---
kind: concept
slug: density-map-regression
title: Density map regression
summary: >-
  Density map regression predicts a value per pixel whose sum over a region is the object count, trained against Gaussian-smoothed point annotations. One card covers it so far: DecideNet blends a detector's head boxes with a regression map through a learned per-pixel attention. The trade-off: the detector needs box labels and runs in every training step.
tags: [counting, density-map, crowd-counting]
papers: ["1712.06679"]
related: [object-detection, attention-mechanism]
status: draft
updated_at: 2026-11-01
---

## Overview

A density map model outputs one value per pixel, and the count in a region is
the sum of those values over the region [arxiv:1712.06679 §3.1]. The training
targets are built the same way. A Gaussian is placed at each annotated head and
the results are summed, so the total of the map equals the head count
[arxiv:1712.06679 §3.1] [eq. 1]. The regression branch does not locate heads
[arxiv:1712.06679 §3.2]. Only one card covers this slug so far, so the
comparison below is an ablation inside one paper, not a survey of the field.

## Variants

### Regression map alone

A five-layer fully convolutional network produces the density map directly from
the image patch, with a ReLU after the last layer [arxiv:1712.06679 §3.2]
[eq. 2]. Its ablation MAE is 3.37 on Mall and 42.85 on ShanghaiTech PartB
[arxiv:1712.06679 tab. 4]. This is the simplest variant on the page, and it
needs only point annotations.

### Attention-weighted fusion with a detector (DecideNet)

A Faster R-CNN head detector also contributes a density map. Each detected box
centre gets a fixed Gaussian, and the sum forms the detection map
[arxiv:1712.06679 §3.3] [eq. 3]. A small attention network reads both maps and
the image, and outputs a per-pixel weight in (0, 1). The final density is the
weighted sum of the two maps [arxiv:1712.06679 §3.4] [eq. 4]. Card:
[DecideNet](/papers/1712.06679).

An extra loss term pushes the attention toward the detector's score map, which
the paper calls quality-aware [arxiv:1712.06679 §4] [eq. 8]. This term lowers
the ablation MAE from 1.83 to 1.52 on Mall and from 24.93 to 21.53 on
ShanghaiTech PartB [arxiv:1712.06679 tab. 4].

## When to use which

The page has one paper, so these rows follow the constraints the paper states,
not a wider comparison.

| Situation | Use | Why |
|---|---|---|
| Point annotations only, no box labels | Regression map alone | DecideNet's detector needs boxes. The paper hand-labels boxes on part of the training heads and uses one average box size for all heads [arxiv:1712.06679 §4]. |
| Training compute is tight | Regression map alone | The detection map runs the detector on every training patch in every iteration, and the paper reports no training or inference cost [arxiv:1712.06679 §5.1]. |
| Crowd density varies a lot within one image and a detector is available | Attention-weighted fusion | In the paper's ablation it reduces the late-fusion average from 3.93 to 1.83 MAE on Mall [arxiv:1712.06679 tab. 4]. The paper reports that the attention block roughly halves the late-fusion MAE [arxiv:1712.06679 §5.5]. |
| Lowest reported MAE on ShanghaiTech PartB | Not DecideNet in this paper | The paper's own table lists CP-CNN at 20.1 and DecideNet+R3 at 20.75, both below DecideNet at 21.53 [arxiv:1712.06679 tab. 2]. |

## Implementation sketch

The core is the target construction, the count as a sum, and a pixel-wise
loss. The fusion variant adds a detector map and an attention blend on top.

```pseudocode
# Target: a Gaussian at each annotated head, summed  [arxiv:1712.06679 §3.1]
target = zeros(H, W)
for (x, y) in head_points:
    # sigma 4.0 with a 15-pixel window, as in the paper's training setup [§5.1]
    target += gaussian_kernel(center=(x, y), sigma=4.0, window=15)
# The paper states that the total of the target equals the head count [§3.1] [eq. 1]

# Model: a non-negative density per pixel  [arxiv:1712.06679 §3.2]
density = model(image)          # output may be smaller than the input
# Count of a region: sum of the density over its pixels  [§3.1]
count = density[region].sum()

# Loss: pixel-wise squared error, summed and divided by the number of
# training images  [arxiv:1712.06679 §4] [eq. 6]
loss = ((density - target) ** 2).sum() / num_images
```

Open details the paper does not state, which an implementer must choose:

- How the output map is upsampled to image size before fusion [arxiv:1712.06679 §3.4].
- The λ weight of the quality-aware term and the σ of the detector's Gaussian
  [arxiv:1712.06679 §4] [eq. 3] [eq. 8].
- Optimizer momentum, weight decay, batch size and the validation metric
  [arxiv:1712.06679 §5.1].
- Whether the detector's score map is detached from the gradient
  [arxiv:1712.06679 §4].

## Contradictions

The paper's headline claim conflicts with its own tables. The abstract and
conclusion state the best results on all three datasets [arxiv:1712.06679 §6].
The tables show CP-CNN with a lower ShanghaiTech PartB MAE, 20.1 against 21.53
[arxiv:1712.06679 tab. 2]. They also show Cross-scene with a lower MAE on
WorldExpo'10 scenes S4 and S5, 9.30 and 3.10 against 17.40 and 4.75
[arxiv:1712.06679 tab. 3]. The authors attribute the S4 gap to the detector
working poorly on remote crowds [arxiv:1712.06679 §5.4].

## Papers

- [DecideNet](/papers/1712.06679) (arXiv:1712.06679, CC BY 4.0): fuses a detection density map and a regression density map with a learned per-pixel attention.