---
kind: concept
slug: distance-transform
title: Distance transform
summary: >-
  A distance transform gives every pixel a value based on its distance to the nearest annotated point. Covers the Euclidean nearest-point distance, the inverse distance transform, and the focal inverse distance transform (FIDT), which makes each head a sharp peak for crowd localization from point labels. Only one paper card so far.
tags: [localization, point-annotations, loss-function]
papers: ["2102.07925"]
related: [density-map-regression, structural-similarity, signed-distance-field]
status: draft
updated_at: 2026-11-09
---

## Overview

A distance transform turns a set of marked points into a map. Each pixel gets
a value computed from its distance to the nearest marked point. The base
quantity is the Euclidean distance to the nearest head [arxiv:2102.07925 §3.1]
[eq. 1], which the paper calls the Euclidean distance transform [arxiv:2102.07925 §2.3].
A network can be trained to predict such a map, and the marked points can be
recovered from the peaks of the prediction.

So far the only card on this concept is for crowd localization from point
labels, so every variant below comes from one paper. The raw distance is
transformed further, first by an inverse and then by a focal exponent, so
that a head becomes a sharp local maximum and the background stays near zero
[arxiv:2102.07925 §3.1] [eq. 2] [eq. 3].

## Variants

### Euclidean nearest-point distance

The map P(x, y) is the distance from each pixel to the nearest head in the
set of annotated points [arxiv:2102.07925 §3.1] [eq. 1]. Distances are in
pixels, so the map depends on image resolution (see Contradictions).

### Inverse distance transform (IDT)

The inverse 1 / (P + C), with a constant C that avoids division by zero
[arxiv:2102.07925 §3.1] [eq. 2]. The paper notes that this is the K = 1 case
of an iKNN map from Olmschenk et al., which that work did not use for
localization [arxiv:2102.07925 §3.1]. The paper reports that its response decays
too fast near a head and stays too high in background [arxiv:2102.07925 §3.1]
[fig. 3]. Its localization on ShanghaiTech Part A at sigma = 8 is an F-measure
of 75.1%, against 77.6% for the focal version [arxiv:2102.07925 tab. 9].
Card: [Focal Inverse Distance Transform Maps for Crowd Localization](/papers/2102.07925) (IDT is its alpha = 0 case).

### Focal inverse distance transform (FIDT)

The exponent of P grows with P, so the decay is slower near a head and the
background goes to nearly zero [arxiv:2102.07925 §3.1] [eq. 3]. The paper
fixes alpha = 0.02, beta = 0.75 and C = 1 [arxiv:2102.07925 §4] [tab. 9].
Nearby settings reach F-measures of 76.3% (alpha 0.01, beta 0.65) and 76.9%
(alpha 0.03, beta 0.85) on the same split [arxiv:2102.07925 tab. 9]. Heads
are read off the map as local maxima, using a 3x3 max-pool and a threshold
at 100/255 of the peak [arxiv:2102.07925 §3.2.2] [alg. 1]. Card: [Focal Inverse Distance Transform Maps for Crowd Localization](/papers/2102.07925).

## When to use which

| Situation | Use | Why |
|---|---|---|
| Only point labels per object are available | FIDT target with local-maxima decoding | The method needs point labels only [arxiv:2102.07925 §1] [§4.2]. |
| You need object positions, not only a count | FIDT peaks | Each head is a local maximum of the map [arxiv:2102.07925 §3.1]. |
| Dense scenes where density blobs overlap | FIDT instead of Gaussian density maps | The paper argues that overlapping blobs move the peaks off the heads [arxiv:2102.07925 §1]. |
| Only a count is needed and the density regressor already works well | Density maps may be enough | On JHU-Crowd++ the paper's counting MAE (66.6) is above TopoCount's (60.9) [arxiv:2102.07925 tab. 7]. |
| Same regressor, count error matters | FIDT targets, if the paper's comparison carries over | With BL + FPN on NWPU validation, the count error was 65.7 for FIDT and 85.4 for density maps [arxiv:2102.07925 tab. 14]. The paper uses L2 loss only for these runs [§5.4]. |
| Real-time inference is needed | Measure before choosing | At 768 x 1024 on an RTX 3090 the paper reports 7.1 FPS [arxiv:2102.07925 tab. 15], and it names a faster method, GL, with no card here. |

## Implementation sketch

The target is built from the head points, then decoded by local maxima.
Distances are in pixels, so resizing the image changes the target (see
Contradictions).

```pseudocode
# Build the FIDT target from head points  [arxiv:2102.07925 §3.1] [eq. 1] [eq. 3]
# Input:  image size (H, W); head points (x, y), in pixels
# Output: target map (H, W)

def fidt_target(H, W, points):
    # Distance to the nearest head, per pixel  [eq. 1]
    P = nearest_point_distance(H, W, points)
    # Focal inverse with C = 1, alpha = 0.02, beta = 0.75  [eq. 3] [§4]
    return 1 / (P ** (0.02 * P + 0.75) + 1)
```

```pseudocode
# Decode heads from a predicted map  [arxiv:2102.07925 §3.2.2] [alg. 1]
# Input:  predicted map (H, W)
# Output: head coordinates, one row per head

def decode_heads(pred):
    # Keep pixels that equal their 3x3 neighbourhood maximum  [alg. 1]
    local_max = pred * (max_pool3x3(pred) == pred)
    peak = max(local_max)
    # Adaptive threshold: 100/255 of the peak  [§3.2.2]
    keep = local_max >= (100 / 255) * peak
    return coordinates_where(keep)
```

The paper also reports an empty-image rule: if the peak is below 0.10, the
count is zero, and the 0.10 value was chosen from NWPU validation maxima
[arxiv:2102.07925 §3.2.2] [§5.4] [fig. 9]. A model with a different loss
scale needs its own value. The training loss is MSE plus a structural
similarity term computed on a 30 x 30 region for each person, with weight 1
on each [arxiv:2102.07925 §3.3] [eq. 8]. The card records that the released
training script does not include the structural term.

## Contradictions

None known across papers, because only one card exists. Open points within
the one paper:

- **Label types differ in the comparison.** The paper's NWPU localization
  numbers set its point-labelled method against a box-trained baseline,
  TopoCount [arxiv:2102.07925 §5.1].
- **Resolution.** Eq. 1 and eq. 3 use pixel distances, and the paper resizes
  the three large datasets to a longer side below 2048 [arxiv:2102.07925 §4].
  The paper does not discuss how alpha and beta interact with resizing.
- **Reproduction.** The loss ablation compares L2, L2 with SSIM, and L2 with
  the per-person structural term [arxiv:2102.07925 tab. 10]. The card records
  that the released training script does not include the structural term, so
  the ablation cannot be reproduced from the release.

## Papers

- [Focal Inverse Distance Transform Maps for Crowd Localization](/papers/2102.07925): defines the IDT and FIDT maps, the local-maxima decoder and an I-SSIM training term for point-supervised head localization.