11institutetext: Indian Institute of Technology, Patna 22institutetext: Indian
Institute of Technology (Banaras Hindu University), Varanasi

# Broad Neural Network for Change Detection in Aerial Images

Shailesh Shrivastava 11 Alakh Aggarwal and Pratik Chattopadhyay 2222

##  1 Abstract

A change detection system takes as input two images of a region captured at
two different times, and predicts which pixels in the region have undergone
change over the time period. Since pixel-based analysis can be erroneous due
to noise, illumination difference and other factors, contextual information is
usually used to determine the class of a pixel (changed or not). This
contextual information is taken into account by considering a pixel of the
difference image along with its neighborhood. With the help of ground truth
information, the labeled patterns are generated. Finally, Broad Learning
classifier is used to get prediction about the class of each pixel. Results
show that Broad Learning can classify the data set with a significantly higher
F-Score than that of Multilayer Perceptron. Performance comparison has also
been made with other popular classifiers, namely Multilayer Perceptron and
Random Forest.

##  2 Introduction

Change detection in images has become one of the important aspects in recent
years, with a wide range of applications. Various aspects include detection of
changes in motion, change in vegetation cover over an area, extent of damage
caused due to floods and other natural calamities, etc. The basic objective of
change detection is to identify dissimilar regions by comparing a pair of the
images of the same area at different points of time.

Over the past few years, a lot of research work has been carried out in this
area, and several supervised, unsupervised and semi-supervised learning
techniques have been developed, each with some pros and cons. While
unsupervised learning does not need any training information, supervised
learning techniques require substantial amount of ground truth knowledge,
which may not be conveniently available always. On the other hand, the semi-
supervised technique is efficient in a way that it requires only a few labeled
data for training. The few labeled patterns of the changed class can help in
training a classifier by means of self-training.

Recently Broad Learning has been proposed in [1] as an alternative to Deep
Learning. It has been seen that, Broad Learning is immensely successful in
classifying image feature sets, and in some cases it performs even better than
that of Deep Learning. Moreover, training using Broad Learning is
significantly faster than that of Deep Learning. In this work, we use a
variation of the Broad Learning algorithm proposed in [1], and compare its
results with the Multilayer Perceptron and Random Forest classifiers.

##  3 Literature Survey

A change detection method essentially does comparison between two images to
differentiate between changed and unchanged pixels. Early methods are mostly
threshold-based in which a difference image $D$ is first computed from the two
input images say, $X$ and $Y$, and a threshold is applied on the difference
image to detect the changes. The threshold is chosen empirically as explained
by Rosin [2],[3]. This is referred to as‘simple differencing’ [4]. Two
different techniques are usually employed to generate the difference image:
Change Vector Analysis (CVA) [5],[6],[7] and Image ratioing [8]. While the
former uses the modulus of the difference between the feature vectors of the
each pixel, the latter uses the ratio of the pixel intensities to generate the
difference image. However, manual selection of threshold is erroneous and is
not robust in presence of changes in illumination, and noise.

In view of the two hypothesis used in change detection algorithms, i.e., the
null hypothesis $H_{0}$ corresponding to a no-change decision and the
alternative hypothesis $H_{1}$ corresponding to a change decision at a given
pixel, the hypothesis to be chosen to best describe the extent of change at a
given pixel (say, _n_) is based upon the conditional joint probability density
functions $p(X(n),Y(n)|H\textsubscript{0})$ and
$p(X(n),Y(n)|H\textsubscript{1})$, based upon the classical framework of
hypothesis testing [9],[10].

Later work on change detection include [11] in which pixels are softly
classified into mixture components for different change models by employing
the Expectation Maximization algorithm, The algorithm computes the optical
flow field between a pair of images. This approach seems to be an improvement
over the previous techniques as it estimates mis-alignment, as well as the
illumination variation between the input images. Another change detection
algorithm based on the concept of ’self-consistency’ among the regions of an
image is proposed in [12]. This approach uses the Minimum Description Length
(MDL) model [13] for selecting the hypothesis that best describes the image
pair.

More sophisticated change detection algorithms make use of the spatial and
temporal relation between a pixel and its neighboring pixels. The spatial
models consider a polynomial function of a pixel coordinate $n$ upon which the
intensity values of a block of pixels is fitted. Constant, linear and
quadratic models are used on these blocks of pixels via generalized likelihood
ratio tests [14],[15]. The threshold value can be obtained by t-test for
constant models and F-test for linear or quadratic models. Out of the three
models, the quadratic model is found to be a better option compared to the
constant and linear models due to its higher confidence in prediction.

Numerous temporal models have been proposed for detecting changes in sequence
of images. Variation of pixel intensities over time is modeled as an auto-
regressive process [16]. Mean, variance and correlation coefficient were
estimated and used in likelihood ratio tests where the null hypothesis depicts
that the image intensities are dependent and the alternative hypothesis
depicts them as independent [17].

Another algorithm known as the Wallflower algorithm uses a Wiener filter for
predicting a pixel’s value from it previous values [18]. Pixels are classified
as changed if the prediction error is much more than the expected error. This
algorithm can also be considered as a background estimation algorithm. Due to
low accuracy of the linear models, nonlinear models are proposed to study the
relationship between images [19]. The optimal nonlinear function is the
conditional expected intensity value $X(n)$ given $Y(n)$. An unsupervised
approach to estimate the parameters of the nonlinear predictor is proposed
using adaptive neural networks [20]. Those pixels on which the predictor
performed poorly are classified as changed.

The shading models are used in various change detection techniques to produce
illumination-invariant algorithms. These algorithms computed ratio of the
intensity of corresponding pixels in the two images as given by:

$R(n)=\frac{X(n)}{Y(n)}$ (1)

and compare it to an empirically determined threshold. Instead of directly
comparing the ratio $R(n)$, a new form of the difference image was introduced
in [21] :

$D(n)=\frac{1}{N}\sum_{l\in\Omega\textsubscript{n}}{(R(n)-\mu\textsubscript{n})}^{2},$
(2)

where $N$ is the number of pixels in a block of pixels $\Omega_{n}$, $l$ is a
pixel in the pixel block $\Omega_{n}$, and $\mu_{n}$ is given by Equation 3:

$\mu\textsubscript{n}=\frac{1}{N}\sum_{l\in\Omega\textsubscript{n}}{R(n)}.$
(3)

If $D(n)$ is greater than the threshold, it is considered as a changed pixel.

A change detection algorithm based on the shading model is introduced in [22]
that uses the weighted combinations of difference in intensities and textures
and the intensity ratio. This algorithm assumes that the texture information
is less sensitive to variations in illuminations as compared to intensity
difference. Instead of using the intensity ratios, the technique in [23]
compares the circular shift moments to detect the changes in image. The
reflectance component of the intensity is represented by these moments and are
claimed to capture the details of an image more precisely.

An algorithm using semi-supervised Multilayer Perceptron was employed in [24]
for detecting changes in remotely sensed images. Here, the difference image is
generated by the CVA technique using the difference operator. The input
pattern for each pixel is generated using the pixel intensity of the
difference image as well as those within a spatial neighbourhood. Thus the
input vectors contain nine components with gray values of a pixel and gray
values of its eight neighbouring pixels. A few initial labeled patterns were
identified automatically using the K-means clustering algorithm. A neural
network is trained using these labeled patterns initially. Since the input
patterns have nine pixel values, so the input layer has nine units and 1 bias
unit. The output layer has two units for the unchanged and changed classes.
The remaining unlabeled patterns are processed by the perceptron in a semi-
supervised manner by obtain a soft class label for each unlabeled pattern in
every iteration, and next using these soft labels along with the ground truth
to train the network repeatedly till optimal condition is reached. This
approach is highly time-intensive, and suffers from the disadvantage that
incorrect prediction at an initial iteration, propagates to the successive
iterations as well.

Another change detection technique for remotely sensed images is proposed in
[25] that carries out fusion of spectral and statistical indices in an
unsupervised manner. The spectral changes are taken into account via the CVA
technique, while the statistical changes are identified using a similarity
index based on Kullback Leibler distance. The spectral change information and
similarity index are fused using wavelets. A neuro-fuzzy classifier next
classifies the fused image into changed and unchanged classes, overcoming
mixed pixel problems of the satellite images.

In this work, we aim to study the performance of Broad Learning in classifying
the difference image. It has been shown in previous studies [1] that Broad
Learning is faster and also has a similar (sometimes better) level of accuracy
as that of Deep Learning in classifying image feature sets. Moreover, Broad
Learning algorithm is incremental, i.e., with the availability of more ground
truth data, training of the classifier can be accomplished very fast.

##  4 Proposed Approach

A block diagram of the proposed approach is explained using the block diagram
shown in Fig. 1.

Figure 1: System overview of change detection from a pair of aerial images
captured at different time instants

###  4.1 Input Pattern Generation

Consider two co-registered and geometrically corrected RGB images of the same
place taken at two different times. Since L*a*b* color space is known to be
illumination invariant, the two images are initially transformed from RGB
color space to L*a*b* color space. For generating the input pattern, a pixel
is chosen along with its eight neighboring pixels. Thus, the input feature for
each pixel is a vector in nine dimensional feature space. The difference image
is generated via absolute simple difference of the two images for each
channel.

A general characteristic of change detection data set is that, the ground
truth consists of large samples from unchanged class and very few samples of
changed class. Training any learning algorithm with this data set will cause a
high bias for the unchanged class. To account for this problem, we study two
different cost-sensitive data adjustment techniques. First the unchanged class
of the training is down-sampled using random down-sampling. Next the changed
class is up-sampled using a random up-sampling technique, namely, $Synthetic$
$Minority$ $Over$ $Sampling$ $Technique$ ($SMOTE$).

We construct various data sets with different $Imbalance$ $Ratios$ (i.e.,
ratio of the number of majority to the number of minority samples) and study
the performance of the classifier for each such case.

###  4.2 Classification using Broad Learning

Broad Learning network [1] is a recently proposed two-layer deep neural
network which consists of input feature map nodes in the input layer, and
enhancement nodes in the hidden layer. The original version of Broad Learning
has been seen to perform with significant accuracy in classifying some image
data sets. Moreover, training using Broad Learning is very fast. Detailed
architecture of the network, along with its working principle is given in [1].
One drawback of the approach in [1] is that, it does not specify the number of
enhancement nodes to be added for optimum performance. This is important since
we have limited memory space, and adding enhancement nodes without bounds
would cause the memory to overflow.

So, we propose a modification to the existing architecture of the Broad
Learning System. We continue to recursively add new layer of enhancement nodes
till a desired optimum condition is reached. Sparse Autoencoder is employed to
find new layer of enhancement nodes at each recursion level with a pre-
specified compression rate. Finally, all the generated layers are concatenated
with the input layer. The weights from the set of input and enhancement layers
to the output layer is next determined using Moore-Penrose Psuedo-inverse,
similar to that in [1]. The scheme is explained with the help of Fig. 2.

Figure 2: Broad Learning Architecture

At each step after adding a new layer of enhancement nodes, we compute the
cross-validated average F-Score on classifying the training set. Addition of
enhancement nodes is terminated once the average F-Score values in two
successive iteration is less than a small threshold value. At this point,
further addition of new enhancement nodes does not improve classification
accuracy on training set any further, but the network tends to get over-
fitted.

##  5 Results

Evaluation of our algorithm is done on a publicly available data set for
change detection 111{https://computervisiononline.com/dataset/1105138664}
which consists of 1000 pairs of 800$\times$600 images, each pair consisting of
one reference image and one test image, and the 1000 corresponding
800$\times$600 ground truth masks. The images were rendered using the
realistic rendering engine of the serious game Virtual Battle Station 2,
developed by Bohemia Interactive Simulations. The data set consists of 100
different scenes containing several objects (trees, buildings, etc.) and
moderate ground relief. Using the reference and test images, the difference
image is created and the ground truth masks are used for labeling the data
set, and final evaluation of the classifier. To construct the training and
test sets, from the ground truth, we randomly select 70% samples from each
class as the training set and remaining 30% for test set.

We study the effect of various sampling techniques as described above, as well
as the effect of decreasing imbalance ratio on the classification performance
of Broad Learning. Table 1 shows the F-scores for the two classes along with
the Average F Score using the proposed Broad Learning neural network.

Table 1: Table showing the F-scores for the two classes along with the Average
F Score using the Broad Learning Network Broad Learning approach
---
Imbalance Ratio |  Random Undersampling Random Oversampling |  Random Undersampling SMOTE
IR |  Layers |  Compression |  AFS |  F0 |  F1 |  Layers |  Compression |  AFS |  F0 |  F1
1:1 |  3 |  0.9 |  75.8 |  80 |  72 |  3 |  0.9 |  81.6 |  81.6 |  81.6
|  3 |  0.7 |  70 |  73 |  67 |  3 |  0.7 |  81.5 |  81.4 |  81.6
|  5 |  0.9 |  76 |  80 |  72 |  5 |  0.9 |  81.6 |  81.6 |  81.6
|  5 |  0.7 |  70 |  73 |  67 |  5 |  0.7 |  81.5 |  81.4 |  81.6
2:1 |  3 |  0.9 |  79 |  88 |  63 |  3 |  0.9 |  79 |  85 |  66
|  3 |  0.7 |  74 |  88 |  55 |  3 |  0.7 |  77 |  84 |  63
|  5 |  0.9 |  79 |  88 |  63 |  5 |  0.9 |  79 |  85 |  66
|  5 |  0.7 |  74 |  88 |  55 |  5 |  0.7 |  77 |  84 |  63
10:1 |  3 |  0.9 |  90 |  96 |  27 |  3 |  0.9 |  88 |  95 |  10
|  3 |  0.7 |  88 |  95 |  10 |  3 |  0.7 |  87 |  95 |  4
|  5 |  0.9 |  90 |  96 |  27 |  5 |  0.9 |  88 |  95 |  10
|  5 |  0.7 |  88 |  95 |  10 |  5 |  0.7 |  87 |  95 |  4
50:1 |  3 |  0.9 |  97 |  99 |  8 |  3 |  0.9 |  0 |  99 |  0
|  3 |  0.7 |  0 |  99 |  0 |  3 |  0.9 |  0 |  99 |  0
|  5 |  0.9 |  97 |  99 |  8 |  5 |  0.9 |  0 |  99 |  0
|  5 |  0.7 |  0 |  99 |  0 |  5 |  0.7 |  0 |  99 |  0
100:1 |  3 |  0.9 |  99 |  99 |  1 |  3 |  0.9 |  0 |  99 |  0
|  3 |  0.7 |  0 |  100 |  0 |  3 |  0.7 |  0 |  100 |  0
|  5 |  0.9 |  99 |  99 |  1 |  5 |  0.9 |  0 |  99 |  0
|  5 |  0.7 |  0 |  100 |  0 |  5 |  0.7 |  0 |  100 |  0
250:1 |  3 |  0.9 |  0 |  100 |  0 |  3 |  0.9 |  0 |  100 |  0
|  3 |  0.7 |  0 |  100 |  0 |  3 |  0.7 |  0 |  100 |  0
|  5 |  0.9 |  0 |  100 |  0 |  5 |  0.9 |  0 |  100 |  0
|  5 |  0.7 |  0 |  100 |  0 |  5 |  0.7 |  0 |  100 |  0

In the table, AFS represents the Average F Score, F0 is the F Score of class 0
and F1 is the F score of class 1. We observe the effect of varying the number
of layers of Enhancement nodes as well as the compression rate while
generating a new layer of enhancement node. The set of enhancement nodes
generated at each recursion level is termed as a layer, while compression is
the ratio of number of nodes in the newly generated layer to that of the
previous layer.

From the table, we conclude that SMOTE gives better results and is more
effective than random over-sampling. Also, as expected, F-score for minority
class (class 1) tends to zero as the imbalance ratio in data set increases. It
is seen that, the Broad learning classifier is able to perform reasonably well
even with the imbalance ratio of 10:1. Also, a higher compression results in
more number of nodes in the enhancement layer, and hence, fits the data well.
However, the effect of varying the number of layers is not well observed from
the data set used in the study. Fig 3 shows the changed map obtained with two
pairs of input images using the Broad Learning classifier with 10:1 imbalance
ratio.

Figure 3: (a) and (b) the two input images, (c) ground truth, (d) predicted
changed map from 10:1 imbalanced ratio

The first two images shown in Figures 3(a) and (b) are aerial images of the
same place captured at different times, as obtained from the database. Figure
3(c) shows the ground truth, while Figure 3(d) shows the change detection map
obtained by using the proposed method. Visual observation shows that most of
the changed regions are correctly identified by our algorithm, although there
exists few false positive cases possibly occurring due to noise/variation of
illumination conditions.

We also compare the performance of our approach with two other popularly used
classifiers, namely, Random Forest and Multilayer Perceptron. Table 2 shows
the F scores for the two classes along with the Average F Score for both the
class using the Random Forest Classifier.

Table 2: Table showing the F scores for the two classes along with the Average
F-Score using the Random Forest Classifier Random Forest Classifier
---
Imbalance Ratio |  Random Undersampling Random Oversampling |  Random Undersampling SMOTE
IR |  Trees |  AFS |  F0 |  F1 |  Trees |  AFS |  F0 |  F1
1:1 |  5 |  43.39 |  60.88 |  17.89 |  5 |  89.29 |  89.99 |  88.59
|  10 |  37.69 |  67.58 |  7.81 |  10 |  89.40 |  90.17 |  88.63
2:1 |  5 |  52.42 |  82.07 |  22.77 |  5 |  89.21 |  93.38 |  85.03
|  10 |  46.80 |  81.07 |  12.53 |  10 |  88.07 |  92.88 |  83.26
10:1 |  5 |  61.76 |  95.92 |  27.60 |  5 |  82.55 |  97.35 |  67.75
|  10 |  56.99 |  95.68 |  18.31 |  10 |  80.69 |  97.21 |  64.17
50:1 |  5 |  64.66 |  99.14 |  30.18 |  5 |  68.98 |  99.12 |  38.84
|  10 |  61.99 |  99.12 |  24.85 |  10 |  66.24 |  99.12 |  33.36
100:1 |  5 |  62.56 |  99.53 |  25.59 |  5 |  63.77 |  99.48 |  28.06
|  10 |  60.08 |  99.54 |  20.62 |  10 |  59.48 |  99.50 |  19.46
250:1 |  5 |  58.98 |  99.78 |  18.18 |  5 |  56.19 |  99.78 |  12.60
|  10 |  56.51 |  99.80 |  13.22 |  10 |  51.71 |  99.79 |  3.64

Table 3 shows the corresponding results for Multilayer Perceptron classifier.

Table 3: Table showing the F scores for the two classes along with the Average
F-Score using MLP Multi Layer Perceptron
---
Imbalance Ratio |  Random Undersampling Random Oversampling |  Random Undersampling SMOTE
IR |  Alpha |  AFS |  F0 |  F1 |  Alpha |  AFS |  F0 |  F1
1:1 |  0.5 |  0 |  66.67 |  0 |  0.5 |  33.34 |  66.65 |  0.02
|  1 |  36.26 |  5.24 |  67.27 |  1 |  41.48 |  14.51 |  68.45
2:1 |  0.5 |  25.03 |  0.06 |  50.01 |  0.5 |  0 |  80.00 |  0
|  1 |  39.01 |  24.85 |  53.18 |  1 |  0 |  80.00 |  0
10:1 |  0.5 |  0 |  95.24 |  0 |  0.5 |  0 |  95.24 |  0
|  1 |  0 |  95.24 |  0 |  1 |  0 |  95.24 |  0
50:1 |  0.5 |  0 |  99.01 |  0 |  0.5 |  0 |  99.01 |  0
|  1 |  0 |  99.01 |  0 |  1 |  0 |  99.01 |  0
100:1 |  0.5 |  0 |  99.50 |  0 |  0.5 |  0 |  99.50 |  0
|  1 |  0 |  99.50 |  0 |  1 |  0 |  99.50 |  0
250:1 |  0.5 |  0 |  99.80 |  0 |  0.5 |  0 |  99.80 |  0
|  1 |  0 |  99.80 |  0 |  1 |  0 |  99.80 |  0

In each cell, value of ‘0’ implies that there is no correct prediction for
that class. The above results show that both Broad Learning and Random Forest
handle imbalanced data better than a Multilayer Perceptron. Table 2 shows that
performance of Random Forest classifier is almost consistent across the
various imbalance ratios, but its effectiveness is too much dependent on the
choice of a suitable data-adjustment technique. In other words, Random Forest
may not be able to perform well on any given data. On the other hand, although
the effectiveness of Broad Learning depends on the choice of a suitable
imbalance ratio, its performance is consistent for any given data adjustment
technique, i.e., Broad Learning can robustly handle noisy data as long as the
imbalance ratio is small. Similar to Random Forest, an ensemble of Broad
Learners can be employed to study if the classification performance can be
made robust to variation in imbalance ratio as well.

Working with Deep Neural Network requires a large amount of training data. The
feature set for change detection, as described here, is of small dimension,
and is not suitable for training a deep learning model. Moreover, as shown in
[1], Broad Learning performs better and in a more efficient way on low-
dimensional data as compared to that of Deep Learning Approaches. So, we have
deliberately avoided comparison with Deep Learning in the present paper.

##  6 Conclusions and Future Work

In this paper, we describe a technique for detecting changes in aerial images
over a period of time. Usual characteristics of a change detection ground
truth data set is that, the number of pixels in changed category is
significantly less than that in the unchanged category. Due to this fact, a
classifier trained on the available ground truth data gets highly biased
towards accurately classifying the majority class. We handle this problem by
applying a suitable data adjustment technique on the imbalanced ground truth
data to make it balanced. Experimental results show that the Broad Learning
system works quite effectively in terms of the F-Score metric. Moreover, the
results obtained from Random Under-sampling and SMOTE data adjustment
technique is usually better than those obtained from a combination of random
under-sampling and over-sampling technique for any classifier. In future, this
technique can be also extended to change detection in videos that can be an
efficient tool in surveillance. Good performance on the chosen data set also
motivates us to test our approach on real satellite images. Broad Learning
based change detection technique can also be employed in summarizing aerial
videos.

## Acknowledgements

The authors would like to thank Dr. Lipo Wang of Nanyang Technological
University and Dr. C.L. Philip Chen of University of Macau for introducing us
to the concept of Broad Learning and helping us with the initial
implementation stage. Authors also sincerely acknowledge Dr. Prasenjit
Banerjee, Nalanda Foundation Mentor for his support. Authors also share their
sincere gratitude to HP Innovation Incubator program for giving them the
opportunity towards initiation of this project.

## References

  * [1] C. L. P. Chen and Z. Liu, “Broad learning system: An effective and efficient incremental learning system without the need for deep architecture,” IEEE Transactions on Neural Networks and Learning Systems, 2017.
  * [2] P. Rosin, “Thresholding for change detection,” in Computer Vision, 1998. Sixth International Conference on, pp. 274–279, IEEE, 1998.
  * [3] P. L. Rosin and E. Ioannidis, “Evaluation of global image thresholding for change detection,” Pattern recognition letters, vol. 24, no. 14, pp. 2345–2356, 2003.
  * [4] L. Di Stefano, S. Mattoccia, and M. Mola, “A change-detection algorithm based on structure and colour,” in Advanced Video and Signal Based Surveillance, 2003. Proceedings. IEEE Conference on, pp. 252–259, IEEE, 2003\.
  * [5] L. Bruzzone and D. F. Prieto, “An adaptive semiparametric and context-based approach to unsupervised change detection in multitemporal remote-sensing images,” IEEE Transactions on image processing, vol. 11, no. 4, pp. 452–466, 2002.
  * [6] L. Bruzzone and D. F. Prieto, “Automatic analysis of the difference image for unsupervised change detection,” IEEE Transactions on Geoscience and Remote sensing, vol. 38, no. 3, pp. 1171–1182, 2000.
  * [7] W. A. Malila, “Change vector analysis: an approach for detecting forest changes with landsat,” in LARS symposia, p. 385, 1980.
  * [8] A. Singh, “Review article digital change detection techniques using remotely-sensed data,” International journal of remote sensing, vol. 10, no. 6, pp. 989–1003, 1989.
  * [9] H. V. Poor, An introduction to signal detection and estimation.  Springer Science & Business Media, 2013.
  * [10] S. M. Kay, “Fundamentals of statistical signal processing, ser,” Signal Processing Series. Englewood Cliffs, NJ, USA: Prentice-Hall, 1993.
  * [11] M. J. Black, D. J. Fleet, and Y. Yacoob, “Robustly estimating changes in image appearance,” Computer Vision and Image Understanding, vol. 78, no. 1, pp. 8–31, 2000.
  * [12] Y. G. Leclerc, Q.-T. Luong, and P. Fua, “Self-consistency and mdl: A paradigm for evaluating point-correspondence algorithms, and its application to detecting changes in surface elevation,” International Journal of Computer Vision, vol. 51, no. 1, pp. 63–83, 2003.
  * [13] J. Rissanen, “Minimum description length principle,” Encyclopedia of statistical sciences, 1985.
  * [14] Y. Hsu, H.-H. Nagel, and G. Rekers, “New likelihood test methods for change detection in image sequences,” Computer vision, graphics, and image processing, vol. 26, no. 1, pp. 73–106, 1984.
  * [15] Y. Yakimovsky, “Boundary and object detection in real world images,” Journal of the ACM (JACM), vol. 23, no. 4, pp. 599–618, 1976.
  * [16] A. Elfishawy, S. Kesler, and A. Abutaleb, “Adaptive algorithms for change detection in image sequence,” Signal Processing, vol. 23, no. 2, pp. 179–191, 1991.
  * [17] Z.-S. Jain and Y. A. Chau, “Optimum multisensor data fusion for image change detection,” IEEE transactions on systems, man, and cybernetics, vol. 25, no. 9, pp. 1340–1347, 1995.
  * [18] K. Toyama, J. Krumm, B. Brumitt, and B. Meyers, “Wallflower: Principles and practice of background maintenance,” in Computer Vision, 1999. The Proceedings of the Seventh IEEE International Conference on, vol. 1, pp. 255–261, IEEE, 1999.
  * [19] M. J. Carlotto, “Detection and analysis of change in remotely sensed imagery with application to wide area surveillance,” IEEE Transactions on image processing, vol. 6, no. 1, pp. 189–202, 1997.
  * [20] C. Clifton, “Change detection in overhead imagery using neural networks,” Applied Intelligence, vol. 18, no. 2, pp. 215–234, 2003.
  * [21] K. Skifstad and R. Jain, “Illumination independent change detection for real world image sequences,” Computer vision, graphics, and image processing, vol. 46, no. 3, pp. 387–399, 1989.
  * [22] L. Li and M. K. Leung, “Integrating intensity and texture differences for robust change detection,” IEEE Transactions on image processing, vol. 11, no. 2, pp. 105–112, 2002.
  * [23] S.-C. Liu, C.-W. Fu, and S. Chang, “Statistical change detection with moments under time-varying illumination,” IEEE Transactions on Image Processing, vol. 7, no. 9, pp. 1258–1268, 1998.
  * [24] S. Patra, S. Ghosh, and A. Ghosh, “Change detection of remote sensing images with semi-supervised multilayer perceptron,” Fundamenta Informaticae, vol. 84, no. 3, 4, pp. 429–442, 2008.
  * [25] A. Singh and K. K. Singh, “Unsupervised change detection in remote sensing images using fusion of spectral and statistical indices,” The Egyptian Journal of Remote Sensing and Space Science, 2018.


$\sum\limits_{i=1}^{5}(-1)^{i}[t,1-t,1-c.r.(z_{1},\dots,\hat{z}_{i},\dots,z_{5})/t,c_{4},\dots,c_{n}]_{{\mathbb{P}}^{1}}$

is closed. We need to show that this element is equal to zero. By Lemma 4.3
the cohomology $H^{i}(\widetilde{\mathcal{N}}_{\geq m-1}(\mathbb{K},m))$
satisfies Galois descent. So we can assume that $\mathbb{K}$ is algebraically
closed. In Section 4 we showed that the map
$\widetilde{\mathcal{W}}\colon\widetilde{\mathcal{N}}_{\geq
m-1}(\mathbb{K},m)\to\Lambda_{\geq m-1}(\mathbb{K},m)$ is an isomorphism. So
it is enough to prove the corresponding relation in the complex $\Lambda_{\geq
m-1}(\mathbb{K},m)$. This follows from Lemma 6.2. See also [GMS99].

  2. 2.

Let $(t,w)$ be coordinates in ${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$. The
expression is the boundary of the following cycle:

$\left[t,1-t,(1-a/t),\dfrac{(w-b_{1}b_{2})(w-1)}{(w-b_{1})(w-b_{2})},w,c_{5},\dots,c_{n}\right]_{\mathbb{P}^{1}\times\mathbb{P}^{1}}.$

  3. 3.

It is enough to prove that

$\displaystyle d([x_{1},x_{2},(1-x_{1}),(1-x_{2}/x_{1}),(1-a/x_{2}),$
$\displaystyle
c_{5},\dots,c_{n}]_{{\mathbb{P}}^{2}}=[t,(1-t),(1-a/t),a,c_{5},\dots,c_{n}]_{{\mathbb{P}}^{1}}.$

This formula was proved in [BK94], see also [GGL09]. Let $f_{i}$ be the $i$-th
function on the left hand-side. Denote by $Z$ the closure of the zeros of the
functions $f_{i}-1$. Let $U={\mathbb{P}}^{2}\backslash Z$. It is easy to see
that the only irreducible components of the divisors of $f_{i}$ on $U$ are
$x_{1}=x_{2}$ and $x_{2}=a$. Moreover these divisors belong only to the
functions $1-x_{2}/x_{1}$ and $1-a/x_{2}$ correspondingly. This imply that the
cycle on the left hand-side is admissible. For the divisor $x_{1}=x_{2}$ the
term in the differential is zero. For the divisor $x_{2}=a$ we get the element
given in the statement of the lemma.

  4. 4.

This formula is obtained by a direct computation.

∎

###  6.2 The map $\mathcal{T}^{\prime}$

Define a morphism of complexes

$\mathcal{T}^{\prime}\colon\Gamma_{\geq m-1}(\mathbb{K},m)\to\Lambda_{\geq
m-1}(\mathbb{K},m)$

as follows. The element $\\{a\\}_{2}\wedge c_{3}\wedge\dots\wedge c_{m}$ goes
to

$[{\mathbb{P}}^{1},t\wedge(1-t)\wedge(1-a/t)\wedge c_{3}\wedge\dots\wedge
c_{m}].$

The element $c_{1}\wedge\dots\wedge c_{m}$ goes to
$[\operatorname{Spec}\mathbb{K},c_{1}\wedge\dots\wedge c_{m}]$. It follows
from the definition that
$\mathcal{T}^{\prime}=\widetilde{\mathcal{W}}\circ\mathcal{T}$. In particular
$\mathcal{T}^{\prime}$ is a well-defined morphism of complexes.

######  Theorem 6.4 (Theorem 1.8).

Let $\mathbb{K}$ be algebraically closed. The map $\mathcal{T}^{\prime}$ is an
isomorphism.

Let us show that the map $\mathcal{T}^{\prime}$ is surjective. This follows
from the following result:

######  Proposition 6.5.

Let $\mathbb{K}$ be algebraically closed. The group
$\Lambda(\mathbb{K},m)_{m-1}/\operatorname{Im}(d)$ is generated by the
elements of the form

$[{\mathbb{P}}^{1},t\wedge(1-t)\wedge(1-a/t)\wedge c_{4}\wedge\dots\wedge
c_{n}].$

######  Lemma 6.6.

Let $S={\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$. Consider the element

$a=[S,x\wedge(1-x)\wedge\alpha_{2}\wedge\dots\wedge\alpha_{k}\wedge
c_{k+1}\wedge\dots\wedge c_{n}].$

In this formula $x$ is the canonical coordinates on the first
${\mathbb{P}}^{1}$,
$\alpha_{i}\in\mathbb{K}({\mathbb{P}}^{1}\times{\mathbb{P}}^{1})$ and
$c_{i}\in\mathbb{K}$. For any birational morphism
$\varphi\colon\widetilde{S}\to S$, and any divisor $E$ contracted under
$\varphi$, the element $[E,\operatorname{\partial}_{E}(\varphi^{*}(a))]$ can
be represented as linear combination of the elements of the form

$[{\mathbb{P}}^{1},c\wedge(1-c)\wedge\beta_{3}\wedge\dots\wedge\beta_{k}\wedge
c_{k+1}\wedge\dots\wedge c_{n}].$

In this formula $c\in\mathbb{K}\backslash\\{0,1\\}$ and
$\beta_{i}\in\mathbb{K}({\mathbb{P}}^{1})$.

###### The proof of Lemma 6.6.

Denote by $(x,t)$ the canonical coordinates on $S$. Let
$\\{(x_{0},t_{0})\\}\subset S$ be the image of $E$. If $x_{0}=0$ then the the
restriction of the function $\varphi^{*}(1-x)$ to $E$ is equal to $1$ and so
$[E,\operatorname{\partial}_{E}(\varphi^{*}(a))]=0$. The case $x_{0}=1$ is
similar. Let us assume that $x_{0}=\infty$. In this case, the restriction of
the function $\varphi^{*}(x/(1-x))$ to $E$ is equal to $-1$. As the element
$a$ can be rewritten in the form

$a=[S,(x/(1-x))\wedge(1-x)\wedge\alpha_{2}\wedge\dots\wedge\alpha_{k}\wedge
c_{k+1}\wedge\dots\wedge c_{n}],$

this implies that $[E,\operatorname{\partial}_{E}(\varphi^{*}(a))]=0.$

So we can assume that $x_{0}\in\mathbb{K}\backslash\\{0,1\\}$. This implies
that the restriction of the function $\varphi^{*}(x)$ to $E$ is constant.
Denote this constant by $c$. We get

$[E,\operatorname{\partial}_{E}(\varphi^{*}(a))]=[E,c\wedge(1-c)\wedge\operatorname{\partial}_{E}(\alpha_{2}\wedge\dots\wedge\alpha_{k})\wedge
c_{k+1}\wedge\dots\wedge c_{n}].$

The statement follows. ∎

###### The proof of Proposition 6.5.

Denote by $A$ the subgroup of the group
$\Lambda(\mathbb{K},m)_{m-1}/\operatorname{Im}(d)$ generated by the elements
stated in the lemma.

By Theorem 3.18 we know that the group
$\Lambda(\mathbb{K},m)_{m-1}/\operatorname{Im}(d)$ is generated by the
elements of the form

$[{\mathbb{P}}^{1},(t-a_{1})\wedge\dots\wedge(t-a_{k})\wedge
c_{k+1}\wedge\dots\wedge c_{n}].$

Let us prove by induction on $k,k\geq 0$, that this element lies in $A$.

  1. 1.

The case $k\leq 2$ follows from Lemma 4.4.

  2. 2.

Let $k=3$. Denote by $\varphi$ an automorphism of ${\mathbb{P}}^{1}$ given by
the formula $\varphi(t)=a_{1}+(a_{2}-a_{1})t$. We get

$\displaystyle[{\mathbb{P}}^{1},(t-a_{1})\wedge(t-a_{2})\wedge(t-a_{3})\wedge
c_{4}\wedge\dots\wedge c_{n}]=$
$\displaystyle[{\mathbb{P}}^{1},\varphi^{*}((t-a_{1})\wedge(t-a_{2})\wedge(t-a_{3})\wedge
c_{4}\wedge\dots\wedge c_{n})]=$
$\displaystyle[{\mathbb{P}}^{1},(t(a_{2}-a_{1}))\wedge(t(a_{2}-a_{1})-(a_{2}-a_{1})\wedge(t(a_{2}-a_{1})-(a_{3}-a_{1}))\wedge
c_{4}\wedge\dots\wedge c_{n}]=$
$\displaystyle\left[{\mathbb{P}}^{1},t\wedge(1-t)\wedge\left(t-\dfrac{a_{3}-a_{1}}{a_{2}-a_{1}}\right)\wedge
c_{4}\wedge\dots\wedge c_{n}\right]=$
$\displaystyle\left[{\mathbb{P}}^{1},t\wedge(1-t)\wedge\left(1-\left(\dfrac{a_{3}-a_{1}}{a_{2}-a_{1}}\right)/t\right)\wedge
c_{4}\wedge\dots\wedge c_{n}\right].$

  3. 3.

Let $k\geq 4$. Consider the element

$\displaystyle[{\mathbb{P}}^{1}\times{\mathbb{P}}^{1},x\wedge(1-x)\wedge(x-(t-a_{1})/(t-a_{2}))\wedge$
$\displaystyle(t-a_{3})\wedge\dots\wedge(t-a_{k})\wedge
c_{k+1}\wedge\dots\wedge c_{n}].$

It follows from Lemma 6.6 that the differential of this element is equal to
$x+y$, where $x\in A$ and

$y=(t-a_{1})/(t-a_{2})\wedge(1-(t-a_{1})/(t-a_{2}))\wedge(t-a_{3})\wedge\dots\wedge(t-a_{k})\wedge
c_{k+1}\wedge\dots\wedge c_{n}.$

By inductive assumption this implies that the element

$[{\mathbb{P}}^{1},(t-a_{1})\wedge\dots\wedge(t-a_{k})\wedge
c_{k+1}\wedge\dots\wedge c_{n}]$

lies in $A$.

∎

It remains to show that the map $\mathcal{T}^{\prime}$ is injective. We recall
that in the previous section for any smooth complete curve over $\mathbb{K}$
we constructed the canonical map
$\mathcal{H}_{\mathbb{K}(X)}\colon\mathrm{EP}^{m+1}(X)\to\Gamma_{\geq
m-1}(\mathbb{K},m)_{m-1}$. Define a morphism of complexes

$\mathcal{CD}\colon\Lambda_{\geq m-1}(\mathbb{K},m)\to\Gamma_{\geq
m-1}(\mathbb{K},m)$

as follows. The image of the element $[X,a]\in\Lambda_{\geq
m-1}(\mathbb{K},m)_{m-1}$ is equal to $-\mathcal{H}_{\mathbb{K}(X)}(a)$. The
image of the element $[\operatorname{Spec}\mathbb{K},a]$ is equal to $a$. It
follows from the results of the previous section $\mathcal{CD}$ is a morphism
of complexes and $\mathcal{CD}\circ\mathcal{T}^{\prime}=id$. This implies that
the map $\mathcal{T}^{\prime}$ is injective. So the map $\mathcal{T}^{\prime}$
is an isomorphism.

###  6.3 The proof of Theorem 1.3

In the beginning of this section we have shown that the map
$\mathcal{T}\colon\Gamma_{\geq
m-1}(\mathbb{K},m)\to\widetilde{\mathcal{N}}_{\geq m-1}(\mathbb{K},m)$ is
well-defined. By the results of [Blo86, Rud21] the groups
$H^{i}(\widetilde{\mathcal{N}}_{\geq m-1}(\mathbb{K},m)),H^{i}(\Gamma_{\geq
m-1}(\mathbb{K},m))$ satisfy Galois descent. So we can assume that
$\mathbb{K}$ is algebraically closed. In this case, we need to show that
$\mathcal{T}$ is an isomorphism. In Section 4 we have constructed the map
$\widetilde{\mathcal{W}}\colon\widetilde{\mathcal{N}}_{\geq
m-1}(\mathbb{K},m)\to\Lambda_{\geq m-1}(\mathbb{K},m)$ and have shown that
this map is an isomorphism. So it remains to prove that the map
$\mathcal{T}^{\prime}=\mathcal{W}\circ\mathcal{T}$ is an isomorphism. This was
proved in the previous subsection.

## References

  * [BK94] S. Bloch and I. Kriz.  Mixed tate motives.  Annals of mathematics, 140(3):557–605, 1994.
  * [Blo86] S. Bloch.  Algebraic cycles and higher $K$-theory.  Adv. in Math., 61(3):267–304, 1986.
  * [Bol23] V. Bolbachan.  Chow dilogarithm and strong Suslin reciprocity law.  J. Alg. Geom., 32:697–728, 2023.
  * [DJ96] A. J. De Jong.  Smoothness, semi-stability and alterations.  Publications Mathématiques de l’IHÉS, 83:51–93, 1996.
  * [FS02] E. Friedlander and A. Suslin.  The spectral sequence relating algebraic $K$-theory to motivic cohomology.  Annales scientifiques de l’Ecole normale supérieure, 35(6):773–875, 2002.
  * [Ful13] W. Fulton.  Intersection theory.  Springer New York, 2nd edition, 2013.
  * [GGL09] H. Gangl, A. B. Goncharov, and A Levin.  Multiple polylogarithms, polygons, trees and algebraic cycles.  Alg. Geom., Seattle 2005, 80:547–593, 2009.
  * [GMS99] Herbert Gangl and Stefan Müller-Stach.  Polylogarithmic identities in cubical higher Chow groups.  In Algebraic $K$-Theory: AMS-IMS-SIAM Joint Summer Research Conference on Algebraic K-Theory, July 13-24, 1997, University of Washington, Seattle, volume 67, page 25. American Mathematical Soc., 1999.
  * [Gon94] A. B. Goncharov.  Polylogarithms and motivic Galois groups.  Motives (Seattle, WA, 1991), 55:43–96, 1994.
  * [Gon95] A. B. Goncharov.  Geometry of configurations, polylogarithms, and motivic cohomology.  Advances in Mathematics, 114(2):197–318, 1995.
  * [MVW06] C. Mazza, V. Voevodsky, and C. Weibel.  Lecture notes on motivic cohomology, volume 2.  American Mathematical Soc., 2006.
  * [NS90] Yu. Nesterenko and A. Suslin.  Homology of the full linear group over a local ring, and Milnor’s K-theory.  Mathematics of the USSR-Izvestiya, 34(1):121, 1990.
  * [Par75] A. N. Parshin.  Class fields and algebraic $K$-theory.  Uspekhi Mat. Nauk, 30(1):253–254, 1975.
  * [Rud21] D. Rudenko.  The strong Suslin reciprocity law.  Compositio Mathematica, 157(4):649–676, 2021.
  * [Sus91] A. A. Suslin.  $K_{3}$ of a field and the Bloch group.  In Proc. Steklov Inst. Math, volume 183, pages 217–239, 1991.
  * [Sus00] Andrei Suslin.  Higher Chow groups and etale cohomology.  Cycles, transfers, and motivic homology theories, 143:239–254, 2000.
  * [Tot92] B. Totaro.  Milnor $K$-theory is the simplest part of algebraic $K$-theory.  K-theory, 6(2):177–189, 1992.
  * [Voe02] Vladimir Voevodsky.  Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic.  International Mathematics Research Notices, 2002(7):351–355, 2002.
  * [Wof23] Eric Wofsey.  Exterior square of rational function of one variable.  Mathematics Stack Exchange, 2023.  math.stackexchange.com/q/4728781 (version: 2023-07-01).

Vasily Bolbachan
Skolkovo Institute of Science and Technology, Nobelya str., 3, Moscow, Russia,
121205;
Faculty of Mathematics, National Research University Higher School of
Ecnomics, Russian Federation, Usacheva str., 6, Moscow, 119048;
HSE-Skoltech International Laboratory of Representation Theory and
Mathematical Physics, Usacheva str., 6, Moscow, 119048
_E-mail:_<EMAIL_ADDRESS>