Polygon Reconstruct

Version: 1.0.2
Author: VA
Year: 2026

Polygon Reconstruct is an image reconstruction algorithm that recreates any input image using overlapping semi-transparent triangles. The algorithm employs simulated annealing with random mutations to iteratively find the optimal position, size, and color for each polygon.

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Overview

Given a target image, the algorithm builds a reconstruction by placing semi-transparent triangles one at a time. Each triangle is optimized through a simulated annealing process that explores the space of possible vertex positions and colors, accepting improvements and occasionally accepting worse configurations to escape local minima.

The result is a stylized, low-poly approximation of the original image — similar to the low-poly art style, but generated algorithmically.

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Project Structure

PolyRecon/
├── polygonreconstruct.lpr    # Desktop CLI — fixed polygon count
├── polygonrecon_auto.lpr     # Desktop CLI — adaptive stopping (max failures)
├── PolygonWeb.lpr            # WebAssembly version — runs in the browser
├── polygonreconstruct.lpi    # Lazarus project file (desktop)
├── polyrecon_auto.lpi        # Lazarus project file (auto)
├── PolygonWeb.lpi            # Lazarus project file (web)
├── web/
│   ├── index.html            # Web interface
│   ├── PolygonWeb.js         # Compiled Pascal → JavaScript
│   ├── PolygonWeb.js.map     # Source map for debugging
│   ├── poly-gisa.head.png    # Header image
│   └── poly-gisa.logo.png    # Logo image
└── README.md                 # This file

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How It Works

1. Rasterization

Each triangle is rasterized into a binary mask using the edge function (signed area test). For every pixel inside the triangle's bounding box, the algorithm checks whether the pixel lies on the same side of all three edges — if so, the pixel is inside the triangle.

function EdgeFn(ax, ay, bx, by, px, py: double): double; inline;
begin
  Result := (px - bx) * (ay - by) - (ax - bx) * (py - by);
end;

2. Optimal Color (Analytic)

Rather than searching over color space, the algorithm computes the analytically optimal color for a candidate polygon by solving for the color that minimizes the squared error between the current canvas and the target image, weighted by the polygon's alpha blend factor.

3. Simulated Annealing

Each polygon is optimized using a simulated annealing loop:

• Temperature schedule: Temperature and mutation step size (Sigma) decay exponentially from initial to final values.
• Mutations: Four mutation strategies are applied with equal probability:
  • Single vertex jitter (50%): Perturb one vertex with Gaussian noise.
  • Global translation (25%): Shift all vertices by a shared offset.
  • Scaling (12.5%): Scale all vertices away from or toward the centroid.
  • Rotation (12.5%): Rotate all vertices around the centroid.
• Acceptance criterion: A candidate is accepted if it improves the score, or with probability exp(-ΔE / T) if it does not.

4. Adaptive Schedule

The annealing schedule adapts based on reconstruction progress. As the error decreases, fewer steps and smaller mutation magnitudes are used, allowing the algorithm to fine-tune details in later stages.

5. Canvas Composition

Accepted polygons are committed to the canvas using alpha blending:

new_pixel = (1 - α) * current_pixel + α * polygon_color

The canvas is initialized to the mean color of the target image, providing a neutral starting point.

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Building

Prerequisites

• Lazarus IDE with Free Pascal Compiler (FPC)
• BGRABitmap library (for desktop versions)

Desktop Versions

# Build the standard CLI version
lazbuild polygonreconstruct.lpi

# Build the auto-stopping CLI version
lazbuild polyrecon_auto.lpi

Web Version

The web version is compiled from Pascal to JavaScript using the Lazarus Pas2Js toolchain. The compiled output is PolygonWeb.js.

lazbuild PolygonWeb.lpi

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Usage

Desktop CLI — polygonreconstruct

Reconstruct an image with a fixed number of polygons:

polygonreconstruct <input.jpg> <output.png> [max_size] [n_polygons] [n_steps] [save_every]

Parameter	Default	Description
input.jpg	—	Path to the input image
output.png	—	Path to save the reconstructed image
max_size	256	Maximum dimension (px) — image is scaled down
n_polygons	300	Number of polygons to attempt
n_steps	2000	Simulated annealing steps per polygon
save_every	0	Save intermediate frames every N polygons (0 = off)

Example:

polygonreconstruct photo.jpg result.png 512 500 3000 50

Desktop CLI — polyrecon_auto

Reconstruct an image with adaptive stopping based on consecutive non-improvements:

polyrecon_auto <input.jpg> <output.png> [max_size] [max_failures] [n_steps] [save_every]

Parameter	Default	Description
input.jpg	—	Path to the input image
output.png	—	Path to save the reconstructed image
max_size	256	Maximum dimension (px)
max_failures	20	Stop after N consecutive non-improving iterations
n_steps	2000	Simulated annealing steps per polygon
save_every	0	Save intermediate frames every N committed polygons

Example:

polyrecon_auto photo.jpg output/result.png 512 30 3000 10

Web Version

Open web/index.html in a modern web browser. The interface provides:

1. Image upload — Select any image file to reconstruct.
2. Live preview — Side-by-side view of the original and reconstruction.
3. Annealing steps — Slider to adjust mutation attempts per polygon (250–10,000).
4. Consecutive non-improvement count — Slider to set the stopping threshold (1–50).
5. Stop button — Immediately halt the reconstruction.

The web version runs entirely client-side using WebAssembly-compiled Pascal code.

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Algorithm Parameters

Parameter	Default	Range / Notes
Alpha	0.5	Polygon opacity (0 = invisible, 1 = solid)
T_init	auto	Initial temperature (calibrated per image)
T_final	1e-6	Final temperature
Sigma_init	0.3	Initial mutation magnitude (normalized)
Sigma_final	0.005	Final mutation magnitude
N_steps	2000	Annealing iterations per polygon
Epsilon	-0.1	Minimum score delta to accept a polygon

Note: T_init is automatically calibrated at the start of each run by measuring the typical uphill energy change during random mutations.

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Key Data Structures

Type	Description
TFloatColor	RGB color with single precision (0.0–1.0)
TVertex	2D point with double precision
TPolygon	Triangle defined by 3 vertices and a color
TRasterizedPolygon	Polygon + binary mask + bounding box + score delta
TAnnealingSchedule	Temperature and sigma schedule for one polygon

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Dependencies

Dependency	Purpose	Required For
Free Pascal	Compiler	All
Lazarus IDE	IDE / project management	All
BGRABitmap	Bitmap loading, scaling, saving	Desktop versions
Web / JS units	Browser DOM and canvas access	Web version

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License

© 2026 Polygon Reconstruct by VA. All rights reserved.