# Proof of Correctness and Time Complexity Analysis of a Maximum Distance Transform Algorithm

## TL;DR

This paper presents a new average time O(N) algorithm for maximizing the distance transform, analyzing its correctness, runtime, and duality with the minimum transform in computer vision applications.

## Contribution

It introduces an average time O(N) algorithm for maximum distance transform, with a proof of correctness and analysis of duality with the minimum transform.

## Key findings

- The algorithm achieves average linear time complexity.
- Proof of correctness for the maximum distance transform algorithm.
- Discussion of duality between minimum and maximum transforms.

## Abstract

The distance transform algorithm is popular in computer vision and machine learning domains. It is used to minimize quadratic functions over a grid of points. Felzenszwalb and Huttenlocher (2004) describe an O(N) algorithm for computing the minimum distance transform for quadratic functions. Their algorithm works by computing the lower envelope of a set of parabolas defined on the domain of the function. In this work, we describe an average time O(N) algorithm for maximizing this function by computing the upper envelope of a set of parabolas. We study the duality of the minimum and maximum distance transforms, give a correctness proof of the algorithm and its runtime, and discuss potential applications.