phase robustness is empirical and architectural: phase is treated as a stronger structural reference, but the best reported performance still comes from refining both spectra together. This suggests that CrossDenoise is not a replacement of amplitude by phase, but a structured exploitation of their complementarity.