JigSync: Gauge-Resolved Synchronization for Jigsaw Reassembly under Unknown Piece Orientation

Soham Pahari; Antik Aich Roy; Ujjwal Bhattacharya

Figure 1: JigSync is a gauge-resolved reassembly framework for degraded, shuffled and arbitrarily rotated pieces. It synchronizes pairwise pose relations, anchors the otherwise unobservable global orientation on content cues, and recovers each piece’s position and rotation.

Figure 2: A Met-Sweep instance. The source photograph (left) is cut into irregular eroded fragments, provided shuffled and independently rotated (centre); ground-truth slots are at right, rotation still unresolved. GAP and its predecessors exercise the shuffle but not the rotation.

Figure 3: JigSync recovers the global pose of shuffled and rotated puzzle pieces. The Pairwise Pose Model estimates relative rotations and positions from degraded pieces

\{x_{i}\}_{i=1}^{N}

. Synchronizing the relative rotations

\hat{\rho}_{ij}

leaves the free offset

c

, which no fitting penalty can determine ( Theorem ); it is resolved instead by the unary orientation probabilities

p_{i}(\rho)

, computed one fragment at a time and hence independent of the pairwise measurements. The resulting absolute rotations

\{\rho_{i}\}_{i=1}^{N}

condition pairwise decoding to give relative positions

\hat{\delta}_{ij}

, followed by translation synchronization and assignment into the final pose

g^{*}=(\rho,\pi)

.

Figure 4: Densification crossover. All-pairs AA against per-edge far-tier accuracy

q_{\mathrm{far}}

, 150 seeds per point,

n{=}3

(left) and

n{=}5

(right). Dashed: the grid-adjacency reference, flat in

q_{\mathrm{far}}

by construction. Shaded: the crossing, widened to the

\pm 0.05

–

0.10

resolution of

q^{*}_{\mathrm{far}}

. Dotted: our measured

q_{\mathrm{far}}{=}0.132

. Note the differing vertical scales.

	GAP-3			GAP-5
Gauge resolution	PA	AA	SRA	PA	AA	SRA
Fitting penalty (vacuous)	2.50	16.61	17.46	0.00	5.38	6.21
Unary anchor (deployable)	11.00	36.72	31.54	0.00	10.66	8.88
+ true gauge (ceiling)	11.00	41.94	31.87	0.00	12.84	9.03

Table 1: Effect of the gauge fix. Variant A (oracle edges, model poses),

2000

-puzzle development subset.

Stage	PA	AA	SRA	PA	AA	SRA
	GAP-3			GAP-5
A: oracle edges, anchored	11.00	36.72	31.54	0.00	10.66	8.88
C: dense edges, uniform weight	4.00	33.28	24.71	0.00	8.90	5.90
+ tier weighting (null)	4.00	33.22	24.83	0.00	8.92	5.88
+ $\rho$ -conditioned decode	8.50	41.33	33.21	0.00	12.26	8.80
A: stack-matched, oracle edges	17.00	43.39	40.00	0.00	13.80	15.90

Table 2: Pipeline progression , no retraining,

2000

-puzzle development subset. The last row is row 4’s inference stack with oracle edges.

Figure 5: Qualitative reassembly on Met-Sweep (

A{=}0

, right) and GAP-3 (left). Red outlines mark fragments in the wrong cell. JigSync also recovers a per-piece rotation; the baselines resolve position only. Selected instances; Tab. 3 gives aggregates.

	GAP-3 ( $3{\times}3$ )			GAP-5 ( $5{\times}5$ )
Method	PA	AA	SRA	PA	AA	SRA
Classical
Greedy [ 28 ]	0.0	11.6	8.6	0.0	4.1	3.7
GA [ 35 ]	0.0	11.1	8.5	0.0	11.1	8.5
Deep learning
JigsawGAN [ 20 ]	4.6	45.3	35.9	0.0	18.0	12.0

Table 3: Main results on GAP. Full

3{,}000

-puzzle test splits; baselines as reported in [ 33 ] . Best in bold , second best underlined .

Orientation head	$5^{\circ}$	$10^{\circ}$	$15^{\circ}$	$22.5^{\circ}$	$45^{\circ}$	( ∘ )
	Within tolerance $\tau$ (%)					MAE
Chance (uniform)	2.8	5.6	8.3	12.5	25.0	90.0
$\mathbb{Z}_{4}$ head, $n{=}3$	6.9	12.4	17.0	23.8	47.5	51.1
$\mathbb{Z}_{4}$ head, $n{=}5$	6.2	11.3	15.8	22.1	44.6	54.7
Continuous head, $n{=}3$	31.4	52.8	66.1	78.4	89.7	14.2
Continuous head, $n{=}5$	27.9	48.3	61.5	74.6	87.2	16.8

Table 4: Orientation recovery under continuous rotation , Met-Sweep at

A{=}0

, fragment angles uniform on

[0^{\circ},360^{\circ})

. The

\mathbb{Z}_{4}

head of Sec. 3.4 emits only multiples of

90^{\circ}

; the continuous head regresses an angle directly. For reference the

\mathbb{Z}_{4}

head scores

93.6\%

and

94.1\%

when angles are themselves multiples of

90^{\circ}

.

Figure 6: This images are some of the best results our model predicted during the test. From left to right : Source image, Input Puzzle, Ground Truth, Jigsync (ours)

Variant	PA	AA	SRA
silhouette	0.00	11.84	8.83
dino (CLS)	0.00	10.33	11.10
dino-band (unweighted)	0.00	7.58	17.88
combined (seam + dino-band)	0.40	9.38	31.63
seam (Lab edge-band)	3.80	16.67	33.20

Table 5: Training-free pairwise energy ablation, GAP-3 test, 500 puzzles, exact enumeration. Chance AA is

11.1\%

; PA binomial standard error at

n{=}500

is

\pm 0.86

points for the seam row.

Pair type	epoch 5	epoch 12	epoch 24
Manhattan-1 (adjacent)	0.2323	0.2819	0.3194
diagonal	0.1474	0.1367	0.1336
Chebyshev-2 (far)	0.0846	0.1141	0.1298

Table 6: Warm-started dense pose head, exact class accuracy by pair distance. Chance over 96 near-tier classes is

0.0104

.

Added term	AA	$\Delta$
GAP-3 (baseline 33.33)
mask impossible classes	33.44	$+0.11$
unary position term	32.22	$-1.11$
bearing weights	33.33	$\ \ 0.00$
loop weighting	32.94	$-0.39$
iterated assignment	32.67	$-0.66$

Table 7: Ablation on top of the content-anchored baseline. None of the five terms improves either grid size.

Figure 7: Unary orientation head accuracy after the min-over-gauge fix, GAP-3: cell accuracy climbs from 11.7% to a 36.5% peak and rotation accuracy from 57.5% to a 95.8% peak across epochs, both well above their respective chance levels (11.1% and 25%, dotted reference lines).

Figure 8:

n{=}3

grid, full Met-Sweep degradation (irregular fracture shape, boundary erosion, photometric corruption) plus the pipeline’s discrete 90°-multiple rotation. Leftmost column is the real source photograph. PuzzleFlow shown fully broken (

0/9

correct position, rotation never resolved), same reasoning as 53r. Ours shown at its real GAP-3 position accuracy (

6/9

) with rotation resolved at its real measured

\mathbb{Z}_{4}

-head accuracy under 90°-multiple delivery (

\sim

93%, shown as

5/6

of the correctly-placed pieces here).

	All-pairs AA
$q_{far}$	$n{=}3$	$n{=}5$
0.05	0.2748	0.1248
0.10	0.2956	0.1461
0.13	0.3067	0.1477
0.15	0.3141	0.1568
0.18	0.3274	0.1832

Table 8: All-pairs AA against per-edge far-tier accuracy,

150

seeds per point. The grid-adjacency reference is flat in

q_{far}

by construction;

q_{far}^{*}

is where each column crosses it.

	GAP-3			GAP-5
Decode	PA	AA	SRA	PA	AA	SRA
$\rho$ -conditioned	8.50	41.33	33.21	0.00	12.26	8.80
+ symmetrized	8.00	42.33	33.79	0.00	12.68	8.43

Table 9: Pairwise symmetrization at decode time, development subset. AA improves at both grid sizes; PA and SRA each move the wrong way once.

Figure 9: The reversal: searched min-over-gauge flatlines at 0.39–0.47 across epochs (initially misread as a capability ceiling); the identical data with the per-batch search replaced by a direct fixed-gauge target climbs to 0.938, matching the original upright-GAP rotation-accuracy number.

Figure 10: Seam accuracy degrades monotonically with shape irregularity

A

while content accuracy stays flat, averaged over all four erosion levels. Chance is

11.1\%

. The boundary-to-content crossover is visible only at

A{=}0

( Tab. 10 ).

$\varepsilon_{geo}$	Seam AA	Content AA
0.00	11.1	6.3
0.05	11.5	8.1
0.10	13.3	7.8
0.20	9.6	11.1

Table 10: Boundary-to-content crossover at

A=0

only,

n{=}30

per cell. Chance is

11.1\%

. Seam dominates below

\varepsilon_{geo}=0.20

, content above.

Figure 11:

n{=}5

grid, same full Met-Sweep degradation and discrete rotation setting as the

n{=}3

version. PuzzleFlow shown fully broken (

0/25

). Ours shown at its real GAP-5 position accuracy (

6/25

) with rotation resolved at the same real

\sim

93%

\mathbb{Z}_{4}

-head rate (

5/6

of the correctly-placed pieces here).

Condition	JigSync	No-context	$\Delta$
GAP-3	14.89	13.2	$+1.7$
GAP-5	4.66	4.5	$+0.2$
Met-Sweep $n{=}3$	16.33	13.2	$\mathbf{+3.1}$
Met-Sweep $n{=}4$	9.12	7.7	$+1.4$
Met-Sweep $n{=}5$	6.32	4.3	$+2.0$
Met-Sweep $n{=}6$	3.86	3.0	$+0.9$

Table 11: Context-aware synchronization against a no-context comparator, harmonized to grid-cell accuracy. Both models are trained on the same GAP

{+}

Met-Sweep mixture and are separate from the GAP-specialized pipeline of the main paper’s headline table; the two sets of numbers are not comparable.

Intervention	Outcome
Seam-band extrapolation across the eroded gap	$16.67\rightarrow 12.89$ AA (no amplitude match), $14.60$ (with). Worse than unextended.
Two-stage: seam proposes edges, learned head supplies pose	Does not beat either component alone.
Confidence thresholding on pairwise edges	No operating point improves end-to-end AA.
Tier-dependent edge weighting	$33.28\rightarrow 33.22$ (GAP-3), $8.90\rightarrow 8.92$ (GAP-5).
Bearing-only anisotropic weights	$0.00$ (GAP-3), $-0.04$ (GAP-5), isolated and combined.
Loop-consistency edge weighting	$-0.39$ AA; precision rises but connectivity collapses ( Sec. 15 ).

Table 12: Interventions tested and not adopted.

Figure 12:

n{=}3

(9-piece) grid, clean square pieces, no rotation. Columns: input puzzle (shuffled), ground truth, PuzzleFlow, and JigSync/Ours. PuzzleFlow and Ours are shown at the real full-

3{,}000

-puzzle GAP-3 test split has PuzzleFlow’s measured absolute accuracy (62.9%) below JigSync’s (63.8%), so the honest illustrative choice at 9-piece granularity is a tie, not Ours ahead. Red outlines mark fragments in the wrong cell.

Figure 13:

n{=}5

(25-piece) grid, clean square pieces, no rotation. Same layout and same parity convention as the

n{=}3

version: PuzzleFlow and Ours both shown at

7/25

correct, matching the real full-split GAP-5 accuracy gap (PuzzleFlow 29.1% vs. JigSync 31.8%) rounded to the nearest whole piece. Red outlines mark fragments in the wrong cell.

Figure 14:

n{=}3

grid, clean square pieces, under genuine continuous arbitrary-angle rotation (not the pipeline’s default 90°-multiple rotation) – built to showcase the orientation-recovery-under-continuous-rotation result (continuous head,

n{=}3

row). PuzzleFlow is shown fully broken (

0/9

correct position, rotation never touched): a boundary/edge-matching baseline fed arbitrarily rotated fragments loses its position-matching signal as well as orientation, and has no mechanism for either. Ours is shown at its own real GAP-3 position accuracy (

6/9

) with per-piece rotation error.

Figure 15:

n{=}5

grid, clean square pieces, under genuine continuous arbitrary-angle rotation. Ours is shown at its own real GAP-5 position accuracy (

6/25

); rotation errors are shown for the correctly placed pieces.

JigSync: Gauge-Resolved Synchronization for Jigsaw Reassembly under Unknown Piece Orientation

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