Proximal Mapping Loss: Understanding Loss Functions in Crowd Counting & Localization
Authors: Wei Lin
Jia Wan
Antoni B. Chan
Venue: ICLR 2025 Poster (2025)
Keywords: crowd counting
Primary area: applications to computer vision, audio, language, and other modalities
TL;DR: A loss function for crowd counting

Abstract: Crowd counting and localization involve extracting the number and distribution of crowds from images or videos using computer vision techniques. Most counting methods are based on density regression and are based on an ``intersection'' hypothesis, *i.e.*, one pixel is influenced by multiple points in the ground truth, which is inconsistent with reality since one pixel would not contain two objects. This paper proposes Proximal Mapping Loss (PML), a density regression method that eliminates this hypothesis. {PML} divides the predicted density map into multiple point-neighbor cases through the nearest neighbor, and then dynamically constructs a learning target for each sub-case via proximal mapping, leading to more robust and accurate training. {Furthermore}, PML is theoretically linked to various existing loss functions, such as Gaussian-blurred L2 loss, Bayesian loss, and the training schemes in P2PNet and DMC, demonstrating its versatility and adaptability. Experimentally, PML significantly improves the performance of crowd counting and localization, and illustrates the robustness against annotation noise. The code is available at [https://github.com/Elin24/pml](https://github.com/Elin24/pml).

Published as a conference paper at ICLR 2025
PROXIMAL MAPPING Loss: UNDERSTANDING Loss FUNCTIONS IN CROWD COUNTING & LOCALIZATION
Wei Lin", Jía Wan & Antoni B. Chan'
1 Department of Computer Science, City University of Hong Kong, Hong Kong SAR, School of Computer Science and Technology. Harbin Institute of Technology, Shenzhen elonlingg 2n com, jiawanl 9980gma i com, abchanê cityu. edu .kk
ABSTRACT
Crowd counting and localization involve extracting the number and distribution of crowds from images or videos using computer vision techniques. Most counting methods are based on density regression and are based on an "intersection" hy- pothesis, ie., one pixel is influenced by multiple points in the ground truth, which is inconsistent with reality since one pixel would not contain two objects. This paper proposes Proximal Mapping Loss (PML), density regression method that eliminates this hypothesis. PML divides the predicted density map into multiple neigt-neighbor cases through nearest neighbor, and then dynamically constructs a learning target for each sub-case via proximal mapping, leading to more robust and accurate training. Furthermore, PML is theoretically linked to various existing loss functions, such as Gaussian-blurred L2 loss, Bayesian loss, and the training schemes in P2PNet and DMC, demonstrating its versatility and adaptability. Ex- perimentally, PML significantly improves the performance of crowd counting and localization, and illustrates the robustness against annotation noise. The code is available at https: giubbb. wwoom/Einn/////
INTRODUCTION
Automated crowd scene understanding has undergone extensive exploration over the years, primar- ily due to its broad applicability in video surveillance and public safety services (Wang et al.) 2020c
Ma & Chan 2015). The fundamental task within this research domain is crowd counting (Chan & Vasconcelos 2009 2011), wherein statistical data related to crowds, crowd count and distribution, extracted from images or videos using computer vision technologies. In recent times, the advent of deep learning has significantly enhanced the precision of counting methodologies (Wang et al.) 2019 Ma et al.] (2019) Lin et al.] [202aa] Shu al., 2002). with some well- cell-performing methods also being applied to crowd instance localization (Song et al. 2021 Wan al. [021] 2 Lin & Chan 2023 Han et al. 2023). Concurrently, diverse domains, including traffic congestion control (Wang al.
2018 Lu al. [018 Lin et al., 2022b) and marine environmental monitoring (Sun et al.) 2023
have also benefitted from advances in crowd counting algorithms.
Supervised crowd counting is typically formulated counting model aims to predict a density map whose distribution and count closely align with the ground truth (GT) point map, in which a pixel with value of one denotes a person's location. A pseudo GT density map is created by convolving a Gaussian kernel with the GT point map (Lempit-) sky & Zisserman 2010 Zhang al. 2016). Subsequently, pixel-wise L2 loss is computed between this intermediate map and predictions to supervise the counting model. In contrast to L2 loss that uses an intermediate GT density map based on the GT point map. Bayesian Loss (BL) (Ma al., 2019) offers an alternative method for supervision based purely on the GT point map. Specifically, BL. computes the posterior probability of each pixel having GT' point to establish an intermediate point map, which is then compared with the GT for supervision. Both models make theoretical sense, and positive results have been obtained through experiments. Although L2 loss and BL are developed based on different principles, Wan et al. (202I) demonstrates that both L2 and BL are spe- cial cases of a generalized loss (GL), which computes a transport plan matrix between the predicted density map and GT point map using unbalanced optimal transport (UOT) (Peyré et al. 2019). The transport matrix represents both an intermediate density map, by marginalizing over the columns, point map, by marginalizing over the rows, thus providing a theoretical connection between GL.
regression task. Given crowd image, the

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Published as a conference paper at ICLR 2025
(a)
(b)
(c)
(d)
(c)
Figure I: Design principle of counting methods. The top row presents 2D representation, while the boltom
row showcases the corresponding ID representation. (a) A synthetic input with three humans. (b) Density
regression with the intersection hypothesis (using Gaussian prior) or D2CNet (Cheng et al. 2021). where pixel may correspond multiple objects. (c) Point prediction. (d) Head region segmentation. (c) Density regression without the intersection hypothesis (proposed method), where one pixel corresponds to one object. The intersection hypothesis in (b) includes the Gaussian-blurred L2 loss, BL, DMC, and GL. Conversely, (c)- (e) assume no intersections among instances.
and both Gaussian-blurred L2 loss and BL. The effectiveness of GL is also substantiated experimen- tally, where a GL-trained model estimates more accurate density map than with other losses.
Looking closely at these methods, we note that they are all designed with an intersection hypothesis: the value of foreground pixel in the defined learning objective is influenced by multiple instances, i.e., the mixed Gaussian kernel in Fig. T a). However, methods based on point detection, such as P2PNet (Song et al.) (02T) and PET (Liu et al., 2023). have shown that it possible to represent pedestrian's s location with just one pixel, as shown in Fig. (.). Additionally, LSC-CNN (Sam et al.) 2020), TopoCount (Abousamra et al. (02T), and IIM (Gao et al., 2020) localize and count people by clearly defining the boundaries of head regions around each pedestrian in training and then seg- menting them out during inference (Fig. d)). iKNN (Olmschenk et al.) and FIDTM (Liang et al.) 2022b) localize crowds by detecting the local maximum of predicted inverse distance transform maps (IDTM) (Fig. Te). These methods assume there is no interaction between pixels of different pedestrians, but achieve counting results comparable to those of density-regressin models, suggest- ing that the intersection hypothesis may not be necessary. Moreover, not considering the intersection hypothesis leads better localization performance, because there are clear boundaries between each instance (Fig. T (ce)). In contrast, the intersection hypothesis causes the predicted density maps to overlap (Fig. [b)), making it difficult to define clear boundaries between instances and iden- tify each independent individual, especially in dense scenarios where the density representation of crowds blends together. D2CNet (Cheng et al. 202T) also follows Fig. [b), but uses the ma x rather than the sum operator when embedding Gaussian kernels to each GT point when constructing the learning objective, which learns to output overlapping probability maps for localization.
Premised on the above successful experience, we argue for eliminating the intersection hypothe- sis in the density regression framework, so that a model can be trained to achieve both advanced counting and localization performance. Specifically, we propose to divide the predicted density map into multiple non-overlapping irregular patches and then process each patch independently when computing the loss value, following the philosophy of divide and conquer. The dividing stage is implemented via nearest neighbor assignment, which assigns each pixel in the density map to its nearest point in the ground truth. Consequently, the conquer stage, we only need compute the loss within each point-neighbor case separately and then sum these values to obtain the final loss value. However, new obstacles emerge when processing the sub-problems after assignment: these point-neighbor cases vary in the number of neighbors. which introduces difficulty in training with constant targets like Gaussian- kernel density maps (Lempitsky & Zisserman, 2010). This is because there is no effective way to determine hyper-parameters, i.e., the variance in the Gaussian distribu- tion. To address this issue, we propose the Proximal Mapping Loss (PML.), which uses proximal mapping (Drusvyatskiy 2017) to dynamically construct a learning target for each sample during training. This target lies between the predicted distribution and the delta function established by the GT point parameters. Several experimental results illustrate that our PML outperforms most current approaches, exhibiting superior accuracy in crowd count estimation and capturing crowd distributions.
In conjunction with our empirical advancements, we find that the derived PML theoretically en- compasses many existing loss functions when processing the point-neighhor case. Specifically, it
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can be transformed into the Gaussian-blurre L2 loss by incorporating certain constraints. PML can also be instantiated as a dynamic L2 loss for the counting model, sharing a similar formula- tion with BL (Ma et al., 2019). Moreover, by restricting the space of the learning objective to the delta function, we derive the matching strategy for the Hungarian algorithm used in P2PNet (Song et al.) (.2TT. By modifying a regularizer in our PMI., the loss function based on balanced optimal transport in DM-Count (DMC) (Wang et al. 2020a) also be derived.
To summarize, the contributions of this paper are three-fold:
1. We introduce Proximal Mapping Loss (PML), which can effectively train a crowd counting model based on density regression without the intersection hypothesis. It follows the phi- losophy of divide and conquer - dividing the training sample into many point-neighbor cases and then computing sub-loss by comparing the difference between the prediction and a dynamic learning target defined via proximal mapping.
2. Theoretically. we establish the connection between PML and several widely- used counting methods, including the Gaussian-blurred L2 loss, Bayesian Ioss (Ma et al. (2019). and the training schemes in P2PNet (Song et al. 2021) and DMC (Wang et al., 2020a).
3. Empirically, we demonstrate the effectiveness of the proposed PML in crowd counting and localization. The counting performance is significantly improved compared to previous models, and the localization results outperform all previous methods.
2 RELATED WORK
The design of crowd counting methods primarily focuses on the model structure, loss function for training. and localization processing.
Crowd Counting with intersection hypothesis. Most crowd counting methods are trained with the haussian-blurres L2 loss and follows the scheme of density regression. Early publications such as MCNN (Zhng et al. (016).. Switch CNN (Babu Sam et al. 2017), CSRNet (Li et af., 2018). RAZNet (Liu al.) 2019) focus addressing scale information in crowds by integrating infor- mation from CNN modules with different receptive fields. In contrast, later approaches (Gao et al.) 2019 Shi al. 2019: Yang al. 2020) aim to extract perspective information from the input image and subsequently allocate detailed attention to regions with dense crowds. MAN (Lin et al., 2022a) introduces an effective approach to embed transformers (Vaswani et al. 2017) into crowd counters, enhancing counting performance.
Instead of designing network modules enhance the effectiveness of crowd counters, Ma et al. 2019); Wan et al.) (202T); Wang et al. (2020a); Ma et al. (2021) focus on training counting model with loss functions based on different principles. Typically, most counting models are trained with Gaussian-blurred L2 loss following MCNN (Zhang et al. 2016), using GT density maps obtained by convolving a Gaussian kernel with GT point maps. In contrast, BL (Ma et al.) 2019) directly considers these point maps priors, computing the posterior probability of density pixels corre- sponding to the points, leading to a more accurate prediction. Subsequently, DMC (Wang et al.] 2020a) introduces balanced optimal transport (OT) to establish the mapping relations between GT points and predicted pixels, while UOTCC (Ma et al., 202T) and GL. (Wan et al.) 2021) utilize unbalanced OT (UOT). Notably, GL demonstrates that BL and Gaussian-blureed L2 loss are spe- cial cases of GL by substituting a half Sinkhorn iteration, providing solid theoretical contributions. Although the Gaussian-basee pseudo density map is later replaced by posterior probability or opti- mal transport, these methods for density regression still operate under the intersection hypothesis. There are no works that design a density regression method without the intersection hypothesis. In D2CNet (Cheng et al. 2021). the learning objective is probability map, where each pixel's 's value the output of Gaussian function with the distance to its nearest GT point as the argument. This approach differs from the use of density maps in constructing the learning objective, as it uses the max operator instead of the sum operator to embed Gaussian kernels at each GT point's location.
Crowd Counting without intersection hypothesis. Frameworks that do not use the intersection hypothesis, like point detection, head segmentation, and inverse distance transform map (IDTM) regression, have also been explored. P2PNet (Song et al., 2021) is the first one proposing to count and localize pedestrians via point detection, treating each pedestrian as a point and predicting their locations through single-stage detection framework. This approach distinguishes itself from pre-
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vious density-basee methods by directly acquiring localization information. uollow-up works, such as CLTR (Liang et al.] 2022a) and PET (Liu et al., 2023), adopt a similar strategy for predicting points but employ distinct network structures. Apart from these methods, head detection is another popular trend for counting without the intersection hypothesis. LSC-CNN (Sam et al. 2020) and
pseudo and coarse box annotations supervise the de- tection network, while IIM (Gao et al. (020)) adopts accurate box annotations provided in NWPU- Crowd (Wang et al. 2020c) train the counting model. IDTM is also investigated for counting without the intersection hypothesis. FIDTM Liang et al. (002b) and iKNN (Olmshhenk et al.) de- fine the learning objective of each pixel as the inverse value the distance to the k-nearest point in the ground truth, and locate pedestrians by finding peak positions in 3 x 3 patch during inference.
TopoCount (Abousamra et al.] (02T))
The above methods without the intersection hypothesis demonstrate outstanding localization per- formance, but their counting performance is not as good as density regression methods. However, the above methods have inspired us to design a density regression method without the intersection hypothesis. We show that our proposed method encompasses many existing crowd counting loss functions, including BL (Ma et al. 2019) and the widely-used Gaussian-blurred L2 loss, as well as the training schemes presented in P2PNet (Song al. 2021) and DMC (Wang al. 2020a).
3 PROXIMAL MAPPING LOSS (PML)
To remove the intersection hypothesis existing in current density regression methods, we adopt nearest neighbors split the density map into multiple irregular patches without overlapping. In this way, the loss computation is divided into multiple simpler sub-problems, as each point-neighbor case is handled independently. For each case, we compute the loss in a dynamic way based on proximal mapping. Overall, PML is based on the principle of divide and conquer.
3.1 DIVIDE STAGE: NEAREST NEIGHBOR
Mathematically, we represent the predicted density map with pixels as which IR+ represents the predicted density value at pixel located at 2; E R.. Similarly, the
{aa. (e))?-n in
corresponding ground truth with m pedestrians represented as B = {(1, yj } 1==1 with yj R.2
represents the GT location of the j-th annotation. The "1" in each element of B implicates that the count equals to for each point in GT. Accordingly, the divide stage in PML is formulated as:
L(A,B) -. b,), Āς {aa x 3rEE, b, - (1, y,),
= x v < I|. yull2, Vyk E B5}
(2)
where Xj collects the indices of all nearest neighboring pixels Y/ and A, represents the cor- responding density pixel set. The divide stage splits the construction of loss value L(A, B) into independent sub-loss functions. Thus, we only have to analyze any of these sub-problems, [(A,, by), to ease the theoretical analysis.
3.2 CONQUER STAGE: PROXIMAL MAPPING OF TRANSPORT COST
We next conquer the sub- sub-problem C(Ay, by) presented the subscript j we represent the GT location as y E R. corresponding to by. We assume that there are à neighbor pixels collected in Ay, which we denote as À = {(a,, )- Defining a = [a, -= we construct the learning objective by minimizing the transport cost f(a) = c a, where
To simplify the notation, we remove
8 measures the cost when moving unit mass from æ: y, which is commonly defined as Euclidean distance ||æ; - y/|2 or its exponential (e.g.. in GL (Wan et al. 2021 Here we consider the objective f(a from the perspective of proximal gradient descent. Specifically, after the 1-th round of parameter updates, the new prediction, az+1, of the counting network should be close to the proximal mapping (Parikh et al. 2014) of the previous a,:
a++1 arg f(a.) + (f(..) (p-a:) + S - 12
(3)
linear approximation of f(a,
regularizer
where 7 is a hyperparameter controlling the size of the neighborhood for mapping, and V f(a)) = c the gradient w.rl a,, The regularizer in (3) is required prevent d++1 from straying far from at
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loss function
D.(p. &
p ((
1.2 loss (Zhang et al.
2016
0
p
N(
Bayesian loss (Ma et al.
2019
p
1 C+
P2PNet (Song et al. 202T
P a
8())
( min Cj Ta,)
DM-Count Wang et al
2020a
00
KL(pI
a/ aa
Different crowd counting loss functions implemented from PML. and D. (p, a) are defined in { and p" are the distribution formula and the analytical solution of learning objective.
Table I:
since the linear approximation only works well the neighborhood of the current a₁ Compared with directly computing the loss between prediction and point GT, the proximal mapping in (3)
points out a reachable and soft target for optimization since the estimated a++1 is an intermediate learning object between the current prediction a, and GT. Thus, we can design loss function
according to (3) by adding a constraint that p sum to 1, and dropping terms that do not depend on p:
C(Á, b) minpeç c 1 c'p+ -1pp-al1
(4)
where E C ( 1,p R²), and Vf(a;) is replaced by its value c. Furthermore, the squared nomm can be replaced with other distances, e.g., Bregman divergence, to achieve different implementations. Therefore, the general formulation for PML
[(Á, b) minpeg p a + (D, a),
(5)
where Dp(p, a) is the Bregman divergence w.r.t. strietly convex function sp. Note that the squared L.-norm is special case: D.p Ip all2 when cp()) 113
RELATIONSHIP BETWEEN PML AND OTHER LOSS FUNCTIONS
In this section, we show that PML encompasses previous loss functions, including 1.2 loss, BL. al. 2019). P2PNet (Song et al. 2021). and DMC (Wang al. 2020a).
Ma
4.1
FROM DYNAMIC 1.2 Loss TO GAUSSIAN-BI.URRE 1.2 Loss
We derive the L2 loss from PML with pp-) = 111 1| i.c. Eq. Specifically, is rewritten as:
L2 minp e ' 3 [pp all2: s.l. p -
1=1
Applying Lagrange multipliers, the analytical solution p= can be derived as
p a -
1/ (a :))
Note that takes the role as "filler" such that p= 1 1. Substituting 7 into (6 and taking the derivative of L2 with respect to a:
L2 = a - detach(p")IF
(8)
0a,
2
p" in (8) is detached from the computation graph in implementation, and the RHS is the loss function for training crowd counting network.
From 66) and (7). we note that p* is an intermediate distribution between (3y, and Á. and controls how close p* IS to the latter. In extreme cases, we have lim, p' 8(y) and lim, p* (1 a). However, traditional L.2 loss is likely to formulate p" as Gaussian distribution, which can be achieved by limiting within a Gaussian prior, i.e., E (NHHE)), projecting p" into the feasible region after obtaining p":
(9) Note that 9 is still dynamic solution depending on Á. Fitting the original Gaussian-blurred L2 loss tightly, i.c., removing the dynamic property, can be achieved by setting so that p' 8(y), and limiting Σ with lower bound (i.e., Σ > "0 IZ p" <- I(
pd N( (a (m- 14),
1 p"
However, Compared with fixed Gaussian kernels, adapting the learning target (e.g., the intermediate representation between the density map and ground truth) can lead better counting performance by better fitting the properties of the data, as explored numerous studies (Bai et al. 2020; Wan &
Chan 2019 Wan al. 2020: 2021
Song al. 2021
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(a) image & density map
(b) Bayesian loss
(c) generalized loss
(d) proximal mapping loss
Figure 2: Visualization of the weight item of the background loss in (b) Bayesian loss (a al. 2019). (c) generalized loss (Wan et al. 202I), and (d) our proximal mapping loss. (a) displays the predicted density map overlayed on the imput image. The weight map of PML has clearer boundaries of instances, thus resulting in better localization results.
4.2 RELATIONSHIP WITH BAYESIAN LOSS
Bayesian loss (a et 2019 is formulated as:
(;)0, + E,n j== Σn th ( [,,)a; -1/.
L Bagesian
(10)
where { (z)) with > 0 represents the posterior probability of æ; having the label Yy while ( (x,) models the background pixels. the point-neighbor case, i.e., m = 1, the posterior probability g(3) (2;) 1 since all predicted density values are assigned to the same target y1- Therefore, (I0p is simplified as:
Lb
aaa
a
(11)
background loss
count loss
in which q (3 ax x=)) Similar the previous section, the density map Á = E (x)(); and a = [a:]" In BL, the absolute term in (11) is called the count loss, which forces the sum of to be close to 1, while the background loss q forces the distribution of a to be close o (yy) This achieved by backpropagating greater gradients to pixels in A that are far from y. Specifically, the i-th element g is defined based on Bayes' rule with the Gaussian distribution as the prior:
q: q x;;)
q(an, (Bo) a( (+1:)
( (Bo)
def.
1 exp V2na
(d-c:)² 20
(12)
where d = max, G. The LHS of 12 shows q g(v,/1o). while the RHS holds / x q. Thus the order-preserving is transitive: q xc c, revealing that there must be q > qb} if Cj-
The relationship between [1] and (4) is not apparent, but there is connection between BL and the analytical solution of dynamic L2 loss presented in (8), which means both L2 and BL instantiate (p, a) with the L.2-nom. Borrowing the conclusion from (8) and substituting 1) into 77) we have:
OL2 da;
1 c
c;
(a
+
L2 -= cc-
(a 1a111)
(13)
2n
background loss
count Juss
1 1 the squared item in (13) could be considered as the count loss to force the sum of close Meanwhile, (c 2) a is the background loss since greater a; must result in greater gradient c; 7 for a; via the background loss item. Fig. 2 demonstrates the weight item, i.e., the multiplier of a in the background loss part of BL, GL, and our PML. There is no big difference apart from the sharp boundaries present with PML, which are due to the nearest neighbor regions.
where ē=
D ii=i C is the average value of elements in c. Without loss of generality,
Note that the L2-norm in 13 can be transformed into the Ll-norm by setting dynamically cording to the absolute error, I.C., T = detach (2/ a - 1l). which results in a new solution:
L'Bagesian = (c a 1
p' p'=a-
( (
11) c+
(14)
shows that using Ll-norm the count loss is the same as letting p* change according dynamic 7. When the predicted count is close to 1, p" also close to the distribution a, which nearly has contribution to parameter updates. On the contrary, p" will be close to o(y) if the

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Published as a conference paper at ICLR 2025
predicted count is far from GT. Therefore, using the L1-norm is an implicit way to set dynamic T that is correlated with the absolute difference between predicted and GT count. This observation explains the reason why the Ll-norm performs better than the L2-norm while playing the role of counting loss in both BL. and GL. a dynamic T is robust and able to mitigate noise in annotation. Later, will also show how the L.1- and 1.2-norms affect our PML in experiments.
Despite the similarities between PML and BL, differences also exist. Specifically, q in is constructed using only positive numbers. Therefore, the background loss in BL is expected to reduce all density values, regardless of whether they belong to a pedestrian or the background. In contrast, allocates positive weights pixels that are close GT points, as long as their distance oa is smaller than the average distance č. In other words, the background loss in PML segments the density map into "foreground" and "background," expecting density pixels in the foreground to be promoted while background parts are lowered. This is in contrast to BI., which considers all pixels belong the background and lowers their values.
4.3 RELATIONSHIP WITH P2PNET
Here we build the connection between (5) and the training scheme in P2PNet (Song et al. 2021). We instantiate D, with L1 L1-norm, and limit p subspace o Sp2p = (6(x;) an; E A}. Accordingly, can be instantiated as:
Lyep min c c'p+ - allı, -alli,
Spap = {o(x;) E A},
(15)
which can be rewritten
bypp = minpes c_ + [p a) + p) a)
(16)
milpEE.2p (c 2Ta)
P + (p (p 1
(17)
matching stratogy
constant (1+1|a/h)
whose solution is p- darg min; o 2Ta,), which is exactly the way to compute the cost matrix for the Hungarian algorithm in P2PNet when defining the i-th element in c as c, || -y112:
C(A, = ra; Yjil2 al.EN.JEM-
(18)
By setting
1 2- the matching strategy in 17 the same as (18)
4.4 RELATIONSHIP WITH DM-COUNT (DMC)
2020a) is another approach to compute the loss for crowd counting models. In contrast GL, DMC uses balanced optimal transport match normalized predictions and normal- ized GT. In the point-neighbor case, the DMC loss is simplified as:
DMC Wang et al.
Lame
(19)
OT loss
count loss
The OT loss is simplified in since all normalized mass in a ||l]] are transported to the same
GT point y. Here we show that DMC is special case when Dxp is the generalized KL divergence (L(p)a) (Peyré et al. 2019). whose variables are defined on [
c x
x log x x:
Dz a) =p log 2 - --ppp
1
(20)
By substituting (00 into we have:
Lame minp c + T(p log - - s.t. 11
(21)
The close-form solution p" is obtained by applying Lagrange multipliers to (1 resulting in
p. a, exp(-cs/7)
n n= -, aj cxp((cc/7))
(22)
7
Here, also serves as "filler", ensuring the sum of elements in equals 1. As T + oc, becomes the normalized density map: p |al Replacing p with the normalized a (1 the simplified loss function is:
(+) dmc
c
(1 - log (1 aT)
(23)
|alli
OT
count loss
The OT loss is the same as that in (19 except that the count loss takes a KL divergence formulation:
f() x log æ 1, which is convex function and reaches the minimum the solution x" 1.
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ShTech
ShTTch B MSE
UCF-QNNF
JHU ++
NWPU
METHOD
(backbonc)
MAE
MSE
MAE
MAE
MSE
MAE
MSE
MAE
MSE
MONN /hhng al.
2016
714.6
110.2
173.2
277.0
426.0
188,9
232.5
714.6
CSRNee (Li at.
2018
(V06-10)
68.2
115.0
10.6
16.0
110.6
190.1
85.9
309.2
121.3
387.8
SPCN
Wang cl al. 2019
(RoNo-10) 101)
64.8
107.5
7.6
13.0
102.0
171.4
77.5
297.6
105.7
424.1
(VGG-19)
62.8
101.8
7.7
88.7
154.8
75.0
299.9
105.4
454.2
KDMG Wan el ai.
4020
(NGG-19)
63.8
99.2
7.8
12.7
99.5
173.0
69.7
268.3
100.5
415.5
DMC Wangg 0t al
2020g
(VGG-19) 19)
59.7
95.7
7.4
11.8
85.6
148.3
68.4
283.3
88.4
357.6
CoiseCC Wan
(N03-119)
61.9
99.6
7.4
11.3
85.8
150.6
67.7
258.5
96.9
534.2
P2PNeT Song el al.
102T
(VCG-189e)
52.7
85.6
6.3
9.9
85.3
154.5
77.4
362.0
UOTCC Ma cl ml.
2021
(V06-119)
58.1
95.9
6.5
83.3
142.3
60.5
252.7
87.8
387.5
GL. (Wan el
61.3
95.4
7.3
11.7
84.3
147.5
59.9
259.5
79.3
346.1
ChiL (Shu et al. 2022
(9GG-199u)
57.5
94.3
6.9
11.0
80.3
137.6
57.0
235.7
76.8
343.0
PET [Lu et al.
2025
(VOG-15te)
78.8
6.2
144.3
58.5
238.0
74.4
328.5
STEERER
(Ran ot ml.
2025
54.5
86.9
5.8
8.5
74.3
128.3
54.3
238.1
63.7
309.3
PML (ours)
(V06-1566)
30.6
80.7
6.1
9.7
79.5
142.7
58.9
249.6
75.7
353.1
(ours)
(VGG-19)
89.0
6.0
9.3
132.2
57.4
227.4
73.6
338.6
PML (ours)
(HRNeo
52.3
84.7
5,4
8.2
73.2
127.5
52.6
230.8
63.8
306.9
Table 2: Comparison of PML with recent crowd counting methods.
EXPERIMENTS
In this section, we present experiments demonstrating the efficacy of our proposed PML in crowd counting and crowd localization.
5.1 IMPLEMENTATION DETAILS
In our experiments, nearest neighbor is employed in the divide stage to associate pixels in the density map with its nearest point in the ground truth. During training, the dynamic L2 loss with LI-norm, as described in (14). is utilized to supervise the counting model, showing superior performance. Additionally, a hyperparameter is introduced to balance these two terms in PML. Therefore, the final loss is formulated by incorporating (I4) into (I) and add e as the weight:
C(A, B) = =a T ee|P=
1111:
Ciz Pij
P ss exp(| v)l12) with Ci₃ = max; Pij exp(1 x; - y
(24)
where Č.j = Cij P.j
In 24 P E 10 denotes the matching matrix computed via nearest neighbor, where P.j
if I. 1y S [z: yk llz for all k S m, and Pij 0 otherwise. Inspired by GL, Ci5 is computed by normalizing the exponential Euclidean distance between æ; an yz, in which the normalization is implemented by dividing the maximum distance between the j-th target y) and pixels in its point- neighbor case. Subsequently, subtracting the average distance from C.j results in Cjp for loss computation, aligning with operation that subtracts the average cost č in (14).
The counting model utilizes VGG19 (Simonyan & Zisserman 2015) and HRNet (Wang et al. 20205) as the backbone. Details and related ablation study on the structure are illustrated in the appendix.
5.2 OVERALL PERFORMANCE ON COUNTING
In Tab. 2 we compare PML with other prominent crowd counting methods. Taking VGG16-bn as backbone, PML achieves similar performance to PET and perform better than P2PNet. Using VGG19 as the backbone, PML achieves significant improvements over BI., DMC, and GL. On the JHU Crowd++ dataset (Sindagi et al., 2020), the MAE of PML closely matches Chfl (57.4 vs. 57.0), while the MSE of PML (227.4) is lower (235.7). When combined with HRNet backbone, PML attains the best performance on Shanghai Part B (Zhang et al., (016) with an MAE of 5.4 and MSE of 8.2. On the UCF-QNRF dataset (Idrees et al. 2018), the estimation errors (MAE: 73.2, MSE: 127.5) are superior to the latest records set by STEERER (Han et al. 2023) (MAE: 74.3; MSE: 128.3). Additionally, outstanding performance with an MAE of 63.8 and MSE of 306.9, showcasing its excellent counting performance on large-scale datasets. Note that due to the small size of Sh A, VGG19 usually overfits and does not achieve good performance, while VGGG16-ba does not overfit and obtains MAE of 50.6 and MSE of 80.7, comparable to PET.
the larger NWPU benchmark (Wang et al.) 2020c). PML demonstrates
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Fl-meas
Prec.
Roc.
MONN
CSRNet
VGG19
HRNett
RAZNet
0.599
0.066
0,545
MAE
MAE
GL-LLM
0.660
0.800
0.562
L2 Less
186.4
283.6
110.6
190.1
98.7
176.1
92.03
157.49
0.683
BI.
190.6
272.3
107.5
184.3
88.8
154.8
85.52
149.55
P2PNet
0.729
0.676
0.685
NoiseCC
177.4
259.0
96.5
163.3
85.8
150.6
0.742
DMC
176. 1
263.3
103.6
180.6
85.6
148.3
82.07
144.84
STEERER-LM
0.770
0.814
0.730
142.8
92.0
84,3
140.23
PMIZVGG 1
GCFL
83.0
139.8
80.3
137.6
(HRNe) -
0.802
0,809
0.795
PML (ours)
138.9
215.7
82.1
139.0
77.6
132.8
73.17
127.45
PML(HRNeD
0.790
0,803
0.777
Table 4: Comparison of loss functions and backbones UCF-
Table 3: Localization on NWPU-Crowd.
QNRF dataset.
CHIP PAPOOSAAKH
Figure 3: Visualization of samples in the NWPU validation set: density map (top) and estimated point map (bottom) via OTM (Lin & Chan (2023). In the bottom row, predicted points annotated green, while ground-truth points in red. The blue ellipses highlight examples where PML is robust annotation noise.
5.3 LOCALIZATION PERFORMANCE ON NWPU-CROWD
We applied OTM (Lin & Chan 2023), which transforms the density map into a point map, to our predictions to localize the people in the crowd, and compared the results with other crowd localiza- tion methods, as shown in Tab. 3 PMI. with VGG19 backbone performs well against GL with OTM (F1: 0.735 vs. 0.683), and its is also better than the ppint-detection-bar P2PNet (Song et al.)
2021. close to the recent PET (Liu et al. 2023).
PML with HRNet as backbone achieves the best localization performance. Although the precision is little lower than STEERER (Han et al.) 2023). the recall is much higher, resulting a better FI of 0.802. For fair comparison with STEERER, we also use the local maximum as the localization algorithm. As displayed in the last row of Tab. the performance is still better than STEERER.
In Fig. 3 we present detailed visualizations of predicted density maps and the corresponding point maps estimated via OTM (Lin & Chan 2023
5.4 COMPARISON WITH OTHER Loss FUNCTIONS
We applied PML to different counting models and compared when training with different loss func- tions, including the traditional coussian-blurred L2 loss, BL, NoiseCC (Wan & Chan 2020). method tackling the problem of noisy annotations, DMC, and GL. Four counting models with differ- ent structures and complexities were selected demonstrate the generality of PMI. The test results on UCF-QNRF (Idrees et al.) 2018) are presented in Tab, 4 PML effectively trains all these models and further enhances their performance over other loss functions. Compared with GL, PML reduces the MAE of MCNN from 142.8 to 138.9, while MSE is also reduced from 227.9 to 215.7. For CSRNet (Li et al.) 2018). VGG19, and HRNet the estimation errors are also considerably reduced. Moreover, our PML outperforms the recent GCFL (Shu et al. 2022 2023), which transforms the density map into the frequency domain and then designs generalized characteristic function loss.
5.5
THE EFFECT OF COUNT LOSS
Here we discuss the robustness of PML with L1-norm (Eq. 1 as the count loss against annotation noise when compared 1.2-norm (Eq. 14 Fig. (b) show the results of adding noise following
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300
PMLL42 - PML-1
242
180
132
0%
175
25
4%
20
42
60
202
100
290
1000
3200
1500
1790
3000
trainiing sepch
Addiigg followtag
Adding follewtngg
destng training
Figure 4; Robustness to annotation error. (a) Adding uniform random noises ranging from a percentage of the image height; (b) Selecting noisy annotations by picking pixel uniformly within circle of specific radius.
LAl lnuas prodution
Cades estuation Eisoard
Moll ontinttttei Elivaan)
Or www.aaaonnkieeeee
C.& wnighberphatee
Undder
1g) Will L.Daata
E.Sevrm
Figure 5: Visualization of the learning objective p when given noisy annotation and different estimation a. (a) shows a prediction that is not aligned with the GT point, and (b) shows the point's ' neighbor pixels. (c-e) display the distribution of p* and with (c) under-estimated (d) well-estimated, and (e) over-estimated when Ll-norm is applied, while (f-h) display the distribution of L2-norm.
ML et al. (2019) and NCC Wan & Chan (2200). PML with the Ll-norm achieves lower estima- tion errors than the one with the L2-norm across all noise degrees, and outperforms BL and NCC. Additionally, we plot how the MAE changes during training in Fig. 4cc) which shows that training with the 1.2-norm is unstable, as evidenced by the presence of spurious peaks of high MAE.
The superior performance of L1-norm compared 1.2nnorm is consistent with the findings in abla- tion studies of BI. (Ma et al. 2019) and GL (Wan et al. (022]) This preference for L1-norm (Eq. [4] may arise from the fact that it defines the hyperparameter т as function of MAE, as opposed to the fixed constant in L2-norm (Eq. 13). This is advantageous for improving robustness since it can mitigate the impact of annotation noise. In Fig. 5 we visualize an example to illustrate this characteristic. When the density map is under- or over-estimated, the learning objective p' covers most density pixels that are close the GT point (distance c; is close to in both Li-norm and 1.2-norm. However, if a is well-estimated, L1-norm will keep the learning objective p' close to the estimation (Fig. di)) even if the prediction is far from the annotation. In such cases, the model will consider the GT point prediction closer to the noisy annotation (Fig. glg)), which is detrimental to training. Furthermore, also showcase the PML's robustness in Fig. 3 the two blue ellipses in the last two columns mark examples where predictions are true positives but do not overlap with the GT noisy annotation.
noisy annotation, In contrast, L2-norm will still attempt move the
CONCLUSION
This paper introduces the Proximal Mapping Loss (PML) as an effective training method for crowd counting models. In contrast to previous density regression methods that are based on the inter- section hypothesis, PML climinates it by dividing the predicted density map into multiple point- neighbor cases via nearest neighbors, and then computes the loss value in each sub-case. The loss function is derived by minimizing the difference between current prediction and its proximal map- ping. After theoretical analysis, we find that PML has close links with many loss functions for object counting when they are considered in the point-neighbor case, including Bayesian loss, and the training scheme in P2PNet and DMC. In the experiments, PML achieves outstanding performance compared to other counting models and crowd localization methods. Fur- thermore, different counting models designed with various principles also achieve lower estimation emror when they are trained with PMI., providing empirical evidence for the effectiveness of PMI.,
cauusian-bbuured L2 loss,
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ACKNOWLEDGEMENTS
This work supported by a Strategic Research Grant from City University of Hong Kong (Project No. 7005840) and National Natural Science Foundation of China (Project No. 62406090).
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ground truth
divided ground truth
prediction
divided prediction
point-neighbour case
(I) Divide Stage: assign each pixel to its nearest GT point
(IT) Conquer Stage: loss computation
Figure Procedure of the divide and the conquer stage PML.
A OVERVIEW
In this appendix, we discuss more details about the relationship between PML and other loss func- tions, to show the generalization of PML. Besides, we also present details about the counting model, and more visualization figures are displayed to show the effectiveness PML.
1. In Sec. B we present figure illustrate how PML is implemented following the principle divide and conquer;
2. In Sec.
demonstrate the difference when PML processes sparse and dense crowd.
3. In Sec. D an improved P2PNet (Song et al. 2021) is designed from the view of PML and
achieves better performance than the vanilla P2PNet;
4. In Sec. FP we discuss the relationship between PML with KL--divergence 21 and
DMC (Wang al. 2020a) in more detail;
5. In Sec. F experimental results are presented to demonstrate the difference between PML
and BL (Ma al.) | 2019)):
6. In Sec. G we show that PML is easier to be understand and is more efficient for training
counting model than GL (Wan al., (02T).
7. In Sec. H the model structure used in our experiments is introduced;
8. Sec. I ablation studies are presented to discuss issues involving hyper parameters in
PML and model structure. Besides, we extend Tab. 4 to more datasets.
9. In Sec. | the details of results on NWPU benchmark are presented, and some prediction
results NWPU validation set visualized.
B VISUALIZATION OF DIVIDE AND CONQUER
Fig. 6 demonstrates how our PML follows the principle of the divide and conquer. In the (I) divide stage, each pixel the prediction is assigned to its nearest point in the GT, construct X; in (2). After that, each point-neighhor case is conquered independently based proximal mapping
C SPARSE vs. DENSE CROWD IN PML
In Fig. aa), we plot figure show the "average count in each point-neighbor case" the "NN distance each point" for PML, GL, and P2P on the UCF-QNRF dataset, while the bar chart in it demonstrates "the number of noint-neighhor cases" distance means the L2 distance between the concerned GT point and its corresponding nearest one in GT, which reflects the sparsity of the crowd near the concerned point. It shows that even the NN
"NN distance of each point." Here NN
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(4) Sparsity
(b) Esample (GT: 2671)
[c] Caussiaan-buuree :T:2
(d) PMU's prediction: 2712
Details
Figure 7: performance difference PML when handling sparse and dense crowds.
distance is very small (0~4)), the average count of PML (0.9441) is close to 1, and it is comparable to GL (0.9400, with intersection hypothesis) and P2PNet (0.9248, point-based counting).
We also give an example in UCF-QNRE dataset that contains both sparse and dense crowd in Fig. (e). Fig. (b)) and (c) present the point location and the corresponding GT map, while Fig. d) demonstrates the prediction of PML. In sparse crowd, PML can predict density map similar to point map. In dense region, the density map is similar to Gaussian-blured density map. There is rectangle region clearly demonstrating how the prediction changes from point-similar density-similar case. These evidences illustrate that PML based NN is able to capture the density information in both sparse and dense scene.
clussian-bluree
D IMPROVED P2PNET (SONG ET AL. 2021)
In the main paper, we have proven that P2PNet (Song al. 2021 is also a special case of PMI. However, there is discrepancy between the implemented P2PNet (Song et al. 202T) and the derived [15): the counting network is optimized via binary cross-entropy (BCE) instead L1 norm, which inconsistent with the principle of SGL. To maintain consistency, divergence associated with the binary negentropy function xxx))
define D as the Bregman x log x+ (1 x)) log(1 x),
and then (p, a) is instantiated
Dilp, a) = =p(pp ((a) V (x y)
(25)
=p log (8) + (1 p) log 1-
(26)
which is the sum pixel-wise binary KI. divergence. Thus the BCE loss derived accordingly:
Spep minpes.20 c (D.(p,a)
(27)
millpes,2. c_ - (p" log (1 p) log(1 - aa] + P.
(28)
binary entropy
where Pe = [p logp + (1 p) log(1 p)] is a constant.
From 28 we know how the BCE is derived from (5) by instantiating D with the cumulative binary KL divergence. Based on (88). p should also be re-derived. Specifically, a is modulated by s sigmoi(()) in P2PNet (Song al. 2021) i.e.,
exp(-x))-1 Therefore, (28) could be
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MAE
MSE
MAE
MSE
DMC
WWan et al.
2020a
85.60
148.30
P2PNet (Song et al.
2021
52.74
85.06
in 35
85.56
148.90
P2PNet
52.49
83.02
in 35
84.77
144.36
PML
52.25
83.93
PML (23)
82.35
142.19
P2PNet
uses
30 to estimate p'.
Table Experimental results on UCF-
Table 5: Test results ShTech-A Zhang al. using different P2PNet.
QNRF (Idrees et al. 2018) using different DMC.
2016
further expanded to:
Spp = minpetpap c_p p- PT-p TP log
r1 log - +P
(29)
1- a
min pEEp2p (c Tx) =1 log(1 a) P
(30)
matching stratogy
constant
whose solution is p' = (arg min, 9j- TIT). Compared with p2 in 330 prefers to use the prediction before the activation function instead of that processed by Si gigmo ( to construct the cost matrix in the Hungarian algorithm.
In Tab. 5 we conduct experiments to compare how these two versions of P2PNet perform. P2PNet the one using (30) as the matching strategy estimate p'. The results show that P2PNet can further reduce the MAE to 52.49 from 52.74, and the MSE is also decreased by approximately 2.04. We also train the same model with PMI.. The third row of Tab, [5) demonsttatee that PMI. can decrease the MAE to 52.25, while the MSE is also better than the vanilla P2PNet (83.93 85.06).
E DISCUSSION ON DMC (WANG ET AL., 2020A)
In Sec. 4.4 have shown how to implement PML using KL divergence and presented the relation-
ship between PML and DMC (Wang al.) 2020a) in (21)~22 (00). However, (23) is obtained by letting
approach infinity. Here, we show the solution when positive real number by substituting into 21
Lame c +T (p" pp'
log D -pp ++aa" 1++=a+a" aa
(31)
= c p" +r pp' log exp(-c/r) +a - 1-pp
(32)
= c p" + (- 1 logn -
(33)
T I 1- 1)
(34)
Accordingly, the PML with real postive number is formulated
Lame a TT 11-llog; where rj = a exp(-c/T).
(35)
Tab. presents the experimental results of different DMCs. shows that the performance of (35) is close to the vanilla DMC, but there is still small gap. Setting r = 8 achieves lower estimation emrors than the vanilla DMC. Besides, if let 7 - when computing p", but use constant
2 for the counting loss, ie., (3).. the counting error will decrease significantly and be better than the vanilla DMC.
F DIFFERENCE BETWEEN PML AND BL (MA ET AL.,
2019
In the main paper, we show that the difference between BL Ma et al. 2019 and PML lies in the method used to compute the weight item in the background loss. In BL. q in II is constructed using only positive numbers, which considers all pixels to belong to the background and reduces the corresponding density values. In contrast, considers pixels meeting e foreground and
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background
loss
MAE
MSE
BL Ma al. 2019
q a
88.7
154.8
PML (variant)
85.3
146.1
PML (ours)
(c 2)
76.6
132.2
Table 7: Test results on UCF-QNRF Idrees et al
2018) when using different background loss.
(8)
(b)
(C)
(d)
Figure 8: The difference between vanilla GL Wan et al. 2021> (top) and the proposed PML (bottom). (a) A and B represent the predicted density map and GT point map. GL employs unbalanced optimal transport for dense assignment, but PMI directly assigns the corresponding nearest neighbor pixels in A to each pixel in B. (b) example density map with two GT points b, and b2. (c) Assigned density values each pixel for b b2. (d e) The assigned pixel values or corresponding neighbors for b and b2. From (c-e) we find that the transport in UOT is very close to nearest neighbor assignment.
those with & as background. The foreground pixels are assigned negative weights increase the corresponding density values, while positive weights are assigned to the background to lower the corresponding density values.
Here, we empirically demonstrate that the strategy in PML is superior to BL. In Tab. 7 we test two PMI. models and compare them with BI. The one without - (the second row Tab. 7 similar to BI. since it also considers all pixels as background and assigns positive weights to all pixels in the
background loss part. Its performance is also close to BL. However, PML using (c - 2) a as the background loss can significantly reduce the MAE, demonstrating its effectivvness. Besides, ē in PML has a solid theoretical basis since is derived from proximal mapping.
G CONNECTIONS & DIFFERENCE BETWEEN PML AND GL (WAN ET AL. 2021
The generalized loss (GL) Wan et al. 2021 is formulated
() =
min
(C,P)-eH(P + PP al +2 / P
1111
(36)
PER Pe
entropic OT
marginal penaltics
where C₁ measures the cost associated with moving unit mass from æ; to yj and normally set as Cij æ;; Yjll2. The hyperparameter e is near-zero weight for the entropic term HP))
Eng P.
Poylog log Pty- Additionally, two constants T₁ and Ty are introduced modulate the impact of
these penalties for marginal deviation. Once the optimal transport plan P is obtained, the gradient B.c. is accessible for backpropagation. However, there are no applicable algorithms to solve (3).. The alternative to implement an unbalanced OT with KI. divergence as marginal penalties to
approximate P via Sinkhorn algorithm (Peyré et al. 2019), and then substitute into (36) to obtain E for automatic differentiation.
Connections. Fig. g we visualize how UOT and nearest neighbor perform the assignment in loss
computation. Fig. 8 illustrates the transport plan in PMI. is defined by assigning each density
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Published as a conference paper at ICLR 2025
Piae Shaffte
Cony
Backbone
OTM floralization)
Figure 9: The structure of the trained model in our experiments.
pixel to its nearest GT point. Consequently, PML only needs compute the loss within each point- neighbor group separately, ie., within just one point in GT and its neighbors in the density map. In this case, the Sinkhorn algorithm(Peyré et al.) 2019) is not required, as the mass for each pixel transported completely to the same target point.
easily observed that UOT still obeys the intersection hypothesis, since it "softly" assigns pixels multiple points in GT. However, nearest neighbor only assigns each density pixels to its nearest point, which could be considered as a "hard" assign- ment. The nearest neighbor
Comparing UOT with nearest neighbor,
0, Ti co and T2 = 0. This means the nearest neighbor sacrifices the second marginal penalty, i.e., the count loss 1 to approximate P. However, PMI. (44 implicates the count loss, mending the flaw.
be considered as the solution of (36) when
Drawbacks of GL. We find that GL has two drawbacks. Firstly, the theoretical analysis of UOT that relates it to L.2 and BL. is imperfect. The proof stems from applying a half Sinkhorn iteration to obtain an approximate UOT solution, which means that the equivalence of UOT L2 and BL is based on computing a plan non-iteratively, using a single pass over the inputs. Thus the connection is through a static construction of an approximate P. instead of the whole Sinkhorn iterations, i.e., the full UOT solution. Secondly, the Sinkhorn algorithm in implementation is time-consuming and requires large amount of space to store the transport plan during computation, limiting training efficiency.
Efficiency PML. Theoretically, the computation of UOT is O(tmn), where X is the shape of the cost matrix, and is the number of iterations of the Sinkhorn algorithm. Meanwhile, the nearest neighbor only requires O(mn) to get the index of the nearest point in GT for each pixel in the prediction. In practice, the average loss computation time ratio is St 0.045s as 3.46 on a 3090Ti GPU, with the density map resolution of 256 x 256.
PML 0.013s
H MODEL STRUCTURE
Fig. displays the model structure of our counter. VGG19 (Simonyan & Zisserman 2015) or HRNet (Wang al. (020b) adopted the backbone to extract image features. After that, feature maps outputted from different blocks are fused to generate the density features. Based on this, several convolutional layers are stacked to predict the final density map, and pixel shuffle (Shi et al.) 2016) is applied to interpolate the final prediction to resolution that is half of the input. For VGG19, 19, we fuse features from the last two blocks. For HRNet, we fuse features maps from the first two branches.
I ABLATION STUDIES
The effect of In (4), a balanced hyperparameter is introduced to weigh the two components, enabling a focus on either localization counting. Fig. TO a) shows the impact of e on the counting performance of the trained model. value of e yields a good counting model, but surprisingly, 2 results in even lower estimation errors. However, further increasing this hyperparameter proves detrimental to training, as the model leans excessively towards counting, neglecting vital localization information: setting to 4 and 8 yields degradation in counting performance compared to 1.
The effect of upsampling module. We conducted ablation study the upsampling modules, namely bilinear interpolation and pixel shuffle (Shi et al. (016), to systematically assess their impact model performance. Bilinear interpolation is non-leamable method, while pixel shuffle is
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Published as a conference paper at ICLR 2025
850
1300
GL: 1075
750
1350
76
131
-114
MAE
MSE
bilinear
pixelshuffee
bilinear
pixelsinuffle
(a) Ablation study on 6.
(b) Ablation study on upsampling module.
Figure 10: The effect of in
24 and upsampling module.
METTIO
MAE
MSE
MAE
StLech
MSE
MAE
JRU
MSE
103.0
168.6
16.3
28.2
105.2
358.19
100.4
99.5
95.9
72.9
CSRNei
61.8
111.9
8.1
12.2
69.7
275.3
PML
64.1
109.3
777
11.6
66.5
268.8
Figure 11: The relationship between losses and output resolution.
Table 8: Results on additional models with differ- ents losses.
learnable module often utilized in single-image super-resolution tasks. The detailed structure provided in the Supplementary. The test results in Fig. (T]b) show how the output with different interpolation rates (x J. x 2. and x1) affects counting performance. Here, xr denotes that the output resolution is r times that of the input image. Due to its learnable nature, pixel shuffle consistently achieves lower errors in all experiments when compared to bilinear interpolation. Although the differences in estimation errors between different resolutions are marginal, the model with output resolution rate of 2 has the optimal performance during training.
2.
Fig. II presents the results when different losses are applied different output sizes. Overall, detailed density map is beneficial for counting model since provides a more detailed prediction. However, Fig. T shows that the output size does affect the superiority of PML.
More comparison on loss functions. In Tab. 8 we compare MCNN Zhang et al. 2016) and CSR- Net et al. (2018) on more datasets than the main paper. Without loss of generalization, PML still outpertorms BL (Ma et al. 2019) and GL (Wan et al. 202T) when other networks are adopted.
J DETAILS ON NWPU BENCHMARK
Tab. 9 presents the details on the NWPU benchmark to understand in which parts PML works better than other methods. In the counting section (the top part of Tab, P the overall performance of PML is close to STEERER. However, both models excel at different scene levels. PML performs the best S3, whose crowd size variance the largest (GT count E (500, 5000)), and achieves similar performance to STEERER in other scenes. In the benchmark of localization, FIDTM achieves superior performance in AO (head area 10) and Al (head area e (10, 100)) typical dense scenes. However, the Recalls of PML in AO and Al are the second best ones, and PML obtains the best recall from A2 to A4, where the head area ranges from 100 to 10S. the extremely sparse scene A5 (head area 10*), the segmentation-ban based method, TopoCount (Abousamra al. (202)). achieves the best results. These findings indicate that different models excel in different scenarios, but PML demonstrates strong performance in the localization task.
This section also presents some visualization results are presented in Fig. 12 to Fig. 15
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overall
Scene Level (MAE
MSE)
Method
(backhone)
50
S1
FIDIM
(HRNe0)
86.0
312.5
21.67129.
13,707 20.03
55.6717499
217.17391.9
(G45472288,
P2PNet
Igg thea
72.6
331.6
34.7 202.7
11.3/ 17.0
31.5/52.3
161.0/ 165.0/265.8
2311.6/ 2960.7
GL
(sge19)
79.3
92.47 461.5
8.2/13.9
35.4/65.6
179.2/299.2
2228.3/ 2865.8
STEERER (HRNc0
63.7
309.8
48.3 /327.3
6.0/
25.9/514
/
1814.5/ 2636.9
PML
(12819)
73.6
338.6
39.672257
6.7712.0
32.8793.4
179.67 296.9
2141.7 2934.3
PML
(HRNeO
63.8
306.9
26.0 198.4
7../13.3
27.4/177
151.7 262.5
1983.0 2707.7
overall (x 100)
Head Area (Rec. x X100)
Method (backbonc)
F1.
Prec.
Rec.
A0
A1
A2
A3
A4
A5
TopoCount
(vgg16)
69.2
68.3
70.1
5.7
39.1
72.2
85.7
87.3
89.7
FIDTM
(HRNet)
75.5
79.8
71.7
22.8
66.8
76.0
71.9
37.4
10.2
IIM
(HRNet)
76.0
82.9
70.2
11.7
45.3
73.4
83.0
64.5
16.7
GL
(ggg19)
66.0
80.0
56.2
3.6
21.2
56.2
79.4
79.4
51.2
STEERER
(HRNeO)
77.0
81.4
73.0
12.0
46.0
73.2
85.5
86.7
64.3
PML
(vgg19)
73.5
77.6
69.8
5.5
40.7
72.3
84.1
82.7
62,4
PML
(HRNe))
80.2
80.9
79.5
18.4
63.8
80.5
88.2
88.3
69.9
Table 9: Details the NWPU counting (top) and localization (bottom) benchmark.
YA
R
STAD
YA
R STAD
Figure 12: Density map and localization map visualization.
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Published as a conference paper at ICLR 2025
3 @
L @ $
Figure 13: Density map and localuization map visualization.
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Published as a conference paper at ICLR 2025
Figure 14: Density map and localuization map visualization.
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Published as a conference paper at ICLR 2025
18
Orward State
18
18
Onward State
Figure 15: Density map and localuization map visualization.
23


Reviews:
[{"review_id": "oAs6FsqnuE", "reviewer": "Reviewer_JtJ9", "summary": "The paper introduces Proximal Mapping Loss (PML) to improve crowd counting and localization and challenges the basis of prevailing loss functions used in density regression. PML divides the predicted density map into smaller, non-overlapping point neighbors and is processed independently to construct a target for learning. The paper also links various existing loss functions used in crowd counting to PML, showcasing the generalizability across existing methods.", "questions": "Major Comments\n\n1. There are existing density-based crowd-counting methods that already don’t rely on the intersection hypothesis, such as STEERER and CrowdDiff. STEERER achieves non-overlapping density through scaling, and CrowdDiff uses narrow kernels to prevent overlapping. Both methods use Gaussian kernels and avoid overlapping in density kernels. In that case, why would the non-overlapping nature of PML work better than non-overlapping Gaussian kernels?\n2. The paper lacks a comparison against CrowdDiff, the state-of-the-art NWPU crowd-counting benchmark.\n3. How does eliminating the intersection hypothesis improve crowd counting and localization performance?\n4. Fig. 3 shows the density maps obtained with or without applying PML. If it is the former, how is localization achieved from the density maps? Also, in some cases, more than one density kernel is predicted on a single face. How are these false positives filtered for localization and counting?\n5. How does the nearest neighbor approach in the divide and conquer stage impact performance in dense versus sparse crowd scenes?\n6. The counting loss was tested using the nearest neighbors in CLTR. How does the PML’s nearest neighbor assignment compare to CLTR?\n\nMinor comments\n\n7. How does PML handle edge cases where crowd members are extremely close together or partially occluded? Especially with partial occlusions, even though the pixel values are not influenced by neighboring objects, the density kernels could have some overlapping as this is a distribution predicted over image coordinates conditioned on pixel values.\n8. The detailed results from the NWPU evaluation could help address these edge cases for the crowd-counting community.\n9. Missing citation for RAZNet in line 135.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The paper demonstrates limited originality through its approach, which addresses a key limitation, the intersection hypothesis, in existing methods. The method offers a fresh perspective on how crowd density can be modeled. The paper provides good theoretical foundations for the proposed method and empirical results compared to existing methods. The paper is well-structured and progresses coherently to deliver the proposed method.", "weaknesses": "While the paper demonstrates that PML outperforms several existing loss functions, it does not explain why PML works better for density-based crowd-counting methods. The method’s reliance on nearest-neighbor assignment for dividing the predicted density map into point-neighbor cases could be problematic, particularly in dense crowd structures. NN algorithms may not capture the correct relationships between nearby points, leading to inaccurate density estimations in such edge cases. The paper does not address how this issue is handled or provide evidence that PML performs well in extreme cases.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "2LAzxU9r6e", "reviewer": "Reviewer_cGtL", "summary": "This paper proposes the Proximal Mapping Loss (PML), a novel density regression loss for crowd counting and localization. PML studies the so-called intersection hypothesis of classic density map and addresses the intersection issue by dividing the predicted density map into multiple patches through nearest neighbor analysis. It then constructs a dynamic learning target for each case, leading to more tractable and robust training. In addition, the paper also shows the theoretical connection between the PML loss and other closely related loss functions. Experiments were conducted to demonstrate the efficacy of PML in both crowd counting and localization.", "questions": "- Please consider rephasing the intersection hypothesis. It does not make sense to me that in reality one pixel would not contain two objects. If two objects occlude, one pixel can certainly contain part of the two objects.\n- Further ablation experiments should be conducted to justify the effect of noise on the model and the effectiveness of PML in coping with it.\n- The paper has demonstrated its work on removing the intersection hypothesis. Experiments, however, has shown nothing related to the hypothesis and mainly focus on the efficacy of PML. Could you explain through experiments or visualizations whether PML really addresses such an issue?\n\n- Table 4 only shows plug-and-play results for MCNN and CSRNet; they are somewhat outdated. It is better to report the use of PML on recent SOTA models. \n\n- It would be good to showcase how much PML can actually eliminate the intersection hypothesis? Can it be shown in a quantitative way?", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "+ **Good motivation**. The paper targets an appropriate problem in crowd localization, featured by the intersection hypothesis. The problem sounds.\n+ **A novel loss**. To evade the sum property of density map, a new loss function is proposed to constrain the regression target by dividing the training sample into many point-neighbor cases and then computing sub-loss by comparing the difference between the prediction and a dynamic learning target defined via proximal mapping. According to Fig. 2, it does advance the prior Bayesian Loss and Generalized Loss in discriminating close individuals.\n+ **Theoretically sound**. The derivations in Sec. 3.2 show how the loss function works in a relatively intuitive way during training, and its connections with several loss functions and training schemes have been proven, which is theoretically sound.\n+ **SOTA performance**. Through evaluations on several benchmark datasets, PML is shown to outperform current state-of-the-art methods in general.", "weaknesses": "- **Some closely related work is missing**. For example, the same problem of intersection hypothesis has been discussed in [a]. While the solution differs, the way for addressing such an issue in [a] could be discussed and compared in Fig.1 as well.\n\n[a] Decoupled two-stage crowd counting and beyond, IEEE Transactions on Image Processing, 2021.\n\n- **Inappropriate baselines**. Since the main focus of the work is the loss function, it should demonstrate its effectiveness on SOTA models instead of constructing a new one (the HRNet based model). It would be difficult to interpret where the real improvement comes from.\n- Some important experiments are missing such that some results are not directly comparable. For example, the proposed approach should report the results of vgg-16bn as well.\n\n- **A proof-of-concept experiment is missing**. The initial claim of the paper is to solve the intersection hypothesis problem, but it turns out to focus on how to better predict points. There lacks some comparative experiments to prove the soundness of the initial claim.\n- The paper claims that PML is robust to label noise. Though visualization results are given to demonstrate its robustness, there lacks evidence to support this. \n\nMinor issues：\n- Some cited results may be wrong. E.g., CSRNet in Table 2 uses vgg-16 rather than vgg-16bn.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "dUjly4C9Ag", "reviewer": "Reviewer_opsr", "summary": "This paper proposes a proximal loss function for crowd counting and localization. The key idea is to divide the predicted density map into multiple non-overlapped patches and compute loss for each patch via proximal mapping. Such an operation eliminates the intersection hypothesis in density regression-based methods, which improves the robustness and accuracy of the counting models. Evaluations on several crowd counting datasets demonstrate the superiority of the proposed loss over previous loss functions.", "questions": "* Did the authors use the same output resolution to train different models in Table 4? As shown in the paper, spatial resolution could affect the performance.\n* Does PML converge faster than existing methods? This is important when applying the proposed method to large-scale datasets.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "* This paper presents a new proximal mapping loss, eliminating the intersection hypothesis in existing density-map based methods.\n* The authors comprehensively discuss the relationship between the proposed loss and existing loss functions.\n* Extensive experiments validate the effectiveness of the proposed loss.", "weaknesses": "* It would be better to give a quantitative comparison of training efficiency between the proposed loss and existing losses. While the authors mention that the loss computation of PML is more efficient than GL, it is unclear whether the convergence speed of PML is faster than existing loss functions.\n* It appears that retaining the spatial resolution is important to make PML work as occluded objects could degenerate into a single point when downsampling. In this case, it is beneficial to analyze the characteristics of the proposed loss under different crowd densities.\n* Following the previous comments, it is suggested to discuss in what scenarios PML is better than previous methods.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "u7ZoKOG9GN", "reviewer": "Reviewer_fWss", "summary": "This paper argues that the intersection hypothesis in existing density-map based counting methods is consistent with reality, where one pixel is influenced by multiple ground-truth points. To eliminate such a hypothesis, the authors propose a proximal mapping loss, which follows the philosophy of divide-and-conquer. Specifically, the predicted density map is first divided into non-overlapping irregular patches. Then, proximal mapping loss (PML) is computed within each point-neighbor case. Experimental results on several crowd counting datasets demonstrate the effectiveness of the proposed method. Additionally, this paper also shows that the proposed PML compasses many existing loss functions.", "questions": "1. Does the proposed method suffer from slow convergence speed? As shown in Fig. 4, the training epoch reaches 2000. \n2. Did the authors use different epsilon for different datasets? Fig. 8 shows that the choice of epsilon does affect the performance.", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. This paper presents an interesting argument about the intersection hypothesis in existing density-regression counting methods.\n2. The paper proposes a divide-and-conquer strategy to eliminate the intersection hypothesis, featuring a proximal-mapping loss.\n3. Experiments show that the proposed method achieves state-of-the-art counting and localization results.", "weaknesses": "1. The main concern is the training efficiency. The proposed method not only needs to compute the nearest neighbor, but also constructs learning targets for each sub-problem presented in Eq. 1. This could be computationally expensive when dealing with congested scenes.\n2. The robustness to noisy annotations needs further clarifications, e.g., performance comparisons with existing methods would be sufficient.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)