Yu-Ting Chang1 Qiaosong Wang2 Wei-Chih Hung1 Robinson Piramuthu2 Yi-Hsuan
Tsai3 Ming-Hsuan Yang14 University of California, Merced eBay Inc. NEC Labs
America Google Research Mixup-CAM: Weakly-supervised Semantic Segmentation

# Mixup-CAM: Weakly-supervised Semantic Segmentation via Uncertainty
Regularization

###### Abstract

Obtaining object response maps is one important step to achieve weakly-
supervised semantic segmentation using image-level labels. However, existing
methods rely on the classification task, which could result in a response map
only attending on discriminative object regions as the network does not need
to see the entire object for optimizing the classification loss. To tackle
this issue, we propose a principled and end-to-end trainable framework to
allow the network paying attention to other parts of the object, while
producing a more complete and uniform response map. Specifically, we introduce
the mixup data augmentation scheme into the classification network and design
two uncertainty regularization terms to better interact with the mixup
strategy. In experiments, we conduct extensive analysis to demonstrate the
proposed method and show favorable performance against state-of-the-art
approaches.

##  1 Introduction

Semantic segmentation is one of the fundamental tasks in computer vision, with
a wide range of applications such as image editing and scene understanding. In
order to obtain reliable models and achieve promising performance, recently
deep neural network (DNN) based methods [helhamer and Darrell(2016), Chen et
al.(2016)Chen, Papandreou, Kokkinos, Murphy, and Yuille, Yu and Koltun(2016)]
are learned from fully-supervised data that requires pixel-wise semantic
annotations. However, acquiring such pixel-wise annotations is usually time-
consuming and labor-intensive, which limits the application potentials in the
real world. As a result, numerous approaches tackle this issue via training
models only on weakly-annotated data, e.g., image-level [Ahn and Kwak(2018),
Kolesnikov and Lampert(2016), Pathak et al.(2015)Pathak, Krahenbuhl, and
Darrell, Pinheiro and Collobert(2015)], bounding box [Papandreou et
al.(2015)Papandreou, Chen, Murphy, and Yuille, Dai et al.(2015)Dai, He, and
Sun, Khoreva et al.(2017)Khoreva, Benenson, Hosang, Hein, and Schiele], point-
level [Bearman et al.(2016)Bearman, Russakovsky, Ferrari, and Fei-Fei],
scribble-based [Lin et al.(2016)Lin, Dai, Jia, He, and Sun, Vernaza and
Chandraker(2017)], or video-level [Chen et al.(2020)Chen, Tsai, Lin, and Yang,
Zhong et al.(2016)Zhong, Tsai, and Yang, Tsai et al.(2016)Tsai, Zhong, and
Yang] labels. In this paper, we focus on utilizing the image-level label,
which is the most efficient scheme for weak annotations but also a challenging
scenario.

Existing weakly-supervised semantic segmentation (WSSS) algorithms mainly
operate with three steps: 1) localizing objects via a categorical response
map, 2) refining the response map to generate pseudo annotations, and 3)
training the semantic segmentation network using pseudo ground truths. Recent
methods [Ahn and Kwak(2018), Huang et al.(2018)Huang, Wang, Wang, Liu, and
Wang, Wang et al.(2018)Wang, You, Li, and Ma, Wei et al.(2018)Wei, Xiao, Shi,
Jie, Feng, and Huang] have achieved significant progress for WSSS, but most of
them focus on improving the latter two steps. Since the success of these
sequential steps hinges on the quality of the initial response map generated
in the first step, in this paper we present an effective solution to localize
objects.

One common practice to produce the initial response map is using class
activation map (CAM) [Zhou et al.(2016)Zhou, Khosla, Lapedriza, Oliva, and
Torralba]. However, since CAM is typically supervised by a classification loss
that could be sufficiently optimized through seeing only a small portion of
objects, the generated response map usually only attends on partial regions
(see Figure 1(a)). To tackle this issue, recent methods [Lee et al.(2019)Lee,
Kim, Lee, Lee, and Yoon, Jiang et al.(2019)Jiang, Hou, Cao, Cheng, Wei, and
Xiong] make efforts to improve the response map via using the dropout strategy
that increases the model uncertainty or aggregating maps produced at different
stages to see more object parts. However, there remains a challenge whether
there are better loss function designed to explicitly facilitate the model
training and produce better response maps, which are not addressed in prior
works.

In this paper, we propose a principled and end-to-end trainable network with
loss functions designed to systematically control the generation of the
response map. First, inspired by the mixup data augmentation in [Zhang et
al.(2018a)Zhang, Cisse, Dauphin, and Lopez-Paz], we observe that including
mixup could effectively calibrate the model uncertainty on overconfident
predictions [Thulasidasan et al.(2019)Thulasidasan, Chennupati, Bilmes,
Bhattacharya, and Michalak] and in return enables the model to attend to more
object regions. However, it is challenging to control the mixup augmentation
process and the model uncertainty, due to non-uniform response distributions
(see Figure 1(b)), which may affect subsequent response refinement steps.
Therefore, we introduce another two loss terms to the mixup process by
regularizing the class-wise uncertainty and the spatial response distribution.
We refer to our model as Mixup-CAM and show that the produced response map is
more complete and balanced across object regions (see Figure 1(c)), which
facilitates the latter response refinement and segmentation model training
steps.

We conduct quantitative and qualitative experiments to demonstrate the
effectiveness of the proposed Mixup-CAM method on the PASCAL VOC 2012 dataset
[Everingham et al.()Everingham, Van Gool, Williams, Winn, and Zisserman]. To
the best of our knowledge, our algorithm is the first to demonstrate that
mixup could improve the WSSS task on complicated multi-labeled images, along
with other designed loss functions to produce better response maps. In
addition, we present the ablation study and more analysis to validate the
importance of each designed loss. Finally, we show that our method achieves
state-of-the-art semantic segmentation performance against existing
approaches.

Figure 1: Comparisons of (a) the original CAM method; (b) CAM + the mixup data
augmentation; and (c) the proposed Mixup-CAM framework that integrates the
mixup scheme and the uncertainty regularization. Compared to (a) and (b), our
final response map (c) attends to other object parts with more uniformly
distributed response.

##  2 Related Work

#### Initial Prediction for WSSS.

Initial cues are essential for segmentation tasks since they provide reliable
priors to generate segmentation maps. The class activation map (CAM) [Zhou et
al.(2016)Zhou, Khosla, Lapedriza, Oliva, and Torralba] is a common practice
for localizing objects. It highlights class-specific regions that are served
as the initial cues. However, since the CAM model is trained by a
classification network, it tends to attend to small discriminative parts of
the object, leading to incomplete initial masks. Several methods have been
developed to alleviate this problem. Approaches like [Singh and Lee(2017), Wei
et al.(2017)Wei, Feng, Liang, Cheng, Zhao, and Yan, Zhang et al.(2018b)Zhang,
Wei, Feng, Yang, and Huang, Li et al.(2018)Li, Wu, Peng, Ernst, and Fu]
deliberately hide or erase the regions of an object, forcing the model to look
for more diverse parts. However, such strategies require iterative model
training and response aggregation steps. After gradual expansion of the
attention regions, non-object regions are prone to be activated, which leads
to inaccurate attention maps. Other algorithms [Hou et al.(2018)Hou, Jiang,
Wei, and Cheng, Zeng et al.(2019)Zeng, Zhuge, Lu, and Zhang] use both object
and background cues to prevent the attention map from including more
background regions, yet pixel-level saliency labels are used.

Instead of using the erasing scheme, recently the FickleNet approach [Lee et
al.(2019)Lee, Kim, Lee, Lee, and Yoon] introduces the stochastic feature
selection to obtain a diverse combination of locations on feature maps.
Moreover, the OAA method [Jiang et al.(2019)Jiang, Hou, Cao, Cheng, Wei, and
Xiong] adopts an online attention accumulation strategy to collect various
object parts discovered at different training stages. By aggregating the
attention maps, it could obtain an initial cue that contains a larger region
of the object. Unlike methods that mitigate the problem by discovering
complementary regions via iterative erasing steps or consolidating attention
maps, our proposed approach aims at harnessing the uncertainty in end-to-end
classification learning. In addition, by regularizing both class-wise
uncertainties and spatial response distributions, our approach averts the
attention from focusing on small parts of the semantic objects, hence
producing much improved response maps.

#### Response Refinement for WSSS.

Various approaches [Ahn and Kwak(2018), Fan et al.(2019)Fan, Zhang, and Tan,
Fan et al.(2018)Fan, Hou, Cheng, Yu, Martin, and Hu, Huang et al.(2018)Huang,
Wang, Wang, Liu, and Wang, Kolesnikov and Lampert(2016), Wang et
al.(2018)Wang, You, Li, and Ma, Wei et al.(2018)Wei, Xiao, Shi, Jie, Feng, and
Huang] are proposed to refine the initial cue by expanding the region of
attention maps. Other methods [Ahn and Kwak(2018), Fan et al.(2019)Fan, Zhang,
and Tan, Fan et al.(2018)Fan, Hou, Cheng, Yu, Martin, and Hu] are developed
using affinity learning. The recent SSDD scheme [Shimoda and Yanai(2019)]
proposes a difference detection module to estimate and reduce the gap between
the initial mask and final segmentation results in a self-supervised manner.
However, the performance of these methods is limited as initial seeds are
still obtained from CAM-like methods. If these seeds only come from the
discriminative parts of the object, it is difficult to expand regions into
non-discriminative parts. Moreover, the initial prediction may produce wrong
attention regions, which would lead to even more inaccurate regions in
subsequent refinement steps.

#### Label-preserving _vs._ Non-preserving Augmentations.

Data augmentation is a common regularization technique in both supervised and
unsupervised learning [Cubuk et al.(2019)Cubuk, Zoph, Mane, Vasudevan, and Le,
Lim et al.(2019)Lim, Kim, Kim, Kim, and Kim, He et al.(2020)He, Fan, Wu, Xie,
and Girshick]. Conventional data augmentation techniques such as scaling,
rotation, cropping, color augmentation, and Gaussian noise can change the
pixel values of an image without altering its labels. These label-preserving
transformations are commonly applied in training deep neural networks to
improve the model generalization capabilities.

Recent work has demonstrated that even non-label-preserving data augmentation
can be surprisingly effective. Explicit label smoothing has been adopted
successfully to improve the performance of deep neural models. The Mixup
method [Zhang et al.(2018a)Zhang, Cisse, Dauphin, and Lopez-Paz] is proposed
to train a neural network on a convex combination space of image pairs and
their corresponding labels. It has been proven effective for the
classification task and increases the robustness of neural networks. Numerous
Mixup variants [Berthelot et al.(2019)Berthelot, Carlini, Goodfellow,
Papernot, Oliver, and Raffel, Summers and Dinneen(2019), Yun et al.(2019)Yun,
Han, Oh, Chun, Choe, and Yoo, Verma et al.(2018)Verma, Lamb, Beckham, Najafi,
Mitliagkas, Courville, Lopez-Paz, and Bengio, Thulasidasan et
al.(2019)Thulasidasan, Chennupati, Bilmes, Bhattacharya, and Michalak, Guo et
al.(2019)Guo, Mao, and Zhang] have been proposed to extend mixup for better
prediction of uncertainty and calibration of the DNNs. These methods exhibit
shared similarity of producing better-generalized models.

#### Entropy Regularization.

Aside from mixup schemes for predictive uncertainty, another common
uncertainty measure is entropy, which could act as a strong regularizer in
both supervised and semi-supervised learning [Pereyra et al.(2017)Pereyra,
Tucker, Chorowski, Kaiser, and Hinton, Grandvalet and Bengio(2005)]. In
particular, [Pereyra et al.(2017)Pereyra, Tucker, Chorowski, Kaiser, and
Hinton] discourages the neural network from being over-confident by penalizing
low-entropy distributions, while [Grandvalet and Bengio(2005)] utilizes
entropy minimization in a semi-supervised setting as a training signal on
unlabeled data. In this paper, we also adopt the entropy-based loss to
regularize the uncertainty, coupled with the mixup data augmentation for
producing better response maps on objects.

Figure 2: Overview of Mixup-CAM. We perform mixup data augmentation on input
images with their corresponding labels via (2) and pass the mixed image
through the feature extractor $E$ and the classifier $G$ to obtain the
probability score $P^{c}$ for each category $c$. For loss functions, in
addition to the classification loss $\mathcal{L}_{cls}$ on mixup samples, we
design two terms to regularize class-wise entropy ($\mathcal{L}_{ent}$ via
(3)) and spatial distribution on CAM ($\mathcal{L}_{con}$ via (4)).

##  3 Weakly-supervised Semantic Segmentation

We first describe the overall algorithm and introduce details of the proposed
Mixup-CAM framework with loss functions designed to improve the initial
response map. We then detail how to generate the final semantic segmentation
results.

###  3.1 Algorithm Overview

One typical way to generate response maps for annotated object categories is
to use CAM [Zhou et al.(2016)Zhou, Khosla, Lapedriza, Oliva, and Torralba].
However, these response maps tend to focus on discriminative object parts,
which are less effective for the WSSS task. One reason is that CAM relies on
the classification loss, which only requires partial object regions to be
activated during training. As a result, when the objective is already
optimized with high confidence, the model may not attempt to learn other
object parts.

In this paper, we propose to integrate the idea of mixup data augmentation
[Zhang et al.(2018a)Zhang, Cisse, Dauphin, and Lopez-Paz], thereby calibrating
the uncertainty in prediction [Thulasidasan et al.(2019)Thulasidasan,
Chennupati, Bilmes, Bhattacharya, and Michalak] as well as allowing the model
to attend to other regions of the image. Although we find that adding mixup
could improve the response map, sometimes the response could diverge too much,
resulting in more false-positive object regions. To further regularize this
uncertainty, we introduce two additional loss terms: the spatial loss and the
class-wise loss. We illustrate the overall model and loss designs in Figure 2
and provide more details in the next subsection.

After receiving the initial response map, we utilize the method in [Ahn and
Kwak(2018)] to expand and refine the response. Finally, we generate pseudo
ground truths from the refined response and train a semantic segmentation
network to obtain the final segmentation output. Note that while we focus on
the first step of the initial response map in this paper, the succeeding two
steps could be replaced with alternative modules or models.

###  3.2 Mixup-CAM

#### CAM Generation.

We first describe the CAM method for producing the initial response map as our
baseline (see Figure 1(a)). The base network begins with a feature extractor
$E$, followed by a global average pooling (GAP) layer and a fully-connected
layer $G$ as the output classifier. Next, given an input image $I$ with its
image-level labels $Y$, the network is trained with a multi-label
classification loss $\mathcal{L}_{cls}(Y,G(E(I)))$ following [Zhou et
al.(2016)Zhou, Khosla, Lapedriza, Oliva, and Torralba]. After training this
classification network, the activation map $M^{c}$ for each category $c$ is
obtained by applying the $c$-channel classifier weight $\theta_{G}^{c}$ on the
feature map $f=E(I)$:

$M^{c}=\theta_{G}^{{}^{c}\top}f.$ (1)

Finally, the response is normalized by the maximum value of $M^{c}$.

#### Mixup Data Augmentation.

Since the original classification network could easily obtain high confidence,
in which the generated CAM only attends to small discriminative object parts,
we utilize mixup data augmentation to calibrate the uncertainty in prediction
[Thulasidasan et al.(2019)Thulasidasan, Chennupati, Bilmes, Bhattacharya, and
Michalak]. Given an image pair $\\{I_{1},I_{2}\\}$ and its label
$\\{Y_{1},Y_{2}\\}$ randomly sampled from the training set, we augment an
image $I^{\prime}$ and its label $Y^{\prime}$ via:

$\displaystyle I^{\prime}$
$\displaystyle=\lambda_{mix}I_{1}+(1-\lambda_{mix})I_{2},$ $\displaystyle
Y^{\prime}$ $\displaystyle=\lambda_{mix}Y_{1}+(1-\lambda_{mix})Y_{2},$ (2)

where $\lambda_{mix}$ is sampled from the Beta$(\alpha,\alpha)$ distribution
following [Zhang et al.(2018a)Zhang, Cisse, Dauphin, and Lopez-Paz]. Using
this augmented data, we feed it into the classification network to minimize
the loss $\mathcal{L}_{cls}(Y^{\prime},G(E(I^{\prime})))$ and follow the same
procedure in (1) to produce the response map (see Figure 1(b)).

Compared to the original CAM generation, our network no longer receives a pure
image but a mixed image that could have multiple objects with their weights
based on $\lambda_{mix}$ as in (2). Therefore, the predictive uncertainty
could be enhanced, leading to smoother output distributions and enforcing the
model to pay attention to other regions in the image in order to satisfy the
classification loss $\mathcal{L}_{cls}(Y^{\prime},G(E(I^{\prime})))$.

#### Uncertainty Regularization.

Although mixup could improve the response map by looking at other parts in the
image, sometimes the response could become too divergent and thus attend to
pixels non-uniformly, e.g., Figure 1(b). This is attributed to the difficulty
in controlling the quality of mixed images, especially when the model faces
more complicated images such as PASCAL VOC, e.g., an object could appear at
various locations of the image with noisy background clutters.

To further facilitate the mixup process, we propose to self-regularize the
uncertainty via class-wise loss and spatial loss terms. The first term is to
directly minimize the entropy in output prediction from the classifier to
reduce uncertainty:

$\mathcal{L}_{ent}(G(E(I^{\prime})))=-\frac{1}{HW}\sum_{h,w}\sum_{c\in
C}P^{c}(h,w)\log P^{c}(h,w),$ (3)

where $C$ is the category number and $P^{c}\in\mathbb{R}^{H\times W}$ is the
output probability for category $c$. Since our classifier $G$ outputs multi-
label probability, we concatenate the probabilities and normalize them by the
maximum value, then calculate the final $P$ with softmax.

Although the first term has the ability to minimize the uncertainty, it does
not explicitly operate on the response map. To better balance the distribution
on the response, we utilize a concentration loss similar to [Hung et
al.(2019)Hung, Jampani, Liu, Molchanov, Yang, and Kautz] and apply it directly
on CAM for each category (i.e., $M^{c}$), which encourages activated pixels to
be spatially close to the response center:

$\mathcal{L}_{con}(M)=\sum_{c\in\bar{C}}\sum_{h,w}||\langle
h,w\rangle-\langle\mu_{h}^{c},\mu_{w}^{c}\rangle||^{2}\cdot\hat{M}^{c}(h,w),$
(4)

where $\mu_{h}^{c}=\sum_{h,w}h\cdot\hat{M}^{c}(h,w)$ is the center in height
for category $c$ (similarly for $\mu_{w}^{c}$), $\hat{M}^{c}$ is the
normalized response of $M^{c}$ to represent a spatially distributed
probability map. Note that, here we only calculate the concentration loss on
presented categories $\bar{C}$ as provided in the image-label $Y$ to avoid
confusing the model with invalid categories.

#### Overall Objective.

We have described our proposed Mixup-CAM framework, including mixup data
augmentation in (2) and two regularization terms, i.e., (3) and (4). To train
the entire model in an end-to-end fashion, we perform the online mixup
procedure and jointly optimize the following loss functions:

$\mathcal{L}_{all}=\mathcal{L}_{cls}(I^{\prime},Y^{\prime})+\lambda_{ent}\mathcal{L}_{ent}(I^{\prime})+\lambda_{con}\mathcal{L}_{con}(M).$
(5)

For simplicity, we omit the detailed notation inside each loss term. We also
note that $M$ is produced online via computing (1) on valid categories in each
forward iteration.

###  3.3 Implementation Details

#### Classification Network.

Similar to [Ahn and Kwak(2018)], we use the ResNet-38 architecture [Wu et
al.(2019)Wu, Shen, and Van Den Hengel] as our classification network, which
consists of 38 convolution layers with wide channels, followed by a $3\times
3$ convolution layer with 512 channels for better adaptation to the
classification task, a global average pooling layer, and two fully-connected
layers for classification. In training, we adopt the pre-trained model on
ImageNet [Deng et al.(2009)Deng, Dong, Socher, Li, Li, and Fei-Fei] and
finetune it on the PASCAL VOC 2012 dataset. Typical label-preserving data
augmentations, i.e., horizontal flip, random cropping, random scaling, and
color jittering, are utilized on the training set.

We implement the proposed Mixup-CAM framework using PyTorch with a single
Titan X GPU with 12 GB memory. For training the classification network, we use
the Adam optimizer [Kingma and Ba(2015)] with initial learning rate of 1e-3
and the weight decay of 5e-4. For mixup, we use $\alpha=0.2$ in the
Beta$(\alpha,\alpha)$ distribution. For uncertainty regularization, we set
$\lambda_{ent}$ as 0.02 and $\lambda_{con}$ as 2e-4. Unless specified
otherwise, we use the same parameters in all the experiments. In the
experimental section, we show studies for the sensitivity of different
parameters.

#### Semantic Segmentation Generation.

Based on the response map generated by our Mixup-CAM, we adopt the random walk
approach via affinity [Ahn and Kwak(2018)] to refine the response and produce
pixel-wise pseudo ground truths for semantic segmentation. In addition,
similar to existing methods, we adopt dense conditional random fields (CRF)
[Krahenbuhl and Koltun(2011)] to further refine the response and obtain better
object boundaries. Finally, we utilize the Deeplab-v2 framework [Chen et
al.(2016)Chen, Papandreou, Kokkinos, Murphy, and Yuille] with the ResNet-101
architecture [He et al.(2016)He, Zhang, Ren, and Sun] and train the
segmentation network

##  4 Experimental Results

In this section, we present our main results of the proposed Mixup-CAM method
for the WSSS task. First, we show that our approach achieves better initial
response maps and further improves the subsequent refinement step. Second, we
demonstrate the importance of each designed component. Finally, we provide
evaluations on final semantic segmentation outputs in the PASCAL VOC dataset
[Everingham et al.()Everingham, Van Gool, Williams, Winn, and Zisserman]
against the state-of-the-art approaches. More results can be found in the
supplementary material. We will make our code and models available to the
public.

###  4.1 Evaluated Dataset and Metric

We conduct experiments on the PASCAL VOC 2012 semantic segmentation benchmark
[Everingham et al.()Everingham, Van Gool, Williams, Winn, and Zisserman] with
21 categories, including one background class. Following existing WSSS
methods, we use augmented 10,528 training images [Hariharan et
al.(2011)Hariharan, Arbeláez, Bourdev, Maji, and Malik] to train our network.
For evaluation of response maps of the training set, we use the set without
augmentation with 1,464 examples, following the setting in [Ahn and
Kwak(2018)]. For final semantic segmentation results, we use 1,449 images in
the validation set to compare our results with other methods111Although there
is a test set that can be evaluated on the official PASCAL VOC website, by the
submission deadline the website is still out of service for returning the
evaluated performance.. In all experiments, the mean Intersection-over-Union
(mIoU) ratio is used as the evaluation metric.

Figure 3: Sample results of initial responses. Our approach often produces the response map that covers more complete region of the object (i.e., attention on the body of the animal), while the initial cue obtained by CAM [Zhou et al.(2016)Zhou, Khosla, Lapedriza, Oliva, and Torralba] is prone to focus on small discriminative parts. Table 1: IoU results of CAM and its refinement on the PASCAL VOC training set. Method | CAM | CAM + Refinement
---|---|---
AffinityNet [Ahn and Kwak(2018)] | 48.0 | 58.1
Mixup $\mathcal{L}_{cls}$ | 49.3 | 60.5
Mixup $\mathcal{L}_{cls}$ \+ $\mathcal{L}_{ent}$ | 49.5 | 61.6
Mixup $\mathcal{L}_{cls}$ \+ $\mathcal{L}_{con}$ | 49.9 | 61.7
Mixup $\mathcal{L}_{cls}$ \+ $\mathcal{L}_{ent}$ \+ $\mathcal{L}_{con}$ | 50.1 | 61.9

###  4.2 Ablation Study and Analysis

#### Improvement on Response Map.

We first present results of the initial and refined response maps. In Table
LABEL:table:_ablation_loss, we show the performance for the original CAM used
by the baseline AffinityNet [Ahn and Kwak(2018)], our CAM using the mixup data
augmentation (Mixup $\mathcal{L}_{cls}$), and our final Mixup-CAM with mixup
and uncertainty regularization (Mixup $\mathcal{L}_{cls}$ \+
$\mathcal{L}_{ent}$ \+ $\mathcal{L}_{con}$). In both results of CAM and its
refinement, our IoU improvements are consistent after gradually adding the
mixup augmentation and regularization. In addition, Figure 3 shows some
example results of the initial response, which illustrates that our Mixup-CAM
is able to make the network attend to more object parts and produce more
uniform response distributions on objects.

Figure 4: Enhancement on refinement. Our regularization enforces a more
uniform response on objects, which can facilitate the refinement step. The
examples illustrate that the IoU difference of the resultant refined map is
significantly larger than the one of initial response. Figure 5: Sensitivity
analysis for parameters. (a) $\alpha$ for mixup augmentation; (b)
$\lambda_{ent}$ and (c) $\lambda_{con}$ for uncertainty regularization.

#### Effectiveness of Regularization.

One interesting aspect we find is that adding regularization could enhance the
effectiveness of the refinement step. In Table LABEL:table:_ablation_loss,
compared to Mixup $\mathcal{L}_{cls}$, adding either $\mathcal{L}_{ent}$ (3rd
row) or $\mathcal{L}_{con}$ (4th row) improves the CAM IoU by 0.2% and 0.6%
respectively. Nevertheless, with the refinement, the corresponding
improvements in IoU is 1.1% and 1.2%, which are larger than the ones before
refining the response. This is because our regularization enforces a more
uniform response on objects, which greatly facilitates the refinement step
(e.g., via region expanding). In addition, we illustrate one example in Figure
4, where the IoU difference of initial response is relatively small, but the
resultant refined map could differ significantly.

#### Parameter Sensitivity.

In this paper, we mainly study three parameters in our Mixup-CAM framework,
i.e., $\alpha$ for mixup regularization and
$\\{\lambda_{ent},\lambda_{con}\\}$ in uncertainty regularization. In Figure
5(a), when increasing the $\alpha$ value, the Beta distribution would become
more uniform, which encourages a more uniform $\lambda_{mix}$ in (2) and
results in mixed images that are more challenging to optimize. Nevertheless,
the IoUs of Mixup $\mathcal{L}_{cls}$ under various $\alpha$ are consistently
better than the CAM baseline. For regularization terms, we fix
$\lambda_{con}=2e-4$ and adjust $\lambda_{ent}$ in Figure 5(b), while fixing
$\lambda_{ent}=0.2$ and change $\lambda_{con}$ in Figure 5(c). Both figures
show that these two parameters are robust to the performance under a wide
range.

###  4.3 Semantic Segmentation Performance

After generating the pseudo ground truths using the refined response map, we
use them to train the semantic segmentation network. First, we compare our
method with state-of-the-art algorithms using the ResNet-101 architecture or
other similarly powerful ones in Table LABEL:table:_compare_sota. Note that,
while our method focuses on improving the initial responses on the object,
most methods aim to improve the refinement step or segmentation network
training. In Table LABEL:table:_compare_sota_detail, we further present
detailed performance for each category. We show two groups of results without
(top rows) or with (bottom rows) applying CRF [Krahenbuhl and Koltun(2011)] to
refine final segmentation outputs. Compared to the recent FickleNet [Lee et
al.(2019)Lee, Kim, Lee, Lee, and Yoon] approach that also tries to improve the
initial response map, our proposed Mixup-CAM shows favorable performance in
final semantic segmentation results.

Table 2: Comparison of WSSS methods using image-level labels on the PASCAL VOC 2012 validation set. $\checkmark$ indicates the methods that focus on improving the initial response. The result of $\dagger$ on AffinityNet is re-produced by training the same ResNet-101 as our pipeline. Method | Backbone | Init. Resp. | IoU on Val
---|---|---|---
MCOF CVPR’18 [Wang et al.(2018)Wang, You, Li, and Ma] | ResNet-101 |  | 60.3
DCSP BMVC’17 [Chaudhry et al.(2017)Chaudhry, Dokania, and Torr] | ResNet-101 |  | 60.8
DSRG CVPR’18 [Huang et al.(2018)Huang, Wang, Wang, Liu, and Wang] | ResNet-101 |  | 61.4
AffinityNet CVPR’18 [Ahn and Kwak(2018)] | Wide ResNet-38 |  | 61.7
AffinityNet† CVPR’18 [Ahn and Kwak(2018)] | ResNet-101 |  | 61.9
SeeNet NIPS’18 [Hou et al.(2018)Hou, Jiang, Wei, and Cheng] | ResNet-101 | ✓ | 63.1
Zeng et al ICCV’19 [Zeng et al.(2019)Zeng, Zhuge, Lu, and Zhang] | DenseNet-169 |  | 63.3
BDSSW ECCV’18 [Fan et al.(2018)Fan, Hou, Cheng, Yu, Martin, and Hu] | ResNet-101 |  | 63.6
OAA ICCV’19 [Jiang et al.(2019)Jiang, Hou, Cao, Cheng, Wei, and Xiong] | ResNet-101 | ✓ | 63.9
CIAN CVPR’19 [Fan et al.(2019)Fan, Zhang, and Tan] | ResNet-101 |  | 64.1
FickleNet CVPR’19 [Lee et al.(2019)Lee, Kim, Lee, Lee, and Yoon] | ResNet-101 | ✓ | 64.9
SSDD ICCV’19 [Shimoda and Yanai(2019)] | Wide ResNet-38 |  | 64.9
Ours | ResNet101 | ✓ | 65.6
Table 3: Semantic segmentation performance on the PASCAL VOC 2012 validation set. The bottom group contains results with CRF refinement, while the top group is without CRF. The best three results are in red, green and blue, respectively. Method |  bkg |  aero |  bike |  bird |  boat |  bottle |  bus |  car |  cat |  chair |  cow |  table |  dog |  horse |  motor |  person |  plant |  sheep |  sofa |  train |  tv | mIoU
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
AffinityNet [Ahn and Kwak(2018)] | 88.2 | 68.2 | 30.6 | 81.1 | 49.6 | 61.0 | 77.8 | 66.1 | 75.1 | 29.0 | 66.0 | 40.2 | 80.4 | 62.0 | 70.4 | 73.7 | 42.5 | 70.7 | 42.6 | 68.1 | 51.6 | 61.7
Ours (w/o CRF) | 87.6 | 54.5 | 30.7 | 73.0 | 46.5 | 72.0 | 86.5 | 74.8 | 87.6 | 31.3 | 80.8 | 50.3 | 82.6 | 74.5 | 67.2 | 68.7 | 39.6 | 79.2 | 44.8 | 64.9 | 51.1 | 64.2
MCOF [Wang et al.(2018)Wang, You, Li, and Ma] | 87.0 | 78.4 | 29.4 | 68.0 | 44.0 | 67.3 | 80.3 | 74.1 | 82.2 | 21.1 | 70.7 | 28.2 | 73.2 | 71.5 | 67.2 | 53.0 | 47.7 | 74.5 | 32.4 | 71.0 | 45.8 | 60.3
Zeng et al. [Zeng et al.(2019)Zeng, Zhuge, Lu, and Zhang] | 90.0 | 77.4 | 37.5 | 80.7 | 61.6 | 67.9 | 81.8 | 69.0 | 83.7 | 13.6 | 79.4 | 23.3 | 78.0 | 75.3 | 71.4 | 68.1 | 35.2 | 78.2 | 32.5 | 75.5 | 48.0 | 63.3
FickleNet [Lee et al.(2019)Lee, Kim, Lee, Lee, and Yoon] | 89.5 | 76.6 | 32.6 | 74.6 | 51.5 | 71.1 | 83.4 | 74.4 | 83.6 | 24.1 | 73.4 | 47.4 | 78.2 | 74.0 | 68.8 | 73.2 | 47.8 | 79.9 | 37.0 | 57.3 | 64.6 | 64.9
SSDD [Shimoda and Yanai(2019)] | 89.0 | 62.5 | 28.9 | 83.7 | 52.9 | 59.5 | 77.6 | 73.7 | 87.0 | 34.0 | 83.7 | 47.6 | 84.1 | 77.0 | 73.9 | 69.6 | 29.8 | 84.0 | 43.2 | 68.0 | 53.4 | 64.9
Ours (w/ CRF) | 88.4 | 57.0 | 31.2 | 75.2 | 47.8 | 72.4 | 87.2 | 76.0 | 89.2 | 32.7 | 83.1 | 51.1 | 85.4 | 77.3 | 68.4 | 70.2 | 40.0 | 81.5 | 46.2 | 65.4 | 51.8 | 65.6

##  5 Conclusions

In this paper, we propose the Mixup-CAM framework to improve the localization
of object response maps, as an initial step towards weakly-supervised semantic
segmentation task using image-level labels. To this end, we propose to
integrate the mixup data augmentation strategy for calibrating the uncertainty
in network prediction. Furthermore, we introduce another two regularization
terms as the interplay with the mixup scheme, thereby producing more complete
and uniform response maps. In experimental results, we provide comprehensive
analysis of each component in the proposed method and show that our approach
achieves state-of-the-art performance against existing algorithms.

## References

  * [Ahn and Kwak(2018)] Jiwoon Ahn and Suha Kwak.  Learning pixel-level semantic affinity with image-level supervision for weakly supervised semantic segmentation.  In _CVPR_ , 2018.
  * [Bearman et al.(2016)Bearman, Russakovsky, Ferrari, and Fei-Fei] Amy Bearman, Olga Russakovsky, Vittorio Ferrari, and Li Fei-Fei.  What’s the point: Semantic segmentation with point supervision.  In _ECCV_ , 2016.
  * [Berthelot et al.(2019)Berthelot, Carlini, Goodfellow, Papernot, Oliver, and Raffel] David Berthelot, Nicholas Carlini, Ian Goodfellow, Nicolas Papernot, Avital Oliver, and Colin A Raffel.  Mixmatch: A holistic approach to semi-supervised learning.  In _NeurIPS_ , 2019.
  * [Chaudhry et al.(2017)Chaudhry, Dokania, and Torr] Arslan Chaudhry, Puneet K Dokania, and Philip HS Torr.  Discovering class-specific pixels for weakly-supervised semantic segmentation.  In _BMVC_ , 2017.
  * [Chen et al.(2016)Chen, Papandreou, Kokkinos, Murphy, and Yuille] Liang-Chieh Chen, George Papandreou, Iasonas Kokkinos, Kevin Murphy, and Alan L Yuille.  Deeplab: Semantic image segmentation with deep convolutional nets, atrous convolution, and fully connected crfs.  _CoRR_ , abs/1606.00915, 2016.
  * [Chen et al.(2020)Chen, Tsai, Lin, and Yang] Yi-Wen Chen, Yi-Hsuan Tsai, Yen-Yu Lin, and Ming-Hsuan Yang.  Vostr: Video object segmentation via transferable representations.  _International Journal of Computer Vision_ , 02 2020.
  * [Cubuk et al.(2019)Cubuk, Zoph, Mane, Vasudevan, and Le] Ekin D Cubuk, Barret Zoph, Dandelion Mane, Vijay Vasudevan, and Quoc V Le.  Autoaugment: Learning augmentation strategies from data.  In _CVPR_ , 2019.
  * [Dai et al.(2015)Dai, He, and Sun] Jifeng Dai, Kaiming He, and Jian Sun.  Boxsup: Exploiting bounding boxes to supervise convolutional networks for semantic segmentation.  In _ICCV_ , 2015.
  * [Deng et al.(2009)Deng, Dong, Socher, Li, Li, and Fei-Fei] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei.  Imagenet: A large-scale hierarchical image database.  In _CVPR_ , 2009.
  * [Everingham et al.()Everingham, Van Gool, Williams, Winn, and Zisserman] M. Everingham, L. Van Gool, C. K. I. Williams, J. Winn, and A. Zisserman.  The PASCAL Visual Object Classes Challenge 2010 (VOC2010) Results.  "http://www.pascal-network.org/challenges/VOC/voc2010/workshop/index.html".
  * [Fan et al.(2019)Fan, Zhang, and Tan] Junsong Fan, Zhaoxiang Zhang, and Tieniu Tan.  Cian: Cross-image affinity net for weakly supervised semantic segmentation.  In _CVPR_ , 2019.
  * [Fan et al.(2018)Fan, Hou, Cheng, Yu, Martin, and Hu] Ruochen Fan, Qibin Hou, Ming-Ming Cheng, Gang Yu, Ralph R Martin, and Shi-Min Hu.  Associating inter-image salient instances for weakly supervised semantic segmentation.  In _ECCV_ , 2018.
  * [Grandvalet and Bengio(2005)] Yves Grandvalet and Yoshua Bengio.  Semi-supervised learning by entropy minimization.  In _NIPS_ , 2005.
  * [Guo et al.(2019)Guo, Mao, and Zhang] Hongyu Guo, Yongyi Mao, and Richong Zhang.  Mixup as locally linear out-of-manifold regularization.  In _AAAI_ , 2019.
  * [Hariharan et al.(2011)Hariharan, Arbeláez, Bourdev, Maji, and Malik] Bharath Hariharan, Pablo Arbeláez, Lubomir Bourdev, Subhransu Maji, and Jitendra Malik.  Semantic contours from inverse detectors.  In _ICCV_ , 2011.
  * [He et al.(2016)He, Zhang, Ren, and Sun] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.  Deep residual learning for image recognition.  In _CVPR_ , 2016.
  * [He et al.(2020)He, Fan, Wu, Xie, and Girshick] Kaiming He, Haoqi Fan, Yuxin Wu, Saining Xie, and Ross Girshick.  Momentum contrast for unsupervised visual representation learning.  In _CVPR_ , 2020.
  * [helhamer and Darrell(2016)] Jonathan helhamer, Evan an Long and Trevor Darrell.  Fully convolutional networks for semantic segmentation.  _TPAMI_ , 2016.
  * [Hou et al.(2018)Hou, Jiang, Wei, and Cheng] Qibin Hou, PengTao Jiang, Yunchao Wei, and Ming-Ming Cheng.  Self-erasing network for integral object attention.  In _NIPS_ , 2018.
  * [Huang et al.(2018)Huang, Wang, Wang, Liu, and Wang] Zilong Huang, Xinggang Wang, Jiasi Wang, Wenyu Liu, and Jingdong Wang.  Weakly-supervised semantic segmentation network with deep seeded region growing.  In _CVPR_ , 2018.
  * [Hung et al.(2019)Hung, Jampani, Liu, Molchanov, Yang, and Kautz] Wei-Chih Hung, Varun Jampani, Sifei Liu, Pavlo Molchanov, Ming-Hsuan Yang, and Jan Kautz.  Scops: Self-supervised co-part segmentation.  In _CVPR_ , 2019.
  * [Jiang et al.(2019)Jiang, Hou, Cao, Cheng, Wei, and Xiong] Peng-Tao Jiang, Qibin Hou, Yang Cao, Ming-Ming Cheng, Yunchao Wei, and Hong-Kai Xiong.  Integral object mining via online attention accumulation.  In _CVPR_ , 2019.
  * [Khoreva et al.(2017)Khoreva, Benenson, Hosang, Hein, and Schiele] A. Khoreva, R. Benenson, J. Hosang, M. Hein, and B. Schiele.  Simple does it: Weakly supervised instance and semantic segmentation.  In _CVPR_ , 2017.
  * [Kingma and Ba(2015)] Diederik P Kingma and Jimmy Ba.  Adam: A method for stochastic optimization.  In _ICLR_ , 2015.
  * [Kolesnikov and Lampert(2016)] Alexander Kolesnikov and Christoph H Lampert.  Seed, expand and constrain: Three principles for weakly-supervised image segmentation.  In _ECCV_ , 2016.
  * [Krahenbuhl and Koltun(2011)] Philipp Krahenbuhl and Vladlen Koltun.  Efficient inference in fully connected crfs with gaussian edge potentials.  In _NIPS_ , 2011.
  * [Lee et al.(2019)Lee, Kim, Lee, Lee, and Yoon] Jungbeom Lee, Eunji Kim, Sungmin Lee, Jangho Lee, and Sungroh Yoon.  Ficklenet: Weakly and semi-supervised semantic image segmentation using stochastic inference.  In _CVPR_ , 2019.
  * [Li et al.(2018)Li, Wu, Peng, Ernst, and Fu] Kunpeng Li, Ziyan Wu, Kuan-Chuan Peng, Jan Ernst, and Yun Fu.  Tell me where to look: Guided attention inference network.  In _CVPR_ , 2018.
  * [Lim et al.(2019)Lim, Kim, Kim, Kim, and Kim] Sungbin Lim, Ildoo Kim, Taesup Kim, Chiheon Kim, and Sungwoong Kim.  Fast autoaugment.  In _NeurIPS_ , 2019.
  * [Lin et al.(2016)Lin, Dai, Jia, He, and Sun] Di Lin, Jifeng Dai, Jiaya Jia, Kaiming He, and Jian Sun.  Scribblesup: Scribble-supervised convolutional networks for semantic segmentation.  In _CVPR_ , 2016.
  * [Papandreou et al.(2015)Papandreou, Chen, Murphy, and Yuille] George Papandreou, Liang-Chieh Chen, Kevin Murphy, and Alan L Yuille.  Weakly-and semi-supervised learning of a dcnn for semantic image segmentation.  In _ICCV_ , 2015.
  * [Pathak et al.(2015)Pathak, Krahenbuhl, and Darrell] Deepak Pathak, Philipp Krahenbuhl, and Trevor Darrell.  Constrained convolutional neural networks for weakly supervised segmentation.  In _ICCV_ , 2015.
  * [Pereyra et al.(2017)Pereyra, Tucker, Chorowski, Kaiser, and Hinton] Gabriel Pereyra, George Tucker, Jan Chorowski, Łukasz Kaiser, and Geoffrey Hinton.  Regularizing neural networks by penalizing confident output distributions.  _arXiv preprint arXiv:1701.06548_ , 2017.
  * [Pinheiro and Collobert(2015)] Pedro O Pinheiro and Ronan Collobert.  From image-level to pixel-level labeling with convolutional networks.  In _CVPR_ , 2015.
  * [Shimoda and Yanai(2019)] Wataru Shimoda and Keiji Yanai.  Self-supervised difference detection for weakly-supervised semantic segmentation.  In _ICCV_ , 2019.
  * [Singh and Lee(2017)] Krishna Kumar Singh and Yong Jae Lee.  Hide-and-seek: Forcing a network to be meticulous for weakly-supervised object and action localization.  In _ICCV_ , 2017.
  * [Summers and Dinneen(2019)] Cecilia Summers and Michael J Dinneen.  Improved mixed-example data augmentation.  In _WACV_ , 2019.
  * [Thulasidasan et al.(2019)Thulasidasan, Chennupati, Bilmes, Bhattacharya, and Michalak] Sunil Thulasidasan, Gopinath Chennupati, Jeff Bilmes, Tanmoy Bhattacharya, and Sarah Michalak.  On mixup training: Improved calibration and predictive uncertainty for deep neural networks.  In _NeurIPS_ , 2019.
  * [Tsai et al.(2016)Tsai, Zhong, and Yang] Yi-Hsuan Tsai, Guangyu Zhong, and Ming-Hsuan Yang.  Semantic co-segmentation in videos.  In _ECCV_ , 2016.
  * [Verma et al.(2018)Verma, Lamb, Beckham, Najafi, Mitliagkas, Courville, Lopez-Paz, and Bengio] Vikas Verma, Alex Lamb, Christopher Beckham, Amir Najafi, Ioannis Mitliagkas, Aaron Courville, David Lopez-Paz, and Yoshua Bengio.  Manifold mixup: Better representations by interpolating hidden states.  _arXiv preprint arXiv:1806.05236_ , 2018.
  * [Vernaza and Chandraker(2017)] Paul Vernaza and Manmohan Chandraker.  Learning random-walk label propagation for weakly-supervised semantic segmentation.  In _CVPR_ , 2017.
  * [Wang et al.(2018)Wang, You, Li, and Ma] Xiang Wang, Shaodi You, Xi Li, and Huimin Ma.  Weakly-supervised semantic segmentation by iteratively mining common object features.  In _CVPR_ , 2018.
  * [Wei et al.(2017)Wei, Feng, Liang, Cheng, Zhao, and Yan] Yunchao Wei, Jiashi Feng, Xiaodan Liang, Ming-Ming Cheng, Yao Zhao, and Shuicheng Yan.  Object region mining with adversarial erasing: A simple classification to semantic segmentation approach.  In _CVPR_ , 2017.
  * [Wei et al.(2018)Wei, Xiao, Shi, Jie, Feng, and Huang] Yunchao Wei, Huaxin Xiao, Honghui Shi, Zequn Jie, Jiashi Feng, and Thomas S Huang.  Revisiting dilated convolution: A simple approach for weakly-and semi-supervised semantic segmentation.  In _CVPR_ , 2018.
  * [Wu et al.(2019)Wu, Shen, and Van Den Hengel] Zifeng Wu, Chunhua Shen, and Anton Van Den Hengel.  Wider or deeper: Revisiting the resnet model for visual recognition.  _Pattern Recognition_ , 90:119–133, 2019.
  * [Yu and Koltun(2016)] Fisher Yu and Vladlen Koltun.  Multi-scale context aggregation by dilated convolutions.  In _ICLR_ , 2016.
  * [Yun et al.(2019)Yun, Han, Oh, Chun, Choe, and Yoo] Sangdoo Yun, Dongyoon Han, Seong Joon Oh, Sanghyuk Chun, Junsuk Choe, and Youngjoon Yoo.  Cutmix: Regularization strategy to train strong classifiers with localizable features.  In _ICCV_ , 2019.
  * [Zeng et al.(2019)Zeng, Zhuge, Lu, and Zhang] Yu Zeng, Yunzhi Zhuge, Huchuan Lu, and Lihe Zhang.  Joint learning of saliency detection and weakly supervised semantic segmentation.  In _CVPR_ , 2019.
  * [Zhang et al.(2018a)Zhang, Cisse, Dauphin, and Lopez-Paz] Hongyi Zhang, Moustapha Cisse, Yann N. Dauphin, and David Lopez-Paz.  mixup: Beyond empirical risk minimization.  In _ICLR_ , 2018a.
  * [Zhang et al.(2018b)Zhang, Wei, Feng, Yang, and Huang] Xiaolin Zhang, Yunchao Wei, Jiashi Feng, Yi Yang, and Thomas S Huang.  Adversarial complementary learning for weakly supervised object localization.  In _CVPR_ , 2018b.
  * [Zhong et al.(2016)Zhong, Tsai, and Yang] Guangyu Zhong, Yi-Hsuan Tsai, and Ming-Hsuan Yang.  Weakly-supervised video scene co-parsing.  In _ACCV_ , 2016.
  * [Zhou et al.(2016)Zhou, Khosla, Lapedriza, Oliva, and Torralba] Bolei Zhou, Aditya Khosla, Agata Lapedriza, Aude Oliva, and Antonio Torralba.  Learning deep features for discriminative localization.  In _CVPR_ , 2016.


# Interaction-induced wavefunction collapse

Arnab Acharya  Indian Institute Science Education and Research Kolkata, West
Bengal, India  Pratik Jeware  Soumitro Banerjee<EMAIL_ADDRESS>Indian Institute Science Education and Research Kolkata, West Bengal, India

###### Abstract

Almost a century after the development of quantum mechanics, we still do not
have a consensus on the process of collapse of wavefunctions. Some theories
require the intervention of a conscious observer while some see it as a
stochastic process, and most theories violate energy conservation. In this
paper we hypothesise that the collapse of wavefunctions can be caused by
interactions with other objects (macroscopic or microscopic) and energy is
conserved in that process. To test various hypotheses regarding collapse of
wavefunctions, we propose a model system which is the quantum analogue of a
classical soft-impact oscillator. We propose some alternative postulates
regarding the conditions for and the result of a collapse, and obtain the
implication of each on the behavior of observable quantities, which can
possibly be experimentally tested.

##  1 Introduction

Even though the theoretical structure of quantum mechanics is almost a century
old, we have very little understanding of an important component of that
theoretical structure—the collapse of the wavefunction. This concept is
invoked to explain the results of experiments like the double-slit experiment.
But it is largely ignored (or not needed) in most successful applications of
quantum mechanics [16, 13].

The mainstream Copenhagen interpretation of quantum mechanics claims that,
whenever an observation is made, the wavefunction of a quantum system
collapses instantly to an eigenstate of the observable being measured. Many
questions have been raised [7] regarding what constitutes an observation. “It
would seem that the theory is exclusively concerned about ‘results of
measurement’, and has nothing to say about anything else” wrote John Bell [4].
“What exactly qualifies some physical systems to play the role of ‘measurer’?
Was the wavefunction of the world waiting to jump for thousands of millions of
years until a single-celled living creature appeared? Or did it have to wait a
little longer, for some better qualified system $\cdots$ with a PhD?” These
questions have remained largely unanswered, and are considered to belong to
the domain of philosophy rather than physics.

The founders of quantum mechanics did not provide an explicit mechanism for
the collapse of the wavefunction and later it was termed as the measurement
problem. Einstein believed that a complete description of physical reality
would not be possible in this theory [9]. The famous Einstein-Podolsky-Rosen
gedankenexperiment [10] illuminated the non-local structure of quantum
entanglement and used it as an objection against the completeness of quantum
mechanics. Einstein’s convictions about the determinism of nature inspired
hidden variable theories which tried to rid quantum mechanics of
indeterminacy. John Bell [5] showed that local hidden variable theories are
incompatible with the predictions of quantum mechanics. Since then there have
been several attempts to develop theories that incorporate a mechanism for
wavefunction collapse.

Some have advocated for decoherence [20, 28] as a means to solve the
measurement problem [26]. Even though it explains the absence of macroscopic
superposition and the emergence of the classical world, it remains doubtful
whether it has solved the measurement problem—as the founders of decoherence
theory admit in their seminal papers [1, 19].

Dynamical collapse theories [3], on the other hand, supplant the unitary
evolution with stochasticity and nonlinearity in such a way that all
predictions of quantum mechanics are approximately reproduced in the
microscopic limit while precluding macroscopic superpositions like Schrödinger
cat states. A variety of collapse models have been developed that differ in
their localization basis: while [23, 6] and [2] use the energy, momentum and
spin basis respectively, the Ghirardi-Rimini-Weber (GRW) [11] and the
Continuous Spontaneous Localisation (CSL)[12] operate in the position basis.
The GRW model postulates wavefunction collapse as a random and spontaneous
process. The collapsed wavefunction has a Gaussian form which is in turn
dependent on certain natural constants that the theory introduces. Similar to
it is the CSL model, the only difference being that the wavefunction in CSL
collapses continuously in time [24]. This was followed by the Diosi-Penrose
model [25] which is based on The Quantum Mechanics With Universal Position
Localisation (QMUPL) [8] where gravity is responsible for the collapse of a
wavefunction. These models are also characterized by the noise they use for
generating the stochasticity they need to exhibit collapse.

John Bell had argued in favour of seeing the collapse of wavefunction as a
natural process: “If the theory is to apply to anything but highly idealised
laboratory operations, are we not obliged to admit that more or less
‘measurement-like’ processes are going on more or less all the time, more or
less everywhere? Do we not have jumping then all the time?” [4]

In this paper, we hypothesise that collapse of a wavefunction does not require
the intervention of a conscious observer. An interaction with another physical
object—macroscopic or microscopic—can also cause the collapse of a
wavefunction in the position space. The probability of occurrence of such an
interaction depends on the overlap between their probability density
functions.

If collapse-like processes indeed abound, as they reasonably should, the
violation in energy conservation should be quite observable. The absence of
such observations makes us hypothesise that any valid collapse mechanism must
satisfy energy conservation, at least on average.

In order for a collapse model to be admissible, one has to propose postulates
which have experimentally testable predictions. In this paper, we propose a
model which can be used to test such postulates about interaction-induced
wavefunction collapse, and work out the predictions of the various possible
postulates when applied to that system.

##  2 The model system

Fig. 1 shows the classical analog of the system under consideration—a simple
harmonic oscillator with mass $m$ and spring constant $k_{1}$ which can impact
with a massless wall, cushioned by a spring of constant $k_{2}$. The variable
$x$ is measured from the equilibrium position of the mass, and the wall is at
$x_{\rm wall}$ when the spring $k_{2}$ is relaxed.

Figure 1: The classical soft-impact oscillator.

The classical system, with the inclusion of damping and external forcing, is
known to exhibit a rich variety of dynamical phenomena which are initiated
when the mass grazes the wall [27, 18]. We choose this particular system as
the wall may be modeled either classically or quantum mechanically. This
allows us to investigate the collapse of the wavefunction due to interaction
with either a macroscopic or microscopic system.

The quantum version of the above system will be a particle in a potential
well, which is the same as the harmonic oscillator potential for $x\leq x_{\rm
wall}$ and is given by a different parabolic function for $x\geq x_{\rm
wall}$. The potential function of the system is given by

$V(x)=\left\\{\begin{array}[]{ll}\frac{1}{2}k_{1}x^{2},&x\leq x_{\rm wall}\\\
\frac{1}{2}k_{1}x^{2}+\frac{1}{2}k_{2}(x-x_{\rm wall})^{2},&x\geq x_{\rm
wall}\end{array}\right.$ (1)

We numerically solve the time-dependent Schrödinger equation for this system
using the finite difference method by dividing the range $[-30,30]$ of the
one-dimensional configuration space into 1,500 segments. We start from an
initial wavefunction, which is a Gaussian function centered at $x=-5.0$, and
standard deviation 1.0. This initial state corresponds, in the classical
picture, to releasing the mass from the point $x=-5.0$, which would
subsequently graze the wall located at $x=5.0$. The other parameters are taken
as $m=1$, $k_{1}=1$, $k_{2}=10$. All quantities in this work are in units
where $\hbar=1$. Fig. 2 shows snapshots of the dynamics of the wavefunction
for this system.

Figure 2: Evolution of the probability density distribution of the quantum
particle at four time instants, with $t_{1}<t_{2}<t_{3}<t_{4}$. Dashed line
indicates the equilibrium position of the wall. The potential function,
plotted in blue for the sake of visualization, is not to scale.

To investigate the dynamics in the phase space, we compute the Wigner function

$W(x,p)=\frac{1}{{\pi}\hbar}\int_{-\infty}^{\infty}\psi^{*}(x+y)\psi(x-y)e^{\frac{2ipy}{\hbar}}\,dy$
(2)

which gives a real valued function of the position and momentum, which varies
with time. Fig. 3 shows the plots of the Wigner function in the $x$–$p$ phase
space at four different time instants. Prolonged observation of both
$|\Psi^{2}|$ and the Wigner function shows that the time-evolution of the
system is aperiodic.

Figure 3: The evolution of the Wigner function at four time instants. The
dashed line indicates the equilibrium position of the wall

Although all information about the system is contained in the wavefunction, or
equivalently the Wigner quasiprobability distribution, it is hard to
characterize the type of dynamics using them. A time series of real values is
much more amenable to such analysis. Other investigators have used the
expectation value of an observable for this purpose [22, 17, 21]. Since
different states can have the same expectation value for an observable, we
have used a different quantity, the absolute value of the overlap of the
wavefunction at time $t$ with the initial wavefunction, given as

$\mathcal{O}(t)=\langle\psi(0)|\psi(t)\rangle.$

Figure 4: Plot of the overlap time series.

We find that, when the wall is placed far away from the particle, the behavior
is periodic, like that of a harmonic oscillator. But when it comes closer
(even when the classical system would not make any impact with the wall) the
evolution of the wavefunction becomes aperiodic. Fig. 4 shows the time series
of $\mathcal{O}(t)$ for a wall position corresponding to the classical grazing
condition. We have performed the 0–1 test for chaos [15], and have found that
the behavior is not chaotic. A Fourier transform (Fig. 5) reveals that there
are many discrete frequencies in its dynamics.

Figure 5: Frequency spectrum of the overlap time series.

Therefore, we conclude that, if the wavefunction is allowed to evolve
indefinitely following the Schrödinger equation, the evolution of the
wavefunction would be aperiodic, but is not chaotic as there is no sensitive
dependence on initial condition. Moreover, the frequency spectrum does not
have a continuum of frequencies and is a combination of a countable infinity
of discrete frequencies.

Figure 6: A plot of the first 250 eigenvalues.

A plot of the eigenvalues (Fig. 6) shows that for low values of $n$, the
eigenvalues coincide with those of the harmonic oscillator. But for larger
values of $n$, the graph has a different slope because eigenfunctions
corresponding to large eigenvalues, which would usually be spread over a large
region, cannot extend far beyond the position of the wall. An energy
measurement can yield any of these eigenvalues and the expectation value is
computed to be 13.125 GeV for our choice of initial state.

##  3 Collapse!

In the last section we considered a situation where the wavefunction evolves
solely following the Schrödinger equation. In this section we bring in a new
possibility. Since the wall can be considered to be a macroscopic object, an
interaction of the particle with the wall (classically, an impact) may amount
to a position measurement, which will cause the wavefunction to collapse.
Following a collapse, the wavefunction will continue to evolve according to
the Schrödinger equation until the next collapse. Thus, if this possibility is
considered, the evolution would contain unitary evolution as well as non-
unitary collapse processes.

Unlike the classical impact oscillator, the impact of the particle with the
wall will be a probabilistic event, guided by the pre-collapse wavefunction of
the particle. However, the present knowledge does not allow us to pinpoint a
unique algorithm with which the instant of collapse and the location of the
collapsed wavefunction can be simulated. So we posit different postulates
regarding the mechanism of collapse, and work out the implication of each.

Postulate 1:

If the probability of finding the particle beyond the position of the wall
exceeds a fraction $r$, i.e., if

$\int_{x_{\rm wall}}^{\infty}|\Psi|^{2}dx\geq r,\;\;\;\;\;\;r\in(0,1)$ (3)

then the wavefunction collapses to the position of the wall.

Postulate 2:

The same as Postulate 1, except that the number $r$ is not fixed, and is a
random number between 0 and 1.

Postulate 3:

The same as Postulate 1, except that the wavefunction collapses to a point
given by the pre-collapse probability distribution.

Postulate 4:

The same as Postulate 2, except that the wavefunction collapses to a point
given by the pre-collapse probability distribution.

(a) (b)

Figure 7: (a) Initial Gaussian with mean $-5.0$ and SD=1, (b) the post-
collapse wavefunction—a narrow Gaussian located at the position of the wall
$x=5.0$ and SD=0.25. The potential function is shown in blue (not to scale).

In the following simulations, the initial wavefunction is considered to be a
Gaussian function centred at $x=-5.0$ with standard deviation 1\. The wall is
placed at $x=5.0$, i.e., where the classical oscillator would experience
grazing. For the first and third postulates, the value of $r$ is taken as 0.5.
The post-collapse wavefunction is supposed to be an eigenfunction of the
position operator, i.e., a delta function. However, the numerical routine
would not work with such discontinuous functions. So we consider the post-
collapse wavefunction to be a narrow Gaussian function of standard deviation
0.25 (Fig. 7). The parameter values are taken as: mass of the quantum particle
$m=1$, spring constant of the spring attached to the mass $k_{1}=1$, the
spring constant corresponding to the soft wall $k_{2}=10$, time step $\delta
t=0.1$. We calculate for a total number of 10,000 time steps. The first 150
eigenfunctions have been used for the time evolution.

##  4 Results

Our objective here is to obtain testable predictions of the various possible
mechanisms of evolution of the wavefunction as outlined above. We focus on two
observables: energy and position.

###  4.1 Probability distribution of energy values

An energy measurement may return any energy eigenvalue presented in Fig. 6,
but the probability of finding each eigenvalue would be different for various
postulated situations. For the different postulates, the computed probability
distribution of the energy eigenvalues are presented in Fig. 8 and the
expectation values of energy are tabulated in Table 1.

Figure 8: Comparison of probability distributions for energy eigenvalues for the different collapse postulates.  Postulate |  |  | Expected energy
---|---|---|---
No collapse |  |  | 13.125 GeV
Postulate 1 |  |  | 14.75 GeV
Postulate 2 |  |  | 14.62 GeV
Postulate 3 |  |  | 11.46 GeV
Postulate 4 |  |  | 5.57 GeV
Table 1: Expectation values of energy in the five postulated situations.

Note that the expectation value of energy (Table 1) for unitary evolution of
the wavefunction depends on the initial wavefunction considered and those for
the four collapse postulates depend on the variance of the post-collapse
wavefunction. Since we have considered the standard deviation to be 0.25 in
all cases, one should pay attention to the relative magnitudes rather than the
absolute magnitudes of the expectation values.

The results presented in Fig. 8 and Table 1 show that, if experiments give the
expectation value of energy larger than what is predicted by standard quantum
mechanics for this system, then Postulate 1 or 2 may be true, and if they give
a lower value, Postulate 3 or 4 may be true.

###  4.2 Probability distribution of position values

We have computed the distribution of position values obtained through 10,000
time-steps. These are plotted in Fig. 9.

(a)
(b)
(c)
(d)
(e)

Figure 9: The averaged probability density functions of the position of the particle (a) without collapse, (b) for postulate 1, (c) for postulate 2, (d) for postulate 3, and (e) for postulate 4. Postulate |  |  | Mean |  |  | SD
---|---|---|---|---|---|---
No collapse |  |  | -0.2662 |  |  | 3.4963
Postulate 1 |  |  | 4.8217 |  |  | 1.1296
Postulate 2 |  |  | -0.2630 |  |  | 3.6354
Postulate 3 |  |  | -0.0657 |  |  | 3.1330
Postulate 4 |  |  | -0.0812 |  |  | 2.8818
Table 2: Mean and standard deviations of the time averaged PDFs for the
different collapse postulates.

It is found that if Postulate 1 is true, there is only one peak in the
probability distribution and in all other cases there are two peaks. For
unitary evolution without collapse, the two peaks are of almost the same
height while in the other cases they are of dissimilar heights. Table 2 gives
the mean and standard deviation of the distributions for the five cases shown
in Fig. 9.

##  5 The wall represented by another particle

In the previous section, we considered how a macroscopic object like a spring-
supported soft wall might bring about the collapse of the wavefunction of a
particle. We now ask: What happens if the wall is represented by another
microscopic particle? Can such an interaction between two quantum particles
lead to collapse of their wavefunctions?

We consider the soft wall to be made up of a particle of mass $m_{2}$,
oscillating in its own harmonic potential with spring constant $k_{2}$ (Fig.
10). Classically, this corresponds to the situation where the soft wall is no
longer massless. Each particle is in their own harmonic potential wells and
there is no interaction between them except when they meet. We assume that
they may meet when their wavefunctions have a significant overlap. We
postulate that an interaction of this kind can also lead to the collapse of
their wavefunctions, and work out the implications of such an assumption.

Figure 10: Classical analog of the two-particle system.

The Hamiltonian of the system is

$\hat{H}=-\frac{\hbar^{2}}{2m_{1}}\frac{\partial^{2}}{\partial{x_{1}}^{2}}-\frac{\hbar^{2}}{2m_{2}}\frac{\partial^{2}}{\partial{x_{2}}^{2}}+\frac{k_{1}x_{1}^{2}}{2}+\frac{k_{2}x_{2}^{2}}{2}$
(4)

where $k_{1}$ and $k_{2}$ are the spring constants of the two harmonic
oscillator potentials; $m_{1}$ and $m_{2}$ are the masses of the two
particles. Since there is no interaction term in this Hamiltonian, the state
of the two-particle system remains separable:

$\psi(x_{1},x_{2})=\psi_{1}(x_{1})\leavevmode\nobreak\ \psi_{2}(x_{2})$

The evolution is governed by their single-particle Schrödinger equations

$i\hbar\frac{\partial\psi_{1}}{\partial
t}=\left[-\frac{\hbar^{2}}{2m_{1}}\frac{\partial^{2}}{\partial{x_{1}}^{2}}+\frac{k_{1}x_{1}^{2}}{2}\right]\psi_{1}$
(5)

and

$i\hbar\frac{\partial\psi_{2}}{\partial
t}=\left[-\frac{\hbar^{2}}{2m_{2}}\frac{\partial^{2}}{\partial{x_{2}}^{2}}+\frac{k_{2}x_{2}^{2}}{2}\right]\psi_{2}$
(6)

except at instants when the wavefunctions collapse.

For numerical simulations, the harmonic potentials are taken to be centered at
$-2$ and $+2$, and the initial states of the two particles are Gaussians
centred at $-5$ and $+5$ respectively with a standard deviation of 1.

###  5.1 No collapse

Figure 11: Evolution of the two-particle probability density
$|\psi(x_{1},x_{2})|^{2}$ at four time instants for the no-collapse case. The
dashed line is the locus of the expected position.

According to standard quantum mechanics, a system collapses into an eigenstate
only when a measurement takes place. So, in the scenario we contemplate, the
standard formulation predicts no collapse. To compare our collapse theories
with the orthodox quantum mechanical prediction, we first simulate the system
without wavefunction collapse. The contour plots of the probability
distribution ($|\psi(x_{1},x_{2})|^{2}$) for such an evolution can be seen in
Fig. 11. The distribution periodically spreads and shrinks while its mean
value moves along the dashed line shown. The evolution is completely periodic
for rational ${\omega_{1}}/{\omega_{2}}$ (where $\omega=\sqrt{{k}/{m}}$) and
quasiperiodic otherwise.

###  5.2 Energy conserved collapse

A shortcoming of orthodox quantum mechanics and dynamical collapse theories
like GRW and CSL is that they violate conservation of energy. Dynamical
collapse theories make small but testable predictions for these violations
[14] which can be used to validate and narrow down the free parameters in
these models. Since no such violation has yet been reported, we invoke Occam’s
razor and assume that energy is indeed conserved. We hence postulate two
different collapse processes that respect the conservation of energy. The
first of the two processes conserves energy of each individual particle, while
the second conserves the total energy of the system through the collapse
process. These processes are detailed below. The expectation value of energy
is given by

$\langle
E\rangle=\langle\psi|\hat{H}|\psi\rangle=\int_{-\infty}^{\infty}\psi^{*}(x)\,\hat{H}\,\psi(x)\,\mathrm{d}x$

For a particle described by a Gaussian wavepacket centered at $a$ with SD

Figure 12: A Gaussian wavepacket in a translated harmonic potential. The
centers of the wavepacket and the potential are $a$ and $c$ repectively.

$\sigma$

$\psi(x)=\frac{1}{\sqrt{\sqrt{2\pi}\sigma}}\exp\left[{-\frac{(x-a)^{2}}{4\sigma^{2}}}\right]$

in the translated harmonic potential (Fig. 12) for which the Hamiltonian is

$\hat{H}=-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial
x^{2}}+\frac{1}{2}k(x-c)^{2},$

the expectation value of energy is

$\displaystyle\langle E\rangle=$
$\displaystyle\int_{-\infty}^{\infty}\psi^{*}(x)\,\hat{H}\leavevmode\nobreak\
\psi(x)\,\mathrm{d}x$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\frac{1}{\sigma\sqrt{2\pi}}\mathrm{e}^{-\frac{(x-a)^{2}}{4\sigma^{2}}}\leavevmode\nobreak\
\left[-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial
x^{2}}+\frac{1}{2}k(x-c)^{2}\right]\leavevmode\nobreak\
\mathrm{e}^{-\frac{(x-a)^{2}}{4\sigma^{2}}}\leavevmode\nobreak\ \mathrm{d}x$
$\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\frac{1}{2\sigma\sqrt{2\pi}}\mathrm{e}^{-\frac{(x-a)^{2}}{2\sigma^{2}}}\leavevmode\nobreak\
\left[-\frac{\hbar^{2}\left\\{(x-a)^{2}-2\sigma^{2}\right\\}}{4m\sigma^{4}}+k(x-c)^{2}\right]\leavevmode\nobreak\
\mathrm{d}x$

which can be integrated using Gaussian integrals to give

$\langle
E\rangle=\frac{k}{2}\left\\{\sigma^{2}+(a-c)^{2}+\frac{\hbar^{2}}{4mk\sigma^{2}}\right\\}.$

(a)

(b)

Figure 13: Dependence of the expected energy $\langle E\rangle$ on the mean
position $a$ and standard deviation $\sigma$ of the Gaussian wavepacket (gray)
and surface of constant energy $\epsilon$ (blue) (a), which intersects the
former surface along the curve in (b). $k=m=1$, $c=-2$.

The dependence of the expected energy on the mean $a$ and the standard
deviation $\sigma$ of the Gaussian wavepacket is plotted in Fig. 13(a). If the
expected energy is constrained to a certain value, say $\epsilon$, then the
possible values of $a$ and $\sigma$ get constrained to the intersection
between the surfaces $\langle E\rangle(a,\sigma)$ and $\langle
E\rangle=\epsilon$. The resulting family of curves parametrized by $\epsilon$
contain the families of Gaussian wavepackets having equal expected energy. An
energy conserved collapse model must map points from such a curve onto some
other point on the same curve. The curves in question are given by

$\sigma^{2}+(a-c)^{2}+\frac{\hbar^{2}}{4mk\sigma^{2}}-\frac{2\epsilon}{k}=0.$
(7)

It is seen that the resulting curves (Fig. 13(b)) do not extend over the
entire domain of the mean position, $a$. Coupled with the fact that Gaussian
wavepackets are nonzero everywhere, this implies that an energy conserved
collapse to a Gaussian wavepacket in a harmonic potential imposes a constraint
on the locations where the wavefunctions can collapse, even though the
probability of finding both particles at these points may be non-zero. In that
sense, the energy conserved collapse postulate does not strictly follow Born’s
rule.

The solution of equation (7) produces the two possible values of the variance
that a Gaussian wavepacket with expected energy $\epsilon$ and centered at $a$
must have

$\sigma^{2}=\frac{\epsilon}{k}+\frac{(a-c)^{2}\pm\sqrt{\left(\frac{2\epsilon}{k}-(a-c)^{2}\right)^{2}-\frac{\hbar^{2}}{mk}}}{2}$
(8)

####  5.2.1 Individual particles conserve energy

Figure 14:  Time evolution of the probability distribution of the two
particles at 6 time instants. The first and second particles are represented
in blue and orange respectively, as are the centres of their respective
harmonic potentials using dashed lines. Notice that the overlap between the
distributions shown above the figures achieves a significant value in the
fourth instant, triggering a collapse in the fifth instant.

The criteria we use for collapse is akin to the case of the soft impact
oscillator. At a time instant $t_{i}$, the probability of collapse is the
overlap between the densities of the two particles.

$P(t_{i})=\int_{\infty}^{\infty}|\psi_{1}(x,t_{i})|^{2}\leavevmode\nobreak\
|\psi_{2}(x,t_{i})|^{2}\leavevmode\nobreak\ \text{d}x$ (9)

For each particle to conserve energy individually, their initial expected
energy has to match their expected energy after collapse. From Fig. 13(b) it
is apparent that in order to conserve energy, the wavefunction cannot collapse
at a position beyond the extent of the closed curve along $a$. Within this
limited domain, a collapse can follow Born’s rule.

$P(x)=\frac{{\lvert\psi(x)\rvert}^{2}}{\displaystyle\int_{a_{\text{min}}}^{a_{\text{max}}}{\lvert\psi(x^{\prime})\rvert}^{2}\leavevmode\nobreak\
\text{d}x^{\prime}}\leavevmode\nobreak\
,\;x\in[a_{\text{min}},a_{\text{max}}]$ (10)

The collapse process is stochastic in both position and time, governed by
$P(x)$ and $P(t_{i})$. When a collapse does occur, the wavefunctions of both
particles collapse simultaneously. For a particular position of collapse $a$,
the energy conservation constraint allows for two different values of standard
deviation $\sigma$ for the post collapse wavepacket (see Fig. 13(b)). We
choose the smaller of the two to achieve maximal localization in position.
Between collapses, the two particles undergo unitary evolution according to
their respective Schrödinger equations. The resulting dynamics are depicted in
terms of a few frames at different time instants in Fig. 14.

####  5.2.2 The system as a whole conserves energy

Now we consider the case where each individual particle might lose or gain
energy but the expected energy of the two-particle system remains conserved
through the process of collapse, i.e.,

$\langle E\rangle_{1}+\langle E\rangle_{2}=\langle
E\rangle_{\text{total}}=\text{constant.}$

The conditions for collapse are calculated exactly as in the previous
scenario. $P(t_{i})$ and $P(x)$ dictate when and where the collapse happens.
In this case, however, the energy must also be redistributed between the
particles.

We chose to partition the total energy between the two particles randomly. The
minimum allowed energy for a particle in a harmonic oscillator potential is
the ground state energy. Hence, an allowed partition must allot at least the
respective ground state energy to each particle. This gives us an allowed
range in energy for the partition. The partition is realized as a sample from
a uniform distribution in the allowed energy range (Fig. 15).

Figure 15: Schematic of the partitioning of energy between the two particles
at the time of collapse. The length of the total line segment represents the
total energy. The point of partition is chosen randomly on the allowed
segment. Figure 16: Expectation value of energy for the first particle (blue),
second particle (orange) and the total system (black) for 1000 time steps.
Energy is shuttled back and forth between the two particles, all the while
conserving the total energy.

Fig. 16 shows that the expectation values of energy of the particles vary in
steps whenever collapses happen, while the total energy remains conserved.
This behavior may be experimentally testable.

(a)

(b)

(c)

Figure 17: The time-averaged probability density functions of position of the
two particles for the cases of (a) no collapse, (b) collapse with individual
particles conserving energy, (c) collapse with total system conserving energy.
The particles are distinguishable with the parameters :
$k_{1}=k_{2}=m_{1}=m_{2}=1$. The centres of the respective potentials,
$c_{1}=-2$ and $c_{2}=2$ are marked with dashed lines.

###  5.3 Comparison of the three cases

To quantify the differences between the three cases considered, we plot the
time-averaged probability density in position for the three cases in Fig. 17.
Without collapse, the average density functions have a two-humped form and are
symmetric for both particles. For both the collapse postulates, the densities
have single peaks and are very similar. For the case where the individual
energies are conserved, the distribution functions are more skewed (i.e., the
probability of finding the particles away from each other is higher) than the
case where the total energy is conserved. An observed increase in the average
distance between the particles can serve as a testable confirmation of the
postulated collapse processes. The amount of increase can narrow down the type
of interaction.

(a)

(b)

Figure 18: The energy conserved collapse scenario when the two particles have
dissimilar parameters: $k_{1}=m_{1}=1$, $k_{2}=10$ and $m_{2}=0.1$. (a)
individual particles conserving energy and (b) total system conserving energy.

Fig. 18 shows the case when the $m$ and $k$ values of the two oscillators are
different. In this case the qualitative behavior is similar, but the density
function of the particle of lower mass has an almost flat top.

##  6 Conclusion

In this work we have formulated a few alternate postulates for the collapse of
the wavefunction. We assume that collapse of the wavefunction of a particle
does not depend on the intervention of a conscious observer. Instead, its
interaction with another object, classical or quantum, may collapse its
wavefunction. The probability of the occurrence of such a collapse depends on
the overlap between the wavefunctions of the interacting entities.

For the situation where the particle interacts with a classical object, we
have formulated four different postulates regarding the condition of collapse
and the post-collapse wavefunction. For interaction among two distinguishable
quantum particles, we have formulated two postulates of energy conserving
wavefunction collapse.

We have proposed a model system—the quantum version of a soft-impact
oscillator—and have obtained testable predictions from each postulate
regarding the energy and position distribution in such a system. Our
computations predict that, if an interaction with a classical object induces
collapse of the wavefunction, then the probability distributions of energy and
position would be different from what is predicted by standard quantum
mechanics. If an interaction between two distinguishable quantum particles can
induce energy-conserving collapse of their wavefunctions, then the average
distance between them would tend to be larger than the distance between the
centres of the two potential functions. Experimental test of the predictions
would enable us to eliminate the wrong postulates.

Notice that, if any of the postulates regarding collapse of the wavefunction
is supported by experiment, it will have important consequence in the
foundation of quantum mechanics. It will imply that collapse of a wavefunction
is a natural process that does not require conscious observation.

## References

  * Adler [2003] Stephen L Adler.  Why decoherence has not solved the measurement problem: a response to P W Anderson.  _Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics_ , 34(1):135–142, 2003.  doi: 10.1016/S1355-2198(02)00086-2.
  * Bassi and Ippoliti [2004] Angelo Bassi and Emiliano Ippoliti.  Numerical analysis of a spontaneous collapse model for a two-level system.  _Physical Review A_ , 69(1):012105, 2004.  doi: 10.1103/PhysRevA.69.012105.
  * Bassi et al. [2013] Angelo Bassi, Kinjalk Lochan, Seema Satin, Tejinder P Singh, and Hendrik Ulbricht.  Models of wave-function collapse, underlying theories, and experimental tests.  _Reviews of Modern Physics_ , 85(2):471, 2013\.  doi: 10.1103/RevModPhys.85.471.
  * Bell [1990] John Bell.  Against ‘measurement’.  _Physics world_ , 3(8):33, 1990.  doi: 10.1088/2058-7058/3/8/26.
  * Bell [1964] John S Bell.  On the Einstein Podolsky Rosen paradox.  _Physics Physique Fizika_ , 1(3):195, 1964.  doi: 10.1103/PhysicsPhysiqueFizika.1.195.
  * Benatti et al. [1988] Fabio Benatti, Gian Carlo Ghirardi, Alberto Rimini, and Tullio Weber.  Operations involving momentum variables in non-Hamiltonian evolution equations.  _Il Nuovo Cimento B (1971-1996)_ , 101(3):333–355, 1988.  doi: 10.1007/BF02828713.
  * DeWitt [2015] Bryce S DeWitt.  Quantum mechanics and reality.  In _The many worlds interpretation of quantum mechanics_ , pages 155–166. Princeton University Press, 2015.  doi: 10.1515/9781400868056.
  * Diósi [1989] Lajos Diósi.  Models for universal reduction of macroscopic quantum fluctuations.  _Physical Review A_ , 40(3):1165, 1989.  doi: 10.1103/PhysRevA.40.1165.
  * Dolling et al. [2003] Lisa M Dolling, Arthur F Gianelli, and Glenn N Statile.  _The tests of time: Readings in the development of physical theory_.  Princeton University Press, 2003.  doi: 10.2307/j.ctvcm4h07.
  * Einstein et al. [1935] Albert Einstein, Boris Podolsky, and Nathan Rosen.  Can quantum-mechanical description of physical reality be considered complete?  _Physical review_ , 47(10):777, 1935.  doi: 10.1103/PhysRev.47.777.
  * Ghirardi et al. [1986] Gian Carlo Ghirardi, Alberto Rimini, and Tullio Weber.  Unified dynamics for microscopic and macroscopic systems.  _Physical review D_ , 34(2):470, 1986.  doi: 10.1103/PhysRevD.34.470.
  * Ghirardi et al. [1990] Gian Carlo Ghirardi, Philip Pearle, and Alberto Rimini.  Markov processes in Hilbert space and continuous spontaneous localization of systems of identical particles.  _Physical Review A_ , 42(1):78, 1990.  doi: 10.1103/PhysRevA.42.78.
  * Ghirardi and Bassi [2020] Giancarlo Ghirardi and Angelo Bassi.  Collapse Theories.  In Edward N. Zalta, editor, _The Stanford Encyclopedia of Philosophy_. Metaphysics Research Lab, Stanford University, Summer 2020 edition, 2020.
  * Goldwater et al. [2016] Daniel Goldwater, Mauro Paternostro, and PF Barker.  Testing wave-function-collapse models using parametric heating of a trapped nanosphere.  _Physical Review A_ , 94(1):010104, 2016.  doi: 10.1103/PhysRevA.94.010104.
  * Gottwald and Melbourne [2009] Georg A Gottwald and Ian Melbourne.  On the implementation of the 0–1 test for chaos.  _SIAM Journal on Applied Dynamical Systems_ , 8(1):129–145, 2009.  doi: 10.1137/080718851.
  * Griffiths and Schroeter [2018] David J. Griffiths and Darrell F. Schroeter.  _Introduction to Quantum Mechanics_.  Cambridge University Press, 3 edition, 2018.  doi: 10.1017/9781316995433.
  * Haake et al. [1992] Fritz Haake, Harald Wiedemann, and Karol Życzkowski.  Lyapunov exponents from quantum dynamics.  _Annalen der Physik_ , 504(7):531–539, 1992.  doi: 10.1002/andp.19925040706.
  * Ing et al. [2008] James Ing, Ekaterina Pavlovskaia, Marian Wiercigroch, and Soumitro Banerjee.  Experimental study of impact oscillator with one-sided elastic constraint.  _Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences_ , 366(1866):679–705, 2008.  doi: 10.1098/rsta.2007.2122.
  * Joos and Zeh [1985] Eric Joos and H Dieter Zeh.  The emergence of classical properties through interaction with the environment.  _Zeitschrift für Physik B Condensed Matter_ , 59(2):223–243, 1985.  doi: 10.1007/BF01725541.
  * Joos et al. [2013] Erich Joos, H Dieter Zeh, Claus Kiefer, Domenico JW Giulini, Joachim Kupsch, and Ion-Olimpiu Stamatescu.  _Decoherence and the appearance of a classical world in quantum theory_.  Springer Science & Business Media, 2013.  doi: 10.1007/978-3-662-05328-7.
  * Laha et al. [2020] Pradip Laha, S Lakshmibala, and V Balakrishnan.  Bifurcations, time-series analysis of observables, and network properties in a tripartite quantum system.  _Physics Letters A_ , 384(23):126565, 2020.  doi: 10.1016/j.physleta.2020.126565.
  * Mendes [1991] R Vilela Mendes.  Sensitive dependence and entropy for quantum systems.  _Journal of Physics A: Mathematical and General_ , 24(18):4349, 1991.  doi: 10.1088/0305-4470/24/18/021.
  * Milburn [1991] GJ Milburn.  Intrinsic decoherence in quantum mechanics.  _Physical Review A_ , 44(9):5401, 1991.  doi: 10.1103/PhysRevA.44.5401.
  * Pearle [1989] Philip Pearle.  Combining stochastic dynamical state-vector reduction with spontaneous localization.  _Physical Review A_ , 39(5):2277, 1989.  doi: 10.1103/PhysRevA.39.2277.
  * Penrose [1996] Roger Penrose.  On gravity’s role in quantum state reduction.  _General relativity and gravitation_ , 28(5):581–600, 1996.  doi: doi.org/10.1007/BF02105068.
  * Pessoa [1997] Osvaldo Pessoa.  Can the decoherence approach help to solve the measurement problem?  _Synthese_ , 113(3):323–346, 1997.  URL http://www.jstor.org/stable/20117696.
  * Peterka and Tondl [2004] František Peterka and Aleš Tondl.  Phenomena of subharmonic motions of oscillator with soft impacts.  _Chaos, Solitons & Fractals_, 19(5):1283–1290, 2004.  doi: 10.1016/S0960-0779(03)00335-7.
  * Schlosshauer [2007] Maximilian A Schlosshauer.  _Decoherence: and the quantum-to-classical transition_.  Springer Science & Business Media, 2007.  doi: 10.1007/978-3-540-35775-9.