[Paper Review] Artifacts of Quantization in Distance Transforms

Bryce A. Besler, Tannis D. Kemp+2 more|arXiv (Cornell University)|Nov 17, 2020

Digital Image Processing Techniques25 references4 citations

TL;DR

This paper identifies a quantization artifact in distance transforms of sampled binary images, where numerical gradients exhibit banding due to discrete sampling, independent of sampling period. The authors propose a dithering and reinitialization algorithm that removes these artifacts while preserving the signed distance property and restoring smooth gradients, enabling second-order accuracy for curve evolution applications.

ABSTRACT

Distance transforms are a central tool in shape analysis, morphometry, and curve evolution problems. This work describes and investigates an artifact present in distance maps computed from sampled signals. Namely, sampling reflects through the distance transform causing quantization in the resulting distance map. Gradients of the quantized distance map show banding, affecting the quality of subsequence processing. Furthermore, this error is independent of the sampling period of the signal and cannot be removed by modifying the number of samples across an objects width. Where needed, distance maps should be computed from representations other than binary images. In the case where exact representations are needed, a dithering and noise removal algorithm is proposed.

Motivation & Objective

• To investigate the origin and impact of quantization artifacts in distance transforms computed from sampled binary signals.
• To demonstrate that these artifacts arise from sampling-induced quantization and are independent of sampling period.
• To propose a practical method for removing quantization artifacts in distance maps without altering the underlying binary representation.
• To enable second-order accurate initialization for curve evolution methods that require high gradient fidelity.
• To show that standard reinitialization fails to achieve second-order accuracy, motivating alternative embedding strategies.

Proposed method

• The authors model the distance transform of a sampled signal as a quantized approximation of the true continuous distance function.
• They use finite difference approximations (forward, backward, central) to compute gradients and detect banding artifacts in the magnitude of the gradient field.
• A dithering procedure is applied to the binary input image with a controlled amplitude (α = 20) to randomize quantization effects.
• The dithered image is then used to compute a new signed distance transform, followed by iterative reinitialization using level set methods to restore the Eikonal equation constraint (|∇φ| ≈ 1).
• The method preserves the zero-level set (original shape) while removing gradient banding and improving curvature estimation.
• The algorithm is validated using a trabecular bone model, comparing mean curvature histograms and surface visualizations before and after correction.

Experimental results

Research questions

• RQ1 Why do numerical gradients in distance transforms exhibit banding artifacts despite high sampling density?
• RQ2 Is the quantization artifact in distance transforms dependent on the sampling period of the input signal?
• RQ3 Can the quantization artifact be removed without altering the shape representation of the original binary image?
• RQ4 Does the proposed dithering and reinitialization method restore second-order accuracy in gradient computation?
• RQ5 Can the method improve curvature estimation in morphological and curve evolution applications?

Key findings

• The distance transform of a sampled binary signal produces a quantized approximation of the true distance function, introducing systematic errors in gradients.
• The resulting gradient banding is independent of sampling period and cannot be eliminated by increasing resolution.
• The proposed dithering and reinitialization algorithm successfully removes gradient artifacts while preserving the original shape and maintaining |∇φ| ≈ 1.
• After correction, mean curvature histograms become continuous and physically plausible, indicating removal of quantization effects.
• The algorithm converges within 100 iterations, and no further improvement is observed, indicating convergence to a stable, artifact-free representation.
• The method enables second-order accurate initialization for curve evolution, which is critical for methods like mean curvature flow.