Training-free Spatially Grounded Geometric Shape Encoding (Technical Report)

Yuhang He
Microsoft Research

Usage from PyPI

1. Install the package from PyPI:

   pip install xshapeenc

2. Shape Geometry Encoding: Encodes a 2-D shape mask into a fixed-length Zernike coefficient vector that captures the shape's geometry.

   import numpy as np
   import xshapeenc
   
   # Build a simple circular mask in polar coordinates (res × res)
   res = 300
   rho   = np.linspace(0, 1, res)
   theta = np.linspace(0, 2 * np.pi, res)
   r, t  = np.meshgrid(rho, theta)
   circle_mask = (r <= 1.0).astype(np.float64)
   
   # Encode – returns a list of `encode_len` floats
   geo_vec = xshapeenc.encode_geometry(circle_mask, n_max=5, res=res, lam=0.6, encode_len=512)
   print(f"Geometry encoding length: {len(geo_vec)}")  # 512
   
   # Decode back to a mask and measure reconstruction error
   encoder = xshapeenc.ShapeGeometryEncoder(n_max=5, res=res, lam=0.6, encode_len=512)
   recon   = encoder.decode(encoder.encode(circle_mask))
   mse     = float(np.mean((circle_mask - recon) ** 2))
   print(f"Reconstruction MSE: {mse:.6f}")

   If your mask is a regular Euclidean image (e.g. loaded from a PNG), pass mask_in_euclidean=True and the library handles the polar re-sampling automatically:

   from PIL import Image
   import numpy as np
   import xshapeenc
   
   img  = np.array(Image.open("my_shape.png").convert("L")) / 255.0  # H × W float
   vec  = xshapeenc.encode_geometry(img, encode_len=512, mask_in_euclidean=True)

3. Pose encoding: Encodes a 1-D spatial pose vector (e.g. x/y coordinates, scale, orientation) into a set of Zernike radial coefficients.

   import xshapeenc
   
   pose_vec   = [0.2, 0.5, 0.7, 0.9, 0.4]   # arbitrary length
   pose_coeffs = xshapeenc.encode_pose(pose_vec, encode_len=128)
   print(f"Pose encoding length: {len(pose_coeffs)}")  # 128

4. Joint geometry-pose encoding: Fuses shape geometry and spatial pose into a single vector. The beta parameter controls the emphasis:

   • beta=0 → pure pose
   • beta=1 → equal weight (default)
   • beta=2 → pure geometry

   import numpy as np
   import xshapeenc
   
   res = 300
   rho   = np.linspace(0, 1, res)
   theta = np.linspace(0, 2 * np.pi, res)
   r, _  = np.meshgrid(rho, theta)
   circle_mask = (r <= 1.0).astype(np.float64)
   
   pose_vec  = [0.2, 0.5, 0.7, 0.9, 0.4]
   joint_vec = xshapeenc.encode_geopose(circle_mask, pose_vec, encode_len=512, beta=1.0)
   print(f"Joint encoding length: {len(joint_vec)}")  # 512

Quick Test

1. To build up the environment pip install requirements.txt.

2. Run quick_test.py to experience shape geometry encoding, shape pose encoding and their joint encoding.

   #Run all tests::
   python quick_test.py
   #Run a specific subset with debug output::
   python quick_test.py --tests pose geometry --verbose

3. XShapeCorpus dataset generation, go to README.md.

XShapeEnc Summary

Method	arbitrary shape?	high frequency?	training-free?	task-agnostic?	spatial-context?
AngularSweep	✗	✗	✓	✗	✓
Poly2Vec	✗	✗	✗	✗	✓
Space2Vec	✗	✗	✗	✗	✓
DeepSDF	✓	✓	✗	✓	✗
2DPE	✗	✗	✗	✓	✓
ShapeEmbed	✓	✓	✗	✓	✗
ShapeDist	✓	✗	✓	✓	✗
XShapeEnc (Ours)	✓	✓	✓	✓	✓

As shown in the table above, XShapeEnc encodes an arbitrary 2D geometric shape associated with a spatial position (e.g., x-, y- coordinate, scale) within a unified framework. It is totally training-free, task-agnostic and frequency-rich, while enjoying the advantage of controllable emphasis between shape geometry and shape pose encoding. In summary, it provides flexible encoding:

Options	Can XShapeEnc do?
Just Shape Rotation Invariant Feature	Yes
Just Shape Rotation Variant Feature	Yes
Just Shape Geometry Encoding	Yes
Just Shape Pose Encoding	Yes
Shape Geometry and Pose Joint Encoding	Yes

XShapeEnc Pipeline

XShapeEnc encoding pipeline is shown in the Figure above. It can independently encode shape geometry, shape pose or geometry-pose jointly. The whole encoding framework is based on Zernike basis. The shape pose requires to be first converted into harmonic pose field so as to be encodable by Zernike basis.

Experiment

1. Inter-Shape Polygon-Polygon Topological Relation Classification

   

   We run experiment on spatially grounded polygon pair topological relation classification. As shown in the figure above, we classify 5 main relations: Disjoint, Within, Overlap, Touch and Equal. The polygon shapes are from Singapore and New York building bird-eye-view (BEV) map. The result is shown in the table below,

   Method	Singapore	New York
   PointSet	0.670	0.564
   ShapeContexts	0.581	0.525
   AngularSweep	0.606	0.546
   Space2Vec	0.706	0.632
   ResNet18	0.674	0.753
   ViT	0.669	0.752
   CLIP	0.700	0.779
   Poly2Vec	0.702	0.684
   XShapeEnc (Ours)	0.760	0.768

   From this table, we can see that XShapeEnc maintains highly competitive performance.

2. Invertibility Visualization

   

   We test XShapeEnc's invertibility property by running encoding-to-shape inversion from various commonly used encodng length: 64, 128, 256, 512, 1024, 2048, 4096. The result is shown in the Figure above, from which we can observe that larger encoding length leads to higher-fielity shape recovery.

3. Shape Geometry Clustering Visualization

   

   We test XShapeEnc's shape geometry encoding inter- and intra- geometry discriminability by running t-SNE clustering on augmented 4 complex shapes. As shown in the figure above, XShapeEnc maintains the discriminability while most of the comparing baselines loosing such discriminability.

Cite XShapeEnc

@inproceedings{yuhheXShapeEnc2026,
title={Training-free Spatially Grounded Geometric Shape Encoding (Technical Report)},
author={He, Yuhang},
booktitle={arXiv:2604.07522},
year={2026}}

Contact

Email: yuhanghe[at]microsoft.com