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A Study of PHOC Spatial Region Configurations for Math Formula Retrieval

Matt Langsenkamp, Bryan Amador, Richard Zanibbi

TL;DR

The paper tackles visual retrieval of mathematical formulas using symbol-label-and-location representations (PHOC) and investigates whether higher PHOC levels add unique information by introducing concentric rectangles as a new region type and by skipping PHOC levels. It extends XY-PHOC with concentric rectangles and analyzes level-skipping (odd levels and last-level only) on the ARQMath-3 benchmark, comparing against a Bag-of-Words baseline. Findings show that rectangular PHOC variants outperform earlier PHOC configurations and that some higher levels are redundant for retrieval, while PHOC-based methods remain competitive with state-of-the-art despite their simplicity. The work highlights practical trade-offs between precision and ranking depth, and suggests future integration with neural embeddings and cross-domain graphic retrieval.

Abstract

A Pyramidal Histogram Of Characters (PHOC) represents the spatial location of symbols as binary vectors. The vectors are composed of levels that split a formula into equal-sized regions of one or more types (e.g., rectangles or ellipses). For each region type, this produces a pyramid of overlapping regions, where the first level contains the entire formula, and the final level the finest-grained regions. In this work, we introduce concentric rectangles for regions, and analyze whether subsequent PHOC levels encode redundant information by omitting levels from PHOC configurations. As a baseline, we include a bag of words PHOC containing only the first whole-formula level. Finally, using the ARQMath-3 formula retrieval benchmark, we demonstrate that some levels encoded in the original PHOC configurations are redundant, that PHOC models with rectangular regions outperform earlier PHOC models, and that despite their simplicity, PHOC models are surprisingly competitive with the state-of-the-art. PHOC is not math-specific, and might be used for chemical diagrams, charts, or other graphics.

A Study of PHOC Spatial Region Configurations for Math Formula Retrieval

TL;DR

The paper tackles visual retrieval of mathematical formulas using symbol-label-and-location representations (PHOC) and investigates whether higher PHOC levels add unique information by introducing concentric rectangles as a new region type and by skipping PHOC levels. It extends XY-PHOC with concentric rectangles and analyzes level-skipping (odd levels and last-level only) on the ARQMath-3 benchmark, comparing against a Bag-of-Words baseline. Findings show that rectangular PHOC variants outperform earlier PHOC configurations and that some higher levels are redundant for retrieval, while PHOC-based methods remain competitive with state-of-the-art despite their simplicity. The work highlights practical trade-offs between precision and ranking depth, and suggests future integration with neural embeddings and cross-domain graphic retrieval.

Abstract

A Pyramidal Histogram Of Characters (PHOC) represents the spatial location of symbols as binary vectors. The vectors are composed of levels that split a formula into equal-sized regions of one or more types (e.g., rectangles or ellipses). For each region type, this produces a pyramid of overlapping regions, where the first level contains the entire formula, and the final level the finest-grained regions. In this work, we introduce concentric rectangles for regions, and analyze whether subsequent PHOC levels encode redundant information by omitting levels from PHOC configurations. As a baseline, we include a bag of words PHOC containing only the first whole-formula level. Finally, using the ARQMath-3 formula retrieval benchmark, we demonstrate that some levels encoded in the original PHOC configurations are redundant, that PHOC models with rectangular regions outperform earlier PHOC models, and that despite their simplicity, PHOC models are surprisingly competitive with the state-of-the-art. PHOC is not math-specific, and might be used for chemical diagrams, charts, or other graphics.
Paper Structure (5 sections, 1 equation, 3 figures, 3 tables)

This paper contains 5 sections, 1 equation, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Rectangular Regions and Level Skipping
  4. Experimental Setup and Results
  5. Conclusion

Figures (3)

  • Figure 1: Pyramidal Histogram of Character (PHOC) Regions for the Word retrieval. Histograms or binary vectors are computed for regions obtained by evenly splitting the character sequence iteratively. The binary PHOC for 'r' in regions above (left-to-right, levels separated by spaces) is $[ 1~10~110 ]$
  • Figure 2: PHOC Spatial Regions for the Golden Ratio. Shown is Level 3 for concentric ellipses, X and Y-Axis partitions, and concentric rectangles.
  • Figure 3: Concentric Rectangular 3-Level PHOC (R3) PHOC with levels shown left-to-right. Check marks indicate levels present in R3-full, R3-odd, and R3-last PHOC models.