Translation-Invariant Tile-Based Phase Unwrapping with Residual-Weighted Multipath Averaging

Antoine Moevus; Max Mignotte

eess.IVSep 11, 2026

Translation-Invariant Tile-Based Phase Unwrapping with Residual-Weighted Multipath Averaging

Authors: Antoine Moevus, Max Mignotte

Organizations: D´epartement d’informatique et de recherche opérationnelle, Universit´e de Montr´eal, Montr´eal, QC, Canada

Abstract

Phase unwrapping is a key step in interferometric and coherent imaging, where the physical quantity of interest is carried by a phase that the instrument delivers only modulo 2*pi. The difficulty in two dimensions is to separate the jumps caused by wrapping from those produced by noise, by true discontinuities, by under-sampling or by decorrelation. Spatial-domain and frequency-domain methods have both been studied extensively, each with advantages the other lacks; hybrid schemes combining the two remain scarce. We propose a semi-global tile-based strategy in which every tile is unwrapped in the frequency domain, through the Discrete Cosine Transform (DCT) and the least squares (LS) formalism of Ghiglia et al., the tiles being merged spatially. Unwrapping and noise filtering are performed jointly, which regularizes an otherwise ill-posed inverse problem. An error stays confined to the tile in which it arose; the tiling artifacts are removed by averaging over every shift of the grid and over the symmetries of the square; and that average is weighted by the Poisson residual each pass leaves behind, so that a pass whose tile boundaries fell on a discontinuity does not impose its seam on the result. Experiments on synthetic and real data, against four reference algorithms from four distinct families and under six complementary metrics, show that the proposed method matches or improves on the state of the art. One of these metrics, a corrected cyclic re-wrap residual introduced here, needs no ground truth and therefore remains available on real acquisitions.

Figures & tables

Fig. 1: A seam measurement becomes a tile-offset constraint. (a) Adjacent pixels $p\in\mathcal{T}_{a}$ and $q\in\mathcal{T}_{b}$ supply $d_{e}$ in ( 4 ); the median estimates $\delta_{e}$ , and the median absolute deviation (MAD) determines the ranking weight $w_{e}$ . (b) Each tile is a node and each shared border is an edge. (c) Kruskal retains a spanning tree; offsets propagate with the signed rule $o_{b}-o_{a}=\delta_{e}$ . Panels (b) and (c) use the same illustrative weights, expressed with $c=1$ and 256 units per cycle.

Fig. 2: Moving an

8\times 8

tile grid over a fixed image. (a) Four horizontal offsets and four vertical offsets, spaced two pixels apart, give sixteen placements of the grid. Each point

(h_{x},h_{y})

specifies a shift of the entire tile grid by

h_{x}

pixels horizontally and

h_{y}

pixels vertically: the tile boundaries move while the image stays fixed. The circled points select the placements illustrated in (b) and (c). (b) A schematic

24\times 24

-pixel image, outlined in black, is partitioned at origin

(0,0)

. Colored lines delimit tiles, one of which is highlighted; dashed ends show that the grid continues. (c) The same image and sample pair after moving the grid two pixels right and down. The pair

p,q

now belongs to one tile instead of straddling a tile boundary. The image size is chosen only for illustration; the tile size and stride match the evaluated configuration.

Fig. 3: Spatial transformations and phase-polarity reversal illustrated on a synthetic wrapped phase image. (a) The upper row shows clockwise rotations through $0^{\circ}$ , $90^{\circ}$ , $180^{\circ}$ , and $270^{\circ}$ ; the lower row applies a left–right reflection after each corresponding rotation. The transformed reconstruction is mapped back to the original pixel coordinates before averaging. These eight transformations, each combined with the sixteen grid placements in Figure 2 , form the 128-pass schedule. (b) Negation changes the phase values while leaving pixel positions unchanged. This additional polarity option is shown for clarity but is not included in that schedule.

Fig. 4: Complementary responses of phase-error metrics and the proposed $\mathrm{MAE}_{\mathrm{uw2}}$ on a synthetic ramp. Reference-based errors use mean additive alignment; $\mathrm{MAE}_{\mathrm{uw2}}$ uses the circular offset in ( 17 ). A global offset leaves reference errors unchanged; amplitude scaling and a spatial cycle error remain visible in reference-based metrics. Both a cycle-shifted region and the wrapped identity can have zero $\mathrm{MAE}_{\mathrm{uw2}}$ . Phase and error color scales are shared within their respective rows.

Method	$n$	MAE	RMSE	Grad. RMSE	$F_{>\pi}$	SSIM mm	SSIM add	MAE uw2
Global DCT	202	0.802	1.052	0.640	20.547	0.748	0.830	0.468
SRNCP	202	0.527	0.816	0.783	14.319	0.721	0.859	0.000
SNAPHU	202	0.474	0.729	0.702	13.930	0.783	0.880	0.000
ROMEO	202	0.505	0.726	0.771	15.967	0.736	0.847	0.000
Delaunay L1	202	0.521	0.824	0.750	16.482	0.752	0.850	0.000
DigiHolo	202	0.588	0.952	0.802	18.359	0.728	0.818	0.000

TABLE I: Full-set reconstruction accuracy and completion counts (

n

). A dash denotes an unavailable full-set aggregate, rather than zero error; successful-case results for all methods are reported separately in Table II . Phase errors and MAE uw2 are in radians.

F_{>\pi}

is the mean percentage of pixels above

\pi

; other scores are per-image medians. SSIM mm uses contrast normalization and SSIM add preserves amplitude. The proposed method uses

8\times 8

tiles and 128 passes.

Method	$n$	MAE	RMSE	Grad. RMSE	$F_{>\pi}$	SSIM mm	SSIM add	MAE uw2
Global DCT	192	0.767	1.003	0.615	19.556	0.754	0.834	0.396
SRNCP	192	0.491	0.755	0.761	12.926	0.740	0.865	0.000
SNAPHU	192	0.458	0.714	0.672	14.062	0.783	0.880	0.000
ROMEO	192	0.481	0.703	0.740	14.557	0.755	0.849	0.000
Delaunay L1	192	0.505	0.763	0.716	16.065	0.758	0.855	0.000
DigiHolo	192	0.570	0.889	0.784	17.137	0.734	0.828	0.000

TABLE II: Accuracy on the 192 images for which all eight implementations completed reconstruction. The column

n

gives the number of images. Every row uses the same images; this conditional comparison omits cases on which any implementation failed. Phase errors and MAE uw2 are in radians.

F_{>\pi}

is the mean percentage of pixels above

\pi

; other scores are per-image medians. SSIM mm uses contrast normalization and SSIM add preserves amplitude. The proposed method uses

8\times 8

tiles and 128 passes.

Fig. 5: Reconstructions and reference differences for InSAR (top) and an axial MRI simulation (bottom). The InSAR reference is a phase inferred from a digital elevation model, not an exact measurement of the true interferometric phase; its uncertainty and mismatch with the acquisition contribute to the displayed differences. The MRI reference is known within the QSM simulation model [ 19 ] ; it is not an independently measured in-vivo ground truth. The absence of exact phase truth in experimental acquisitions motivates distinguishing these reference types. Reconstructions share one phase scale per row after mean additive alignment, and signed differences share one symmetric scale per row. Pixels outside the evaluation mask are blank.

Fig. 6: Spatial error patterns on the Tree case, with seven reference algorithms and the proposed $8\times 8$ , 128-pass reconstruction. Top: exact constructed reference and reconstructions after mean-gauge alignment, on one common phase scale. Bottom: wrapped input, followed by signed phase errors on one symmetric cyclic color scale (matching colors at $-2\pi$ and $+2\pi$ ), with a black boundary around $|e|>\pi$ . The final column shows Tiled-DCT $8\times 8$ (ours), including residual weighting. Phase maps use the reference range; error colors saturate outside $[-2\pi,2\pi]$ , marked by the colorbar extensions. All scores use unclipped values. Phase maps are not individually contrast-normalized. The regional errors of the congruent reconstructions and the diffuse least-squares errors motivate complementary metrics.

Fig. 7: Speed–accuracy trade-offs for eight implementations on the 192 common successful images of Table II , with a budget of 12 numerical threads. Columns: SSIM mm , RMSE, MAE, and

C(\pi)

; rows: mean (top) and median (bottom) of per-image scores. Both coordinates use the same images for every method. Dashed lines connect nondominated point estimates, without uncertainty margins; lower-is-better axes are reversed, so the preferred direction is upper left. The logarithmic time axis is common; vertical ranges vary by panel. The inset in (g) enlarges the DigiHolo and tiled DCT points without changing their coordinates. Tiled DCT uses

8\times 8

tiles, 128 passes and 12 workers. These fronts are conditional on successful completion; Table I retains full-set completion and accuracy results.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Fig. 8: Seven constructed phase fields and reconstructions by the eight methods. Columns: wrapped input, exact constructed reference, and reconstructions. The factor $c$ scales the intensity-derived phase range; the noisy Baboon and Lena inputs retain the corresponding clean reference. Every reference and reconstruction in a row shares the reference’s phase scale after mean additive alignment. Colorbar extensions indicate values outside the display range; numerical scores use unclipped outputs. RMSE is in radians and $<\pi$ denotes the percentage of valid pixels below that error threshold. Wrapped inputs share the cyclic scale shown below their column. The proposed method uses $8\times 8$ tiles and 128 passes. A labeled failure panel indicates that no finite reconstruction was available within the time limit; no accuracy score is assigned to it.

Fig. 9: Multimodal reconstruction using references supplied by the source datasets. Rows 1–2: real TanDEM-X interferograms from InSAR-DLPU; their reference phases are generated from Shuttle Radar Topography Mission (SRTM) elevation data [ 21 ] , using terrain information beyond the wrapped phase available to the compared methods. Rows 3–4: holographic acquisitions with references produced by quality-guided phase unwrapping (QGPU), selected by Gontarz et al. for its robustness in their holographic-tomography processing [ 20 ] . These InSAR and holographic references are comparison targets, not exact physical ground truths; their scores measure agreement with them. Row 5 shows an MRI slice with a phase known within the QSM simulation model [ 19 ] . Columns and scores follow Figure 8 , with a common reference-based phase scale within each row.

Fig. 10: Experimental phase maps without an unwrapped reference: an MRI head and two holographic examples. The first column is the wrapped observation; the remaining columns show the eight reconstructions. Each reconstruction is centered by its valid-pixel mean for display, and all reconstructed maps within a row share one phase scale. The reported $\mathrm{MAE}_{\mathrm{uw2}}$ uses circular offset alignment as defined in ( 17 ). No reference-error map or accuracy ranking is inferred from these images.

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