Residual Polar Representation

• Residual polar representation is a modeling approach that parameterizes signals with radial and angular variables while leveraging residual learning or refinement to counteract distortions.
• It is applied in diverse domains such as XL-MIMO channel estimation, LiDAR 3D detection, and instance segmentation by aligning polar grids with underlying geometric invariants.
• The concept distinguishes between explicit additive residual updates, feature re-alignment corrections, and analytic residue calculus in Mellin analysis, clarifying its multifaceted applications.

Residual polar representation denotes a family of constructions in which a polar-coordinate encoding is coupled either to explicit residual learning, to corrective refinement of polar-domain features, or, in a mathematically distinct line of work, to residue calculus on the polar half-plane. The unifying idea is that the underlying signal is parameterized by radial and angular variables—such as center-plus-rays for object contours, angle-distance atoms for near-field channels, or radial–azimuth grids for 3D perception—while the “residual” component may refer to learning the difference between a transformed observation and a clean polar-domain target, or more broadly to compensating the distortions induced by polar discretization. The literature also contains important counterexamples: in several contour-based segmentation methods, the polar representation is direct rather than residual, and in Mellin analysis the relevant notion is “residue,” not residual learning (Lei et al., 2023, Xie et al., 2021, Li et al., 2021, Bardaro et al., 2019).

1. Terminological scope and conceptual boundaries

Across recent work, residual polar representation is not a single standardized formalism. In XL-MIMO channel estimation, it has an explicit meaning: the channel is transformed into a polar domain and a network learns a residual mapping there, with the estimate written as a transformed input minus a learned correction (Lei et al., 2023). In LiDAR 3D detection and occupancy prediction, by contrast, the dominant theme is feature re-alignment or propagation in a polar grid to correct distortion caused by non-uniform radial–angular partitioning; these works emphasize refinement, but they do not define a canonical additive residual equation for the polar representation itself (Nie et al., 2023, Xue et al., 2024).

This distinction matters because several influential polar-mask methods are explicitly non-residual. PolarMask++ predicts a center and a fixed set of ray lengths directly, and the paper states that there is no residual formulation “in the sense of iterative contour correction or delta refinement of rays” in the base pipeline (Xie et al., 2021). SiamPolar likewise predicts absolute radial distances rather than residual distances, and its summary states that there is “no explicit residual term $\Delta d$, no angle regression, and no center-offset regression inside the polar branch” (Li et al., 2021). A separate mathematical tradition employs the nearly homonymous concept of a residue theorem for polar analytic functions, where logarithmic poles and $c$-residues belong to Mellin analysis rather than deep residual learning (Bardaro et al., 2019).

A recurrent misconception is therefore to treat any polar representation with refinement as “residual.” The surveyed literature does not support that equivalence. In some cases residuality is explicit and algebraic; in others it is architectural and corrective; in others it is absent.

2. Core polar encodings across application domains

The simplest and most widely reused polar encoding in vision represents an object by a center $(x_c,y_c)$ and a set of radial distances $\{d_1,\ldots,d_n\}$ emitted at fixed angular intervals. At inference, contour points are reconstructed by

$$x_i=\cos\theta_i \times d_i + x_c,\qquad y_i=\sin\theta_i \times d_i + y_c.$$

Connecting the points in angular order yields a polygonal mask. PolarMask++ uses this formulation for instance segmentation, rotated text detection, and cell segmentation; SiamPolar uses the same center-plus-rays principle for realtime video object segmentation (Xie et al., 2021, Li et al., 2021).

In XL-MIMO, the polar domain is not a contour parameterization but a transform domain for near-field channels. The near-field channel is represented as

$$\mathbf{h}^{\text{near-field}}=\mathbf{D}\mathbf{h}^{\text{near-field}}_{\mathrm{P}},$$

where $$\mathbf{D}=[\mathbf{D}_{Q_1},\mathbf{D}_{Q_2},\ldots,\mathbf{D}_{Q_N}]$$ is a dictionary of steering vectors indexed by angle and distance, and $\mathbf{h}^{\text{near-field}}_{\mathrm{P}}$ is the polar-domain coefficient vector. The paper’s central claim is that near-field channels are sparse or approximately sparse in this angle-distance domain, whereas angular-domain sparsity degrades because the wavefront is spherical and depends on both $\theta$ and $r$ (Lei et al., 2023).

For LiDAR perception, the polar representation is usually a discretized grid. PARTNER forms a BEV feature map

$c$0

where $c$1 and $c$2 are radial and azimuth resolutions. PVP extends this to a 3D tensor

$c$3

with explicit height resolution. Both papers argue that polar binning matches sensing geometry and offers robustness to resolution changes or better detail preservation near the ego region, but both also identify distortion caused by non-uniform cell area as the key representational difficulty (Nie et al., 2023, Xue et al., 2024).

In trajectory prediction and planning, Polaris uses relative polar coordinates rather than absolute Cartesian offsets. It represents map points in the form $c$4 and constructs relative interaction features as

$c$5

Here the polar representation is relational: distance and direction are made explicit, and refinement operates on these relative polar interactions rather than on a static grid (Zhang et al., 15 Aug 2025).

3. Explicit residual polar learning in XL-MIMO

The clearest instantiation of a residual polar representation appears in channel estimation for XL-MIMO systems. The received pilot signal is modeled as

$c$6

with $c$7. The input to the network is a real-valued feature matrix $c$8 formed by stacking the real and imaginary parts of the received signal. Residuality is defined directly in the transformed domain: for angular MRDN,

$c$9

and for the polar-domain variant P-MRDN,

$(x_c,y_c)$0

followed by inverse polar transform. In this setting, “residual” means that the network learns the difference between the transformed input and the desired sparse or clean polar-domain channel representation (Lei et al., 2023).

The architectural baseline is MRDN, which stacks residual dense networks and then applies a Convolutional Block Attention Module (CBAM). The recursion inside the residual dense component is written as

$(x_c,y_c)$1

P-MRDN keeps this residual-dense-plus-attention design but replaces FFT/IFFT by polar transform and inverse polar transform. P-MSRDN further augments P-MRDN with multi-scale ASPP-enhanced residual dense blocks. The ASPP branch is defined as

$(x_c,y_c)$2

and is fused into the terminal layer of an RDN through

$(x_c,y_c)$3

Multiple ASPP-RDN units are then stacked and followed by CBAM (Lei et al., 2023).

The paper reports complexity orders of $(x_c,y_c)$4 for P-MRDN and $(x_c,y_c)$5 for P-MSRDN, versus $(x_c,y_c)$6 for angular MRDN. Empirically, P-MSRDN has the best NMSE and fastest convergence. After 400 epochs, the gain of P-MRDN over MRDN is $(x_c,y_c)$7 dB at $(x_c,y_c)$8 dB, $(x_c,y_c)$9 dB at $\{d_1,\ldots,d_n\}$0 dB, and $\{d_1,\ldots,d_n\}$1 dB at $\{d_1,\ldots,d_n\}$2 dB; the gain of P-MSRDN over MRDN is $\{d_1,\ldots,d_n\}$3 dB, $\{d_1,\ldots,d_n\}$4 dB, and $\{d_1,\ldots,d_n\}$5 dB at the same SNRs. In direct NMSE comparison, P-MRDN improves over MRDN by $\{d_1,\ldots,d_n\}$6 dB at $\{d_1,\ldots,d_n\}$7 dB SNR, and P-MSRDN improves over MRDN by $\{d_1,\ldots,d_n\}$8 dB at $\{d_1,\ldots,d_n\}$9 dB and $$x_i=\cos\theta_i \times d_i + x_c,\qquad y_i=\sin\theta_i \times d_i + y_c.$$0 dB at $$x_i=\cos\theta_i \times d_i + x_c,\qquad y_i=\sin\theta_i \times d_i + y_c.$$1 dB. The stated conclusion is that polar-domain modeling matches XL-MIMO near-field physics better than angular-domain modeling, and that ASPP-based multi-scale extraction further improves reconstruction (Lei et al., 2023).

4. Corrective refinement in polar feature spaces

A second line of work does not define residuality as an explicit transformed-domain subtraction, but instead treats the polar representation as geometrically advantageous yet systematically distorted, requiring learned correction. PARTNER is a canonical example. It begins from the observation that polar voxelization is robust to resolution changes and favorable for streaming, but that identical objects at different ranges and headings have diverse distorted appearances, producing global misalignment in the BEV feature map. Its Global Representation Re-alignment (GRR) module first condenses each azimuth column by applying a radial max filter,

$$x_i=\cos\theta_i \times d_i + x_c,\qquad y_i=\sin\theta_i \times d_i + y_c.$$2

then selecting top-$$x_i=\cos\theta_i \times d_i + x_c,\qquad y_i=\sin\theta_i \times d_i + y_c.$$3 indices per angular column, and then performing cross-attention with relative positional encoding

$$x_i=\cos\theta_i \times d_i + x_c,\qquad y_i=\sin\theta_i \times d_i + y_c.$$4

$$x_i=\cos\theta_i \times d_i + x_c,\qquad y_i=\sin\theta_i \times d_i + y_c.$$5

$$x_i=\cos\theta_i \times d_i + x_c,\qquad y_i=\sin\theta_i \times d_i + y_c.$$6

Angular self-attention with shifted windows then re-aligns features globally, after which reverse cross-attention broadcasts the aligned representation back to the full BEV map. PARTNER complements GRR with a Geometry-aware Adaptive module that predicts a foreground heatmap $$x_i=\cos\theta_i \times d_i + x_c,\qquad y_i=\sin\theta_i \times d_i + y_c.$$7 and a distance/center offset map $$x_i=\cos\theta_i \times d_i + x_c,\qquad y_i=\sin\theta_i \times d_i + y_c.$$8, fuses them into $$x_i=\cos\theta_i \times d_i + x_c,\qquad y_i=\sin\theta_i \times d_i + y_c.$$9, and injects this geometry into query, key, and value formation before the final head (Nie et al., 2023).

The reported effect is not merely architectural embellishment. On Waymo validation, CenterPoint-Polar reaches $$\mathbf{h}^{\text{near-field}}=\mathbf{D}\mathbf{h}^{\text{near-field}}_{\mathrm{P}},$$0 L1 mAP/mAPH and $$\mathbf{h}^{\text{near-field}}=\mathbf{D}\mathbf{h}^{\text{near-field}}_{\mathrm{P}},$$1 L2, while PARTNER-CP reaches $$\mathbf{h}^{\text{near-field}}=\mathbf{D}\mathbf{h}^{\text{near-field}}_{\mathrm{P}},$$2 L1 and $$\mathbf{h}^{\text{near-field}}=\mathbf{D}\mathbf{h}^{\text{near-field}}_{\mathrm{P}},$$3 L2, and PARTNER-CF reaches $$\mathbf{h}^{\text{near-field}}=\mathbf{D}\mathbf{h}^{\text{near-field}}_{\mathrm{P}},$$4 L1 and $$\mathbf{h}^{\text{near-field}}=\mathbf{D}\mathbf{h}^{\text{near-field}}_{\mathrm{P}},$$5 L2. On ONCE, CenterPoint-Cartesian yields $$\mathbf{h}^{\text{near-field}}=\mathbf{D}\mathbf{h}^{\text{near-field}}_{\mathrm{P}},$$6 overall mAP, PolarStream $$\mathbf{h}^{\text{near-field}}=\mathbf{D}\mathbf{h}^{\text{near-field}}_{\mathrm{P}},$$7, and PARTNER $$\mathbf{h}^{\text{near-field}}=\mathbf{D}\mathbf{h}^{\text{near-field}}_{\mathrm{P}},$$8. The paper highlights gains of $$\mathbf{h}^{\text{near-field}}=\mathbf{D}\mathbf{h}^{\text{near-field}}_{\mathrm{P}},$$9 on Waymo and $$\mathbf{D}=[\mathbf{D}_{Q_1},\mathbf{D}_{Q_2},\ldots,\mathbf{D}_{Q_N}]$$0 on ONCE over previous polar-based works, and shows that applying GRR to Cartesian coordinates hurts performance, indicating that the refinement is specific to polar geometry rather than a generic attention insertion (Nie et al., 2023).

PVP extends the same corrective logic to dense 3D semantic occupancy prediction. It argues that polar grids better match non-uniform sensor information distribution than Cartesian voxel grids, but that distortion impairs both local convolution and global feature propagation. Its Global Representation Propagation module condenses local windows by max-selection,

$$\mathbf{D}=[\mathbf{D}_{Q_1},\mathbf{D}_{Q_2},\ldots,\mathbf{D}_{Q_N}]$$1

applies local cross-attention, performs decomposed global attention along radial, azimuth, and height axes, and then propagates the recalibrated representation back into local windows. Its Plane Decomposed Convolution replaces a $$\mathbf{D}=[\mathbf{D}_{Q_1},\mathbf{D}_{Q_2},\ldots,\mathbf{D}_{Q_N}]$$2 kernel by $$\mathbf{D}=[\mathbf{D}_{Q_1},\mathbf{D}_{Q_2},\ldots,\mathbf{D}_{Q_N}]$$3, $$\mathbf{D}=[\mathbf{D}_{Q_1},\mathbf{D}_{Q_2},\ldots,\mathbf{D}_{Q_N}]$$4, and $$\mathbf{D}=[\mathbf{D}_{Q_1},\mathbf{D}_{Q_2},\ldots,\mathbf{D}_{Q_N}]$$5 kernels to treat projection distortion in $$\mathbf{D}=[\mathbf{D}_{Q_1},\mathbf{D}_{Q_2},\ldots,\mathbf{D}_{Q_N}]$$6, scale distortion in $$\mathbf{D}=[\mathbf{D}_{Q_1},\mathbf{D}_{Q_2},\ldots,\mathbf{D}_{Q_N}]$$7, and identity behavior in $$\mathbf{D}=[\mathbf{D}_{Q_1},\mathbf{D}_{Q_2},\ldots,\mathbf{D}_{Q_N}]$$8. The paper explicitly states that there is no residual equation of the form $$\mathbf{D}=[\mathbf{D}_{Q_1},\mathbf{D}_{Q_2},\ldots,\mathbf{D}_{Q_N}]$$9; any interpretation as residual refinement is conceptual rather than architectural. Quantitatively, PVP reports $\mathbf{h}^{\text{near-field}}_{\mathrm{P}}$0 IoU and $\mathbf{h}^{\text{near-field}}_{\mathrm{P}}$1 mIoU for LiDAR-only and $\mathbf{h}^{\text{near-field}}_{\mathrm{P}}$2 IoU and $\mathbf{h}^{\text{near-field}}_{\mathrm{P}}$3 mIoU for camera+LiDAR on OpenOccupancy, with ablations showing $\mathbf{h}^{\text{near-field}}_{\mathrm{P}}$4 for Polar + GRP + PD-Conv versus $\mathbf{h}^{\text{near-field}}_{\mathrm{P}}$5 for the raw baseline (Xue et al., 2024).

Polaris introduces yet another refinement-based polar framework, now for trajectory prediction and planning. It operates entirely in polar coordinates, encodes agents and lane polylines in polar form, uses a Relative Embedding Transformer with relative features $\mathbf{h}^{\text{near-field}}_{\mathrm{P}}$6, decodes proposal trajectories in $\mathbf{h}^{\text{near-field}}_{\mathrm{P}}$7, and then applies polar relationship refinement by re-encoding the proposal trajectories and letting them re-attend to scene context. The endpoint $\mathbf{h}^{\text{near-field}}_{\mathrm{P}}$8 of each trajectory mode serves as a key spatial anchor, and the appendix reports that the best tradeoff is usually $\mathbf{h}^{\text{near-field}}_{\mathrm{P}}$9 refinement modules. Training combines polar and Cartesian supervision: $\theta$0 On Argoverse 2 single-agent test, Polaris reports $\theta$1, $\theta$2, and $\theta$3; on nuPlan Test 14 Hard it reports $\theta$4, $\theta$5, and $\theta$6 (Zhang et al., 15 Aug 2025).

5. Direct non-residual polar contour representations

PolarMask++ is a foundational example of a polar representation whose improvements are substantial but not residual in the strict sense. The object mask is represented by a center point $\theta$7 and $\theta$8 radial distances sampled at uniform angular interval $\theta$9; the paper illustrates $r$0, corresponding to one ray every $r$1. Ground-truth labels are generated by traversing the contour, choosing the maximum distance when multiple contour points map to the same angle, borrowing the nearest available angle if an angle is missing, and assigning $r$2 if no valid intersection exists. The method’s optimization is driven by soft polar centerness and polar IoU loss rather than by box-conditioned mask refinement (Xie et al., 2021).

The original polar centerness is

$r$3

while the softened version partitions the rays into four angular groups $r$4 and defines

$r$5

For coupled regression of the full ray set, the paper defines

$r$6

The total loss is

$r$7

with focal loss for $r$8, Polar IoU loss for $r$9, binary cross-entropy for $c$00, and $c$01. PolarMask++ also adds a Refined Feature Pyramid that resizes $c$02 to $c$03, sums them, applies a non-local block, resizes the refined features back to the original pyramid levels, and injects them through shortcuts. The paper explicitly notes, however, that this is feature-level refinement rather than contour-level residual correction. It reports $c$04 mask AP on COCO with a stronger backbone and larger input, $c$05 F-measure on ICDAR2015 with external pretraining and $c$06 without it, and $c$07 mAP on DSB2018 (Xie et al., 2021).

SiamPolar adopts the same direct contour parameterization for semi-supervised realtime video object segmentation. Its polar head predicts classification score, bounding-box regression, and polar mask regression, where the mask branch outputs radial distances $c$08 for rays sampled at equal angular intervals; the main experiments use $c$09 rays, and the label-generation algorithm loops over $c$10. The method adds contour merging for occluded or multi-part contours, with contour diameter

$c$11

and merges contours when midpoint distance is less than

$c$12

Its improved polar centerness is

$c$13

and the mask loss uses a polar-IoU-style logarithmic ratio. The architecture couples an asymmetric Siamese backbone, repeated cross-correlation, semi-FPN, and the polar head. The paper reports that $c$14 gives the best upper-bound IoU of $c$15 for generated masks on DAVIS, that $c$16 rays are best for TSD-max with average region similarity $c$17, and that ResNet101 + semi-FPN reaches $c$18 at $c$19 fps on DAVIS-2016 (Li et al., 2021).

These two methods are central because they clarify what residual polar representation is not. In both cases the polar mask itself is directly predicted as absolute ray lengths from a center; refinement appears in centerness design, feature pyramids, contour merging, or optional post-processing, not as residual ray correction.

6. Residue calculus for polar analytic functions

The mathematically distinct notion of residue in polar representation arises in polar analytic functions and Mellin analysis. The domain is the polar half-plane

$c$20

A function $c$21 is polar-analytic if the limit defining

$c$22

exists and is independent of approach within $c$23. Polar analyticity is equivalent to the polar Cauchy–Riemann system

$c$24

or, equivalently,

$c$25

The polar Mellin derivative is

$c$26

and higher-order derivatives are defined via Stirling numbers of the second kind (Bardaro et al., 2019).

The paper introduces the logarithmic pole of order $c$27: an isolated singularity $c$28 for which

$c$29

with polar-analytic $c$30 satisfying $c$31. The associated $c$32-residue is

$c$33

This is a genuine residue concept, but it is adapted to the logarithmic polar variable $c$34 and to the Mellin weight $c$35, rather than to a Laurent coefficient in $c$36 (Bardaro et al., 2019).

The paper proves a Cauchy-type representation formula for polar Mellin derivatives,

$c$37

and a residue theorem: $c$38 It also derives a local polar Taylor-type expansion,

$c$39

a Mellin analogue of Boas’ differentiation formula, a Bernstein-type inequality, and a Valiron sampling theorem analogue. In this domain, the polar representation is analytic rather than learned, and the key concept is residue, not residual correction (Bardaro et al., 2019).

7. Comparative synthesis

Taken together, these works suggest that the decisive issue is rarely polar coordinates alone. Performance gains arise when the polar parameterization is matched to the governing invariants or distortions of the problem: spherical-wave angle-distance structure in XL-MIMO (Lei et al., 2023), non-uniform radial sampling in LiDAR and occupancy prediction (Nie et al., 2023, Xue et al., 2024), center-contour geometry in instance and video segmentation (Xie et al., 2021, Li et al., 2021), relative distance-and-direction interactions in motion forecasting and planning (Zhang et al., 15 Aug 2025), and logarithmic geometry in Mellin analysis (Bardaro et al., 2019).

A second consistent pattern is that “residual” is domain-dependent. In channel estimation it is a literal residual map in a transformed polar basis. In perception systems such as PARTNER and PVP, the nearest analogue is corrective re-alignment or propagation that compensates for distortion in polar grids. In Polaris, refinement is iterative and relation-centric rather than additive. In PolarMask++ and SiamPolar, by contrast, the polar representation itself is direct and compact, and the papers explicitly distinguish enhancement modules from residual contour correction (Lei et al., 2023, Nie et al., 2023, Zhang et al., 15 Aug 2025, Xie et al., 2021, Li et al., 2021).

A final conceptual boundary is between residual representation and residue theory. The Mellin–polar literature shows that polar representation also supports an exact analytic calculus with logarithmic poles, $c$40-residues, and sampling formulas, but this should not be conflated with deep residual learning. The same vocabulary therefore spans at least three technical regimes: explicit residual estimation, refinement of distorted polar features, and polar residue calculus.