Topology-Preserving Smoothing - Python Implementation

A Python implementation of the Alternate Sequential Filter controlled by Topology (ASFT) algorithm for smoothing binary images while preserving their topology (number of connected components and holes).

Example: Portrait

Input	Output

The algorithm smooths jagged boundaries while preserving topology - all connected components and holes from the original image are maintained.

More Examples

Example 1: Pixelized stains #1

Input	Output

Example 2: Pixelized stains #2

Input	Output

Example 3: QR-code (Connectivity Comparison)

Input	Output (C8)	Output (C4)

Example 4: Flying heron

Input	Output

Example 5: Trees

Input	Output

Example 6: Einstein with a pipe

Original	Smoothed (ASFT-MED)	Smoothed (ASFT)

Smoothed with: -s 1 -r 3 -c 4 --medial for ASFT-MED and with the same parameters for the pure ASFT.

Reference

Based on the paper:

M. Couprie and G. Bertrand: "Topology preserving alternating sequential filter for smoothing 2D and 3D objects", Journal of Electronic Imaging, Vol. 13, No. 4, pp. 720-730, 2004.

See also the dedicated web-page:

Topological Smoothing by M. Couprie, G. Bertrand

Installation

From PyPI

# Recommended installation (includes numba for 10-100x speedup)
pip install toposmooth[fast]

# If you have issues installing numba (e.g., unsupported platform):
pip install toposmooth

From Source

# Clone the repository
git clone https://github.com/vyastrebov/toposmooth.git
cd toposmooth

# Recommended: install with numba
pip install -e ".[fast]"

# Without numba (if installation fails):
pip install -e .

Required packages

• numpy
• scipy
• Pillow
• numba (included with [fast], provides 10-100x speedup)

Supported Image Formats

The algorithm supports any bitmap format readable by Pillow:

• PNG (.png) - recommended for lossless output
• JPEG (.jpg, .jpeg) - input only (lossy compression not ideal for binary output)
• PGM (.pgm) - portable graymap, matches C implementation
• BMP (.bmp)
• TIFF (.tiff, .tif)
• And many others...

The output format is automatically determined by the file extension.

Usage

Command Line

After installation, you can use the toposmooth command or run as a Python module:

# Using the installed command
toposmooth input.png output.png

# Or using Python module syntax
python -m toposmooth input.png output.png

# Match C asftmed command: ./asftmed input.pgm 4 3 output.pgm
toposmooth input.pgm output.pgm -s 1 -r 3 -c 4

# With all options
toposmooth input.jpg output.png \
    -s 1           # Scale factor (default: 4, use 1 to match input dimensions)
    -r 5           # Maximum smoothing radius (default: 5)
    -c 4           # Connectivity for white value (1): 4 (neighbor) or 8 (next-neighbour)
    -t 128         # Binarization threshold (default: mean intensity, if the input is not binary)
    --medial       # Use medial axis constraints - strictly preserves the form (default, recommended)
    --no-medial    # Use plain ASFT without medial axis constraints (removes small features)
    --save-binary binary.png  # Save the binarized input image

# Show version
toposmooth --version

Python API

from toposmooth import topology_preserving_smooth, load_image, asftmed
from PIL import Image
import numpy as np

# Load and process an image
img = load_image('input.png')
smoothed, binary = topology_preserving_smooth(
    img,
    scale=1,           # Scale factor (1 = no scaling, matches C)
    smooth_radius=5,   # Maximum ASFT radius
    connex=4,          # Object connectivity (4 or 8)
    threshold=None,    # Binarization threshold (None = mean)
    use_medial=True    # Use medial axis constraints (recommended)
)

# Save result in any format
Image.fromarray(smoothed).save('output.png')   # PNG
Image.fromarray(smoothed).save('output.jpg')   # JPEG
Image.fromarray(smoothed).save('output.pgm')   # PGM

Parameters

• scale: Integer scaling factor. The image is upscaled before smoothing to allow sub-pixel smoothing with respect to original pixel size. Use scale=1 to match the original image dimensions.

• smooth_radius (rmax): Maximum radius of disk structuring elements. The algorithm applies opening/closing operations with disks of radius 1, 2, ..., rmax. Larger values produce smoother results but take longer (linear complexity with rmax), but the default value rmax=5 produces already very good results.

• connex: Object connectivity (4 or 8). For 8-connectivity, diagonal neighbors (white pixels, value=1) are considered connected.

• threshold: Binarization threshold. Pixels above this value become foreground (255), others become background (0). If None, uses the mean intensity.

• use_medial / --medial: When enabled (default), uses medial axis constraints to preserve thin features, recommended.

Algorithms

ASFT-MED (default, --medial)

Uses medial axes of both foreground and background as constraints. This prevents:

• Thin features from being eroded away
• Small gaps from being filled in

Plain ASFT (--no-medial)

Standard ASFT without constraints. May lose very thin features.

How It Works

1. Binarization: The input image is converted to binary using the threshold.

2. Scaling (optional): The binary image is upscaled by the scale factor.

3. Medial Axis Computation (ASFT-MED only):

   • Compute medial axis of foreground (skeleton)
   • Compute medial axis of background (inverse skeleton)

4. ASFT: For each radius r from 1 to rmax:

   • Apply homotopic pseudo-closing (fills small holes while preserving topology)
   • Apply homotopic pseudo-opening (removes small protrusions while preserving topology)
   • With medial axis constraints, these operations are bounded to preserve thin structures

5. The result is a smoothed binary image with the same topology as the input.

Performance

• With numba (pip install toposmooth[fast]): ~1-2 seconds for a 520×372 image with scale=1, radius=3
• Without numba: Much slower (minutes instead of seconds) - only use if numba won't install
• The algorithm scales roughly as O(scale² × rmax × width × height)
• First run with numba may take a few seconds for JIT compilation

Validation

The Python implementation produces pixel-perfect results matching the C implementation for Einstein's binary portrait.

# C version
./asftmed test/einstein.pgm 4 3 c_output.pgm

# Python version
toposmooth test/einstein.pgm py_output.pgm -s 1 -r 3 -c 4

Test Shapes

Run python test_shapes.py to generate comparison panels for various test shapes:

1. Rectangle - Grid-aligned rectangle
2. Square - Grid-aligned square
3. L-Shape - Grid-aligned L shape
4. Squares Edge Touch - Two squares sharing a full edge (1 component for both connectivities)
5. Squares Single Point Touch - Two squares touching at exactly one 4-neighbor pixel (1 component for both)
6. Squares Diagonal Touch - Two squares touching only at diagonal (2 components for 4-conn, 1 for 8-conn)
7. Rotated Square 30° - Square rotated by 30 degrees
8. Rotated Square 45° - Square rotated by 45 degrees

Each test shows the original shape and results for:

• Connectivity 4 and 8
• ASFT and ASFT-MED algorithms
• Scale factors 1 and 4

Results are saved to test_results/ folder, including all_tests_overview.png with all shapes combined.

Sample Test Result

Information

• Authors: Claude Opus 4.5 in Cursor environment (mainly), Vladislav A. Yastrebov (marginally)
• Validator: Vladislav A. Yastrebov (CNRS, Mines Paris)
• License: GPL-2.0
• Date: Nov-Dec 2025