Multi-View Structure-from-Motion Enables Oriented Projective Shape Analysis in Three Dimensions

Musab Alamoudi; Robert L. Paige; Vic Patrangenaru

cs.CVSep 7, 2026

Multi-View Structure-from-Motion Enables Oriented Projective Shape Analysis in Three Dimensions

Authors: Musab Alamoudi, Robert L. Paige, Vic Patrangenaru

Organizations: Department of Statistics, Florida State University, Tallahassee, FL, USA. · Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO, USA.

Abstract

Projective shape analysis provides a geometric framework for studying landmark configurations in digital images acquired by pinhole cameras. In the classical projective shape (PS) model, three-dimensional configurations ( $k$ -ads) are represented as points in $(\mathrm{RP}^3)^q$ , $q = k - 5$ . A nonparametric test is developed in Patrangenaru et al. [12], for this framework, to determine whether an object matches a design blueprint, with each configuration reconstructed from a single uncalibrated stereo pair. Such two-view reconstructions are identified only up to a 3D projective transformation, which may reverse orientation, so that the sign-blind PS summary was the only available, before oriented projective shape (OPS) was recently considered. Multi-view Structure-from-Motion (SfM) technology removes this obstruction: its bundle adjustment is identified up to an orientation-preserving projective transformation. In this paper we revisit a well-cited three-cube object, from Patrangenaru et al. [12], with $n = 8$ SfM reconstructions built in Agisoft Metashape Professional 2.3.0, and validate our implementation by reproducing the published stereo construction from the original data. This allows us to perform what is to the best of our knowledge the first three-dimensional OPS analysis, compute its extrinsic total-variance index and perform statistical inference in this novel setting. Due to the high concentration of SfM data, the OPS index is asymptotically one-half the PS index, a structural consequence of concentration rather than a property of the object. Here our blueprint hypothesis is not rejected for any of the $q = 14$ non-frame landmarks, while the SfM reconstructions are about 26 times more concentrated than the stereo ones, a substantial gain in reconstruction precision. Sample-size, photograph-count, and frame-ordering analyses support the robustness of these conclusions.

Figures & tables

LM	$X$	$Y$	$Z$	LM	$X$	$Y$	$Z$
1	0	20	0	11	0	10	6
2	0	20	4	12 †	6	10	6
3	4	20	4	13	6	10	0
4	4	20	0	14	0	10	10
5	0	16	4	15	10	10	10
6	4	16	4	16	10	10	0

Table 1: The blueprint: exact Euclidean coordinates

(X,Y,Z)

of the

19

landmarks of the three-cube object, in object units (after Patrangenaru et al. [12] ). Frame landmarks

\{8,12,17,18,19\}

are denoted by † . The projective blueprint coordinates

\gamma_{s}

used in the test are obtained by applying the map ( 1 ) to these points (Remark 1 ).

Figure 1: Left: the three-cube object with the $19$ landmarks and the Euclidean blueprint coordinates of Table 1 ( $X$ horizontal, $Y$ vertical, $Z$ depth). Right: the oriented projective frame $F=\{8,12,17,18,19\}$ in the order used to build $U_{3}$ and the normalizing point $p_{19}$ . Both panels are rendered from the exact blueprint coordinates of Table 1 in Blender 5.0 [ 4 ] , for legibility; they are not reproduced from Patrangenaru et al. [12] .

Panel of Figure 2 — Figure 2: Representative SfM output from Agisoft Metashape Professional 2.3.0. (a) The estimated camera positions surrounding the object after multi-view photo alignment, with the sparse tie-point cloud at the center. (b) The dense point cloud of the object, colored by point confidence (blue high, green lower), used to assess reconstruction quality before marker placement.

Model	$M_{1}$	$M_{2}$	$M_{3}$	$M_{4}$	$M_{5}$	$M_{6}$	$M_{7}$	$M_{8}$
Photographs	64	95	76	109	77	89	84	114

Table 2: The eight reconstructions and the number of source photographs used to build each.

Figure 3: The eight independent SfM reconstructions of the three-cube object, arranged in two rows of four as in Patrangenaru et al. [12, Fig. 4] . Each was built from a different photograph set ( $64$ to $114$ images). The panels labelled “Model $k$ ” correspond to $M_{k}$ in the text; the bottom face shows the reconstructed ground plane and minor boundary artifacts of the dense surface, which do not involve the $19$ landmarks.

Sample	$\det(U_{3})$	$\kappa(U_{3})$	$\operatorname{sign}(v_{5})$	Status
$M_{1}$	0.291	122.6	$(-,+,+,-)$	OK
$M_{2}$	6.730	118.3	$(-,+,+,-)$	OK
$M_{3}$	1.551	96.0	$(-,+,+,-)$	OK
$M_{4}$	32.991	229.5	$(-,+,+,-)$	OK
$M_{5}$	4.187	187.4	$(-,+,+,-)$	OK
$M_{6}$	7.923	277.1	$(-,+,+,-)$	OK

Table 3: Frame quality and orientation diagnostics. All reconstructions share orientation and are well conditioned, so both PS and OPS are well posed.

LM	$tS_{n}^{\mathrm{PS}}$	$tS_{n}^{\mathrm{OPS}}$	$\widehat{\mathrm{SE}}_{\mathrm{OPS}}$	$T_{s}$	$p$
1	0.000175	0.000087	0.000034	336.9	0.057
2	0.001112	0.000556	0.000250	307.4	0.052
3	0.000759	0.000379	0.000097	57.9	0.185
4	0.000481	0.000241	0.000106	213.0	0.040
5	0.000229	0.000114	0.000060	868.6	0.054
6	0.001412	0.000706	0.000208	440.4	0.056

Table 4: Per-landmark PS and OPS results on the

n=8

SfM data. The Bonferroni cutoff for the blueprint test is

\alpha/q=0.00357

; no

p

-value falls below it. The OPS columns are discussed in Section 5 .

LM	$g_{s}(M)=[x,\,y,\,z,\,w]$	$\lambda_{1}$	dist. to bp (deg)
1	$[+0.192,+0.546,+0.335,+0.744]$	$0.99991$	$2.40$
2	$[+0.756,-0.339,+0.316,-0.462]$	$0.99944$	$4.80$
3	$[+0.489,-0.220,+0.844,+0.019]$	$0.99962$	$2.18$
4	$[+0.236,+0.663,+0.023,+0.711]$	$0.99976$	$4.09$
5	$[-0.132,+0.509,+0.491,+0.694]$	$0.99989$	$2.19$
6	$[-0.211,+0.762,-0.367,+0.490]$	$0.99929$	$7.19$

Table 5: Veronese–Whitney extrinsic sample means of the

q=14

non-frame landmarks (

n=8

SfM reconstructions). Each sample mean is a unit axis

g_{s}(M)=[x,y,z,w]

\mathbb{RP}^{3}

;

\lambda_{1}

is the corresponding leading eigenvalue of

J_{s}

(closer to

1

indicates tighter agreement); the last column is the angular distance to the blueprint coordinate

\gamma_{s}

Figure 4: Left: angular distance of each VW sample mean from the blueprint (average angular distance $2.52^{\circ}$ ). Right: angular spread of the eight reconstructions about their VW sample mean, for the multi-view SfM pipeline (red) and the $2010$ stereo pipeline (gray), under the common convention of Table 7 ; the SfM reconstructions are about $26$ times more concentrated.

Reconstruction	middle/top	bottom/top
$M_{1}$	$1.486$	$2.558$
$M_{2}$	$1.474$	$2.529$
$M_{3}$	$1.551$	$2.601$
$M_{4}$	$1.560$	$2.593$
$M_{5}$	$1.554$	$2.642$
$M_{6}$	$1.559$	$2.636$

Table 6: Scale-free cube-edge ratios of the eight reconstructions against the blueprint proportions

4\!:\!6\!:\!10

. The reconstructed proportions match the design, so the VW-sample-mean-versus-blueprint comparison is well posed.

Quantity	Stereo	SfM (this work)	Ratio
Photographs per reconstruction	$2$	$64$ – $114$	—
Average angular spread (deg)	$18.5$	$0.71$	$\sim\!26\times$
Average PS total-variance index	$0.291$	$4.2\times 10^{-4}$	$\sim\!700\times$
Average OPS total-variance index (anchored signs)	$0.188$	$2.1\times 10^{-4}$	$\sim\!900\times$
Blueprint decision	fail to reject	fail to reject	same

Table 7: Fair, scale-free comparison of the two acquisition pipelines on the same object, computed under one construction applied to both samples: the frame ordering

(17,18,19,12,8)

under which the published values of Patrangenaru et al. [12] are reproduced (Section 4.2 ) and blueprint-anchored signs. “Stereo” is the pipeline of Patrangenaru et al. [12] ; “SfM” is the multi-view pipeline used here. The two sources label the landmarks differently, so the construction is common at the level of indices, not of physical corners; the aggregate quantities shown are unaffected. Total-variance indices are dimensionless; the angular spread is in degrees.

Figure 5: Per-landmark reconstruction dispersion (average angular spread about the sample mean direction, log scale) for the stereo pipeline of Patrangenaru et al. [12] and the multi-view SfM pipeline used here, computed under the common convention of Table 7 . Landmark indices refer to each source’s own labeling, which differ between the two samples, so the per-landmark pairing is not a physical correspondence. The SfM reconstructions are roughly $26$ times more concentrated.

Figure 6: Per-landmark PS and OPS total-variance indices, $n=8$ . The OPS index equals one-half of the PS index at every landmark; Section 5.4 shows this is an algebraic consequence of the high data concentration, not an empirical property of the object.

Figure 7: OPS total variance with $95\%$ confidence intervals, $n=8$ . Delta-method intervals (circles) and percentile bootstrap intervals (squares) agree closely. Landmarks 2 and 6 carry the largest dispersion; landmark 13 the smallest.

Property	PS (Veronese–Whitney)	OPS (oriented sphere)
Shape space	$(\mathbb{RP}^{m})^{q}$	$(\mathbb{S}^{m})^{q}$
Identifies $z$ with $-z$	Yes	No
Embedding	$z\mapsto zz^{\top}$	$z\hookrightarrow\mathbb{R}^{m+1}$
Consistent sign needed	No	Yes (from SfM)
Mirrored configuration	Accepted (same shape)	Flagged: inadmissible $k$ -ad
Covariance inversion in test	Required	Not required

Table 8: Conceptual comparison of PS and OPS. The two total-variance indices coincide up to a factor of two under high concentration (Proposition 2 ); the operational differences lie in orientation and applicability. The flagging of a mirrored configuration operates through the oriented-frame admissibility condition, certified in practice by

\operatorname{sign}\det U_{3}

under the fixed affine-lift, similarity-gauge convention used throughout; it is not carried by the

(\mathbb{S}^{m})^{q}

coordinates, which are reflection-invariant.

$n$	$\overline{tS_{n}^{\mathrm{OPS}}}$	$\overline{\mathrm{SE}}_{\mathrm{OPS}}$	blueprint reject (Bonf.)
4	$2.77\times 10^{-4}$	$1.08\times 10^{-4}$	$0/14$
5	$2.51\times 10^{-4}$	$0.95\times 10^{-4}$	$0/14$
6	$2.26\times 10^{-4}$	$0.84\times 10^{-4}$	$0/14$
7	$2.15\times 10^{-4}$	$0.80\times 10^{-4}$	$0/14$
8	$2.11\times 10^{-4}$	$0.80\times 10^{-4}$	$0/14$

Table 9: Sample-size sensitivity (cumulative, first

n

reconstructions).

Figure 9: Average angular deviation from the blueprint against the number of photographs used in each reconstruction $M_{1},\dots,M_{8}$ . The dashed line is the least-squares fit (slope $-0.017$ deg/photo; $r=-0.68$ , $p=0.066$ ).

Frame ordering	blueprint reject (Bonf.)	smallest $p$	$\overline{tS_{n}^{\mathrm{OPS}}}$
$(8,12,17,18,19)$ (current)	$0/14$	$0.033$	$2.11\times 10^{-4}$
$(17,18,19,12,8)$ (2010 study)	$0/14$	$0.033$	$2.08\times 10^{-4}$
$(19,18,17,12,8)$	$0/14$	$0.033$	$2.08\times 10^{-4}$
$(12,17,18,19,8)$	$0/14$	$0.033$	$2.08\times 10^{-4}$

Table 10: Sensitivity of the blueprint test to the frame ordering, for the fixed frame set

\{8,12,17,18,19\}

. The ordering

(17,18,19,12,8)

is the one under which the published values of Patrangenaru et al. [12] are reproduced (Section 4.2 ). The decision is unchanged across these orderings; the numeric representation is not.

LM	$M_{1}$			$M_{2}$			$M_{3}$			$M_{4}$
	$X$	$Y$	$Z$	$X$	$Y$	$Z$	$X$	$Y$	$Z$	$X$	$Y$	$Z$
1	0.095	0.843	-2.677	1.139	0.410	-6.804	0.879	2.436	-8.575	-1.244	4.063	-14.478
2	-0.212	0.577	-2.590	0.260	-0.396	-6.901	0.166	2.458	-8.581	0.525	4.036	-13.583
3	0.056	0.277	-2.530	1.052	-1.260	-6.942	0.158	2.386	-7.860	1.433	4.091	-15.414
4	0.359	0.539	-2.612	1.931	-0.466	-6.839	0.878	2.364	-7.853	-0.368	4.125	-16.295
5	-0.235	0.473	-2.992	0.316	-0.306	-8.067	0.146	1.738	-8.653	0.531	2.038	-13.641

Table 11: Reconstructed Euclidean marker coordinates (Agisoft model units) for models

M_{1}

–

M_{4}

. Landmarks

1

–

19

; the frame is

F=\{8,12,17,18,19\}

LM	$M_{5}$			$M_{6}$			$M_{7}$			$M_{8}$
	$X$	$Y$	$Z$	$X$	$Y$	$Z$	$X$	$Y$	$Z$	$X$	$Y$	$Z$
1	-0.567	0.978	-7.074	-0.467	1.907	-7.641	-1.726	1.108	-11.060	0.413	1.879	-4.576
2	-0.586	0.004	-7.075	0.607	2.225	-8.113	-0.398	1.094	-10.998	1.264	1.900	-4.501
3	0.405	-0.016	-7.136	0.043	2.793	-9.056	-0.336	1.697	-12.200	1.335	2.087	-5.356
4	0.428	0.972	-7.149	-1.059	2.469	-8.605	-1.681	1.699	-12.278	0.468	2.056	-5.437
5	-0.660	-0.006	-8.072	0.648	1.169	-8.751	-0.373	-0.123	-11.588	1.300	1.063	-4.669

Table 12: Reconstructed Euclidean marker coordinates (Agisoft model units) for models

M_{5}

–

M_{8}

LM	$T_{s}^{\text{here}}$	$T_{s}^{2010}$	LM	$T_{s}^{\text{here}}$	$T_{s}^{2010}$
1	4.03	3.03	9	5.89	1.57
2	3.59	2.65	10	5.32	5.05
3	3.35	0.15	11	1.85	13.74
4	5.27	3.74	13	2.20	2.04
5	2.60	2.62	14	3.75	3.95
6	2.30	1.79	15	3.94	3.70

Table 13: Reproduction of the blueprint test of Patrangenaru et al. [12] .

T_{s}^{\text{here}}

is computed by the present implementation;

T_{s}^{2010}

is the published value. Both lead to “fail to reject

H_{0}^{\text{(bp)}}

” for all

14

landmarks.

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Structure-from-Motion (SfM) is a fundamental tool for sparse 3D reconstruction with broad impact in robotics and vision, supporting mapping, localization, and large-scale scene modeling. However, conventional pipelines often fail under hard visual ambiguity caused by repeated or symmetric structures, and incur heavy computational cost due to redundant cameras and constraints. We present HiSfM, a hierarchical coarse-to-fine SfM framework that improves robustness and efficiency through scaffold construction. HiSfM first forms strong local communities using geometrical induced heuristics, then connects communities with a compact yet strong skeleton by packing edge-disjoint spanning trees (EDST) while verifying skeletal edges with a two-view disambiguator. We reconstruct a stable scaffold on this verified skeleton, serving as an anchor to capture the essence of the scene, and subsequently absorb remaining images via efficient registration and triangulation for further refinements. Experiments on ambiguity-focused benchmarks and general datasets show that HiSfM prevents ambiguity-induced failures while substantially reducing runtime compared to previous methods, and improves completeness over aggressive sparsification methods. Code is available at https://github.com/3dv-casia/HiSfM.

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