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# Textureless Object Recognition: An Edge-based Approach

1st Frincy Clement  dept. of Computing Science
University of Alberta
Edmonton, Canada
<EMAIL_ADDRESS>2nd Kirtan Shah  dept. of Computing Science
University of Alberta
Edmonton, Canada
<EMAIL_ADDRESS>3rd Dhara Pancholi  4th Gabriel Lugo Bustillo  dept. of
Computing Science
University of Alberta
Edmonton, Canada
<EMAIL_ADDRESS>dept. of Computing Science
University of Alberta
Edmonton, Canada
<EMAIL_ADDRESS>5th Dr. Irene Cheng  dept. of Computing Science
University of Alberta
Edmonton, Canada
<EMAIL_ADDRESS>
###### Abstract

Textureless object recognition has become a significant task in Computer
Vision with the advent of Robotics and its applications in manufacturing
sector. It has been challenging to obtain good accuracy in real time because
of its lack of discriminative features and reflectance properties which makes
the techniques for textured object recognition insufficient for textureless
objects. A lot of work has been done in the last 20 years, especially in the
recent 5 years after the TLess and other textureless dataset were introduced.
In this project, by applying image processing techniques we created a robust
augmented dataset from initial imbalanced smaller dataset. We extracted edge
features, feature combinations and RGB images enhanced with feature/feature
combinations to create 15 datasets, each with a size of 340,000. We then
trained four classifiers on these 15 datasets to arrive at a conclusion as to
which dataset performs the best overall and whether edge features are
important for textureless objects. Based on our experiments and analysis, RGB
images enhanced with combination of 3 edge features performed the best
compared to all others. Model performance on dataset with HED edges performed
comparatively better than other edge detectors like Canny or Prewitt.

###### Index Terms:

Computer Vision, Textureless object detection, Textureless object recognition,
Feature-based, Edge detection, Deep Learning

##  I Introduction

In recent years, a lot of labor-intensive activities in manufacturing sector
has been automated using robots. In industrial assembly line, a robotic arm is
used to pick up small objects and place it in its location. These industrial
objects are mostly textureless and are different from textured objects in
terms of lack of discriminative features. Often, they are smaller in size and
can be similar to each other. In real world, they are present along with
multiple objects of different classes in different scene configurations, which
makes the recognition even more challenging.

In the early 2000s, detection and recognition of the textureless objects
seemed to be a daunting issue, even for the state-of-art detectors at that
time, like SIFT (Scale Invariant Feature Transform) and SURF (Speeded UP
Robust Features). This was due to their dependence on the high-featured,
texture-rich informative regions which were sparse in the textureless
occurrences. Ultimately, this led to several works which contributed for the
detection and recognition of textureless objects, which can be thus classified
into three categories: View-based, Feature-based, and Shape-based.

###  I-A View-based

View-based methods work by comparing the object of interest (OOI) and the pre-
calculated 2D views of that object, denoted by aspect graphs. This method can
provide a rough estimate of the pose of the object, but it suffers from
computational load of searching a large spatial domain. To overcome this
issue, Cyr and Kimia [3] proposed a method that grouped similar OOI together
under a single class and similarity scores were calculated to perform
recognition. Eggert et al. [1], Ulrich et al. [8] and Steger et al. [2]
introduced various improvements on the aspect graphs and similarity metric.
This approach, however was not widely used because of its high complexity.

###  I-B Feature based method

Tombari et al. [9] devised a method called BOLD in which they attempt to
tackle the issue of object detection under clutter. In this method,
neighbouring line segments were aggregated to form a feature representation of
the object. These line segments are invariant to rotation, translation as well
as scaling. A limitation of BOLD is that it fails if there is clutter in the
line segment selected for feature description.

Intended for object recognition and tracking using an RGB-D sensor, Jiang et
al. [18] proposed a method that detects and recognises small objects. The
image is divided into various parts using image segmentation. Then it is sent
to a classifier as an input which classifies the object using the features
extracted in the earlier stages.

Hodan et al. [13] proposed a technique for detecting textureless objects using
AR, which used an edge-based detector which gave better results than canny
edge detector. And efficient towards the object detection as it searched
edgelet in stripes instead of wedges.

While the above-mentioned methods do a commendable job of detecting
textureless objects, they were computationally expensive. In 2016 Jacob et al.
[19] introduced BORDER which is an acronym for Bounded Oriented Rectangle
Descriptors for Enclosed Region, in which they use a rectangle to encapsulate
the object so that minimum outliers are detected. It outperformed the previous
state-of-the-art descriptors BOLD and Line2D when compared for images having
occlusion and clutter.

Counting the number of textureless objects present in a scene was still a
challenge until Verma et al. [21] proposed a method which counts the objects
using shape and colour feature which are obtained through morphological
boundary extraction and segmentation. The uniqueness in this method was it was
invariant to scaling and rotation. Further, different models such as SVM, KNN
etc were used for the classification.

To improve accuracy, Thoduka et al. [23] in 2016 came up with an approach
which used RGB-D based features to recognize objects in the industrial
assembly line such as nuts, bolts, bearing large and small black profiles
etc.The approach performed efficiently on most of the objects in dataset but
they had poor performance recognizing smaller objects.

Following upon his scholarly work, Hodan et al. [25] improved upon his
previous research by presenting a new dataset in 2017, “T-less” publicly for
better understanding of 3D object from different dimensions. This dataset
contains industrial objects which have no texture while having feature
similarities of shape/size which makes it quite hard to differentiate.

To overcome the short comings of previous methods, BIND [24] was introduced
during 2017. It is a detector that uses binary nets having multiple layer for
accurate detection of textureless object. To improve the speed of object
detection, in 2018 Hancheng et al. [37] introduced a faster approach to detect
textureless objects even in the cluttered background and transformations.

Fang et al. [33] deduced that for object recognition, the background of an
image affects the performance of neural networks, and hence came up with a
method called DOG, built upon CNN which effectively removed the background of
an input image for improving the efficiency of the classifier.

###  I-C Shape Based Approach

Template matching is a popular method in textureless object recognition,
dating back as early as the 2000s in lieu of the fact that it provides high
mAP and close to real-time detection. But, up until now, templates were
generated using one modality only (depth or colour or point cloud or 2D).
However, in 2011 Hinterstoisser et al. [6] put forth the idea of multi-modal
template matching. The template comprises of both the image cue (colour
information) as well as depth cue (3D) information. The method can detect
objects in real-time as well as under heavy clutter.A caveat of the above
approach is that it is not invariant to translation, rotation, or scaling.

In the following year, Hinterstoisser et al. [7] overcame the afore-mentioned
issue by considering all the gradient orientations for a template instead of
just the dominant ones. Due to this, the method was more robust to
translations. While it does suppress the caveat of the above approach to a
certain extent, it is still not invariant to scaling.

In 2015 Hodan et al. [13] came up with a very efficient way of template
matching, resolving a lot of its computational complexity in performing
searches of each sliding window with each stored template. This method
resulted in real time performance with low false rates and multiple object
detection.

In 2018, an edge-based Hierarchical Template Matching algorithm was proposed
by Tsai et al. [29] based upon Line2D algorithm, which gave an efficient
method for real-time object detection and recognition. As per this paper,
edges are identified as the most stable feature for a textureless object. This
was the baseline information for building our project.

2018 saw many approaches for textureless object detection. PoseCNN (current
state-of-the-art), an approach by Xiang et al., [27] is a convolutional
network for predicting object pose with varied orientations. It predicts
object’s translation matrix by means of calculating object’s distance with
respect to the camera. They also calculate the object’s rotation matrix.
Midway through 2019, Park et al. [35] proposed a method that takes depth image
as an input and a region of interest (ROI) is generated.

In the next section, we introduce the existing work done in the field of
textureless object recognition by using a combination of image processing and
deep learning techniques. Our proposed method is explained step by step in
Section III. Furthermore, Section IV details the experiments performed,
results and analysis. The report ends with Conclusion and future work,
Acknowledgement, Author Biography and References.

##  II Related Work

Our proposed work uses image processing techniques combined with machine
learning techniques for textureless object recognition. We perform image
processing techniques to extract features and create multiple training sets.
We compare these datasets using machine learning techniques. In this section,
previous researches which uses similar approach are summarized.

Jiang et al. [18] proposed an RGB-D based method for the detection of various
types of common objects. They used depth filtering alongside clustering,
coupled with the watershed algorithm on the depth data for the target object
recognition. Once recognized, they apply a tracking method to reduce the
search space and the computational load. Further, the method was evaluated
using a random forest classifier. However, for real-time object detection,
only RGB dataset was used.

Wang et al. [20] proposed a method for object recognition using model-based
CNN learning. They introduced a unified structure composed of automatic object
reconstruction and multitask learning aimed at doing object recognition by
model-based training alone. This method reduced the gap between real images
and synthetically rendered images for better accuracy.

Later, a technique to count textureless objects with shape and colour feature
was developed by Verma et al. [21]. First, the shape and color features were
extracted during pre-processing using morphological boundary extraction and
segmentation via mean hue value (colour features) and Hu-moments (shape
features). Next, SVM, kNN, NN and tree bagging were applied for the
classification of objects. Through experiments, they observed that tree
bagging has the best accuracy of all the classifiers. Finally, the classified
objects were counted by drawing bounding boxes around them.

Thoduka et al. [23] put forth a method for object detection using 3D point
cloud. They defined size, shape, and color of objects by using various metrics
including the bounding box. The extracted features were then passed through
SVM for classification. However, the accuracy of the method is inversely
proportional to the size of the object in each image.

A breakthrough in the field of textureless object recognition happened when
Hodan et al. [25] introduced T-less dataset for 6D pose estimation. The
dataset houses thirty industry-relevant textureless objects with no
discriminative colour or illumination properties. They are not symmetrical
with respect to shape or size. It contains 39K training images along with 10K
testing images that exhibit occlusion and clutter. It also has manually
created 3D CAD model, as well as one which is constructed semi-automatically.
For our project, we used the 2D images taken from 3D printed models of this
dataset as our baseline data.

Hodan et al. [13] came up with a method using purely edge detector which uses
edgelet strips in place of wedges. When the binary map of edges is provided,
this method takes the edglets of equal fix lengths and they are represented by
their midpoints and orientation, which helps in tracking the orientation. Two
images are matched using their distance transformation in orientation which
proves the presence of that object in image even if more than one same object
is present.

A technique called PoseCNN introduced by Xiang et al. [27] used for 6D pose
estimation, estimated the 3D translation and rotation of an object by
predicting the center of the object’s distance from the camera. For this
process, they also introduced a novel loss function called P loss for
objection detection with clutter. Speaking of which, clutter and background
are important aspects when it comes to image segmentation. Fang et al. [33]
devised a method called DOG which removes the background of the images to
improve the object detection accuracy. The method uses DCGAN for
classification.

Tsai et al. [29] came up with a new approach for Template matching which
proved to be an efficient method for object detection and recognition. They
chose edges as they were identified as the most stable feature for a
textureless object. After image enhancement, the input image goes through the
edge-based template matching algorithm. It detects the color edge by finding
the one with maximum magnitude across all color channels. These detected edges
are then quantized into one of 8 bins and one-hot encoded for computational
efficiency. The one-hot encoded edge orientations are then spread using
dilation. The algorithm then checks for similarity with the precomputed
database which has edge templates of reference images and lookup tables of
edge orientations, which will help in efficient matching and calculation of
similarity score.

For the detection of textureless objects, the importance of depth images over
colour images cannot be overstated. Park et al. [35] developed a method called
Multi-task template matching for object detection, segmentation and pose
estimation using depth images. They put forth a method to match the Nearest
Neighbour template using a similarity score. Since shared feature maps were
used to predict the segmentation mask and pose of an object, additional
training was not required for a new object. For real-time object detection and
recognition, Region of Interest (ROI) predicted by ResNet-50 architecture
pretrained offline on the templates, were compared with the ROI of test image
predicted real-time by another ResNet-50.

##  III Proposed Method

Our work focuses on recognition of textureless objects in 30 classes. In this
section, we will discuss how we acquired the intial dataset, what image
processing techniques are applied on the input images to convert it into
features and feature-boosted versions of original images and which classifiers
are selected for object recognition. The “Fig. 1” depicts the block diagram of
our proposed method.

Figure 1: Project Workflow

Project Implementation Structure

Our project was implemented in 3 parts. In Part 1, we focused on applying
image processing techniques to create a balanced augmented dataset from ground
truth. In Part 2, we apply edge operators to create 14 new datasets to perform
feature comparison. In Part 3, we train a classifier and predict results.

Part 1: Creation of Initial Augmented Data

  * •

Image Data Acquisition

  * •

Obtaining Ground Truth data

  * •

Data Balancing and Augmentation

    * –

Manual: Image Processing Techniques

    * –

Automatic: Augmentor API

Part 2: Creation of 14 new datasets for feature comparison

  * •

Creation of Feature Only Dataset

  * •

Creation of Feature Combinations

  * •

Creation of Feature Enhanced RGB Dataset

  * •

Creation of Feature Combination Enhanced RGB Dataset

Part 3: Training and Testing on Multiple Classifiers

Part 1: Creation of Initial Augmented Data

####  III-1 Image Data Acquisition

The dataset we used for our experiments are based on TLess Dataset [25]
introduced by Hodan et al. 30 textureless objects were 3-D printed from the
CAD model and using an Intel RealSense camera, 3D images were captured at the
Multimedia Research Lab, University of Alberta. We used the 3D-rendered 2D
images to start our implementation for object recognition.

####  III-2 Obtaining Ground Truth Data

We started our experiments with 27000 images that contained textureless
objects from 30 different classes. The objects that were captured covered only
small portion of the images. We used regionprops() from sci-kit learn library
to obtain bounding boxes for each object. From those bounding boxes, the
objects were cropped leaving few pixels on all sides. The bounding box
information was also saved as annotations for training any of YOLO versions in
the form: Classname, x-center, y-center, height, width – all normalized to
image height and width. The different sample images captured from Intel
RealSense camera and the output of regionprops() are shown in “Fig. 2”.

Figure 2: Data Preprocessing to obtain ground truth objects from initial
acquired 2D images

####  III-3 Data Balancing & Augmentation

The data provided was captured in 4 different orientations for most of the
classes except for the ones that had unique features. The data was unevenly
distributed among 30 different classes. As seen on “Fig. 3” Whereas most of
the classes had 1̃200 images, some had only 3̃00.

Figure 3: Distribution of data across 30 classes

To balance the dataset, we calculated a balancing factor. While applying
manual augmentation, we considered this balancing factor to remove the
imbalance in dataset.

The number of images captured using Intel RealSense is 1000 per object, which
is too small to efficiently train a classifier. Hence, we need to apply image
augmentation to generate artificial data. We augmented the input image in two
ways: Manually applying image processing techniques and using Augmentor API.

a) Manual Augmentation: As proposed in the recognition method by Huang et al.
[31], we applied the following techniques on the ground truth data with a
balancing factor. The results are shown in “Fig. 4”

  1. 1.

Contrast Enhancement – to improve the contrast of the textureless object with
respect to the background

  2. 2.

Addition of Noise – to avoid overfitting of input data

  3. 3.

Brightness transformation – to simulate the object detection in the lighting
in manufacturing facility

  4. 4.

Addition of Blur – to account for the blurring that can be caused by the
camera movement

Figure 4: Results of Manual Augmentation

b) Automatic Augmentation by using an API: Augmentor [26] by Bloice et al. is
a software package which provides high level API in Python and Julia with a
pipeline-based approach to perform various operations like rotation,
translation, scaling, cropping, shearing etc. We used this package to augment
our dataset to add following:

  1. 1.

Rotation

  2. 2.

Random Erasing (80 % intact)

  3. 3.

Random cropping (80 % intact)

  4. 4.

Flip (Left-Right & Top-Bottom)

  5. 5.

Skew (Magnitude – 0.3)

A sample of the results of automatic augmentation are shown in “Fig. 5”.

Figure 5: Results of Automatic Augmentation using Augmentor API

Upon performing manual and automatic augmentation, we obtained a balanced
dataset that contains similar number of images for all the classes. The new
dataset comprised of 340,000 images belonging to all 30 classes.

Part 2: Creation of 14 new datasets for feature comparison

From literature review, it could be inferred that for a textureless object,
edges are the most important feature. In part 1, we augmented the data from
27,000 (Ground truth) to 340,000 (Augmented data). For obtaining the edge
features, we implemented three different edge detectors on the ground truth
data to obtain the corresponding edge features.

  1. 1.

Canny Edge detector: By applying non-max suppression on the image gradient
intensities, and applying double threshold, it finds the strong edges by
suppressing the weak ones.

  2. 2.

Holistically Nested edge detector (HED): Xie et al. [38] developed a neural
network, which uses a deep neural network to automatically learn hierarchical
edge maps thereby identifying the edges of the objects in images.

  3. 3.

Prewitt operator: By convolving with a 3x3 kernel with the input image, it
calculates the gradient of image intensity at each point by combining vertical
and horizontal derivative approximations.

Canny edge detector thin dotted lines along the edges of the object, but it
was not continuous. HED provided edges of an object which gave thick outlines
around the object. Prewitt gave thick outline for the object and thin curvy
lines like canny for the inside part of the object. Thus, we created 3
datasets of 3̃40,000 images containing Canny features, HED features and
Prewitt features. We also considered Sobel Edge Detector but since it gave a
similar result as Canny, it was not implemented.

The next step was to combine the said features for all permutations to observe
if they can perform better. First, we combined Canny (thin curvy outlines)
with HED (thick outlines) which yielded us good results in the form of well-
defined clear outlines of the edges. We then combined HED (thick outlines)
with Prewitt (thin curvy outlines) which produced good results, but the
outlines were not as well-defined as Canny-HED, especially the inner edges.
For the third experiment, we implemented a combination of Canny (thin curvy
lines) and Prewitt (thin curvy lines) which gave us thin outlines as well as
inner lines of the object. Finally, the combination all three edge detectors –
Canny, HED and Prewitt which yielded us thicker and more prominent edges for
the object. So, we now had a total of 7 datasets, each containing 340,000
images of features and feature combinations. A sample from these 7 datasets
are shown in “Fig. 6”.

Figure 6: Sample of Features Only and Feature Combination dataset

To retain the original RGB channel features, we performed overlay of above
edges and edge combinations on the original RGB images to get edge enhanced
RGB images as seen on “Fig. 7”.

Figure 7: Sample of RGB images enhanced with Features Only and Feature
Combination

Hence, in total we created 15 datasets containing 3̃40,000 images each to
amass a total of 5 million images along with bounding boxes and their
annotations.

Part 3: Training and Testing on Multiple Classifiers

Our research is focused on understanding the significance of edge features for
textureless objects and to see if edges or edge enhanced pixel intensities can
help improve recognition results. Hence, we planned to train our datasets on
multiple models for comparing its performance and to have an unbiased result
irrespective of choice of model.

1) Model Selection Different models were considered based on the following
criteria:

  * •

Support incremental learning as we have large dataset (15 GB per dataset)

  * •

Faster training (4̃5 minutes per dataset), as we had 15 datasets to train and
predict

  * •

Follows similar input/output dimensions for training

  * •

Have similarity in input pre-processing

  * •

Can be implemented efficiently inside a loop without duplicating lines of
codes

With the above selection criteria in mind, we selected to train 4 models from
sci-kit learn library:

  1. 1.

Stochastic Gradient Descent (SGD): It helps implement linear SVM or Logistic
Regression based on the loss type selected, on a large scale. It performs
really well especially when the data is sparse, which is the case of our
dataset, as we are comparing edge features, which will eventually result in a
nxn sparse matrix with only edge pixel locations having ‘1’ and all others
‘0’. We used this model with loss type ‘log’ for Logistic Regression.

  2. 2.

Perceptron: Perceptron is a simple classification algorithm which is faster
than SGD, especially with ‘hinge’ loss and results in comparatively less
sparser models. It does not require a learning rate and is not regularized; it
learns from its own mistakes. Hence, it is easier to perform hyperparameter
tuning with this model.

  3. 3.

Passive-Aggressive Classifier with hinge loss: Passive-Aggressive algorithms
are used mainly for large scale learning. It does not need learning rate, but
has regularization parameter. We have tried the classifier on both hinge loss
and squared hinge loss as it would give us an understanding on what works
better for our data

  4. 4.

Passive-Aggressive Classifier with squared-hinge loss

2) Data Preprocessing

We have created 15 datasets: 7 datasets with edges and edge combinations,
which are binary images and 7 datasets with edge-enhanced RGB images. Before
sending the inputs to the model we have performed following preprocessing
steps:

  1. 1.

Created class labels for each input image

  2. 2.

Created feature matrix for input images as a numpy array, with each row
corresponding to each input image

    1. (a)

For edges only inputs, we did not apply any initial image processing, but for
edge enhanced RGB images, we initially converted them to grayscale

    2. (b)

The result is then resized to 200x200 to have match the dimensions for all
input images

    3. (c)

The resized input is then flattened from 200X200 to 40000X1 array

  3. 3.

From step 1 and 2, we created each dataset with X as input feature matrix of
size 340000 X 40000 x 1 and y as output labels with size 340000 X 1 x 1

3) Fitting and Training the model in batches

Once we have the input dataset, we keep 20% data aside as unseen data for
final testing. The remaining 80% is then passed to the model Pipeline as
batches which gets divided into 75% and 25% to be used for training and
validation.

A batch of 5000 inputs are passed in every iteration into the Pipeline, which
has a StandardScaler() and Model (). The StandardScaler() helps to standardize
the input features and normalize the values to be between 0 and 1. The
standardized features are then passed to each model for training and
validation. The progressive training and validation accuracies are noted and
compared. The final model performance is noted and compared for Accuracy in “
8” and F1 score (balance between precision and recall) in “ 9” . To understand
if the model is overfitted on any specific dataset, we have captured the
difference between average of progressive testing and validation accuracies in
“ 10” .

Figure 8: Comparison of Accuracy for 15 datasets across 4 classifiers

Figure 9: Comparison of F1 Score for 15 datasets across 4 classifiers

Figure 10: Comparison of Progressive Training and Validation accuracy to
evaluate Overfitting for 15 datasets across 4 classifiers

Analysis and Result on Model Performance

  1. 1.

The best performance across all datasets for Accuracy as well as F1 score was
presented by 15th dataset which is the one with RGB images enhanced with all
edges combined

  2. 2.

Features only and its combination dataset has shown considerably higher
accuracy and F1 score for all classifiers compared to feature enhanced RGB
versions. But it is to be noted that as per table 3, features only dataset has
overfitted the model compared to its feature enhanced version, because of the
high level of sparseness in the input feature matrix.

  3. 3.

Across all the three edge detectors used for comparison, edges features using
HED technique performed the best as a single feature and in combination
whereas Canny scored the lowest scores

4) Predicting on test dataset

We have trained 4 models on 15 datasets, which is a total of 60 trained
models. We test all of them on two different datasets: 1) Unseen images which
are similar to the training dataset 2) Random images taken using a phone
camera with different backgrounds.

a. Performance on Unseen Dataset with White background

Figure 11: Comparison of model performance on test set 1 with white background

As seen on “ 11” the model performance on the dataset with white background
was similar to what was expected from the model performance during training.

  1. 1.

The best performance across all datasets for accuracy was presented by 15th
dataset which is the one with RGB images enhanced with all edges combined

  2. 2.

Features only dataset has shown considerably higher accuracy for all
classifiers compared to Feature enhanced RGB versions.

  3. 3.

Unlike training model performance, all features performed almost equally well
for features only dataset

b. Performance on random dataset with different background

As seen on “ 13” the model performance on the random dataset with different
background was different from test set 1.

Figure 12: Sample images from test set 2

  1. 1.

The best performance across all datasets for accuracy was presented by 15th
dataset which is the one with RGB images enhanced with all edges combined.

  2. 2.

Even though there was an improvement in accuracy from initial augmented data
to feature enhanced images, the overall accuracy was way lower for test set 2
compared to test set 1. The reason for this could be that since the model is
trained on the images with white background, it may not be performing well for
the ones with different background. The model only understands the pixels
where there are edges. For a test image with background, its edges will also
appear in the feature matrix, which makes it difficult for model to understand
and predict.

  3. 3.

HED features performed the best as a single feature and in feature
combinations.

Figure 13: Comparison of model performance on test set 2 with different
background

##  IV Summary of Results

We trained and tested 4 models on 15 datasets with features only, features
combination and RGB images enhanced with the same. The results show that the
original image enhanced with combination of all three edges performed the best
overall. Comparing the three edge detectors, performance on datasets with HED
edges as single feature and its combination performed better. This could be
attributed to the clear, thick edge features given by HED.

##  V Conclusion and Future Work

The project aimed at creating a robust, balanced augmented dataset from
initial set of 27,000 ground truth data of textureless objects. Using that as
baseline, we created 14 other datasets each of 340,000 images by using 3 edge
detectors, Canny, Prewitt and HED. The initial seven were 3 single features
and 4 of its combination dataset. The next seven were RGB images enhanced with
the initial seven dataset. The experiments proved that the best performance
was presented by the RGB images enhanced with all features combined. HED
features proved to be performing the best among all features. Since the
initial ground truth was with white background, the models performed really
well on images with white background, whereas it could not perform when there
was a different background. In future work, we would like to perform the
experiments that can generalize the comparison between features irrespective
of background or clutter. This could be achieved by adding background and
clutter to the training set or by removing background or clutter from the test
set.

##  VI Acknowledgement

The authors would like to thank our mentor, Gabriel Lugo Bustillo, Phd
candidate, dept. of Computing Science, University of Alberta for his guidance
and feedback throughout the research. We would also like to thank our advisors
Dr. Irene Cheng and Dr. Anup Basu for their support and motivation to bring
out the best in our research. We would also like to thank the researchers of
previous work in the field of Textureless Object Recognition, which became the
starting point for our research.

##  VII Resources

Source Code: https://tinyurl.com/v7bbrkb

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##  VIII Author Biography

|  Frincy Clement is an Artificial Intelligence enthusiast who is currently
pursuing her MSc. in Computing Science at the University of Alberta. She
received her BTech in Computer Science and Master in Business Administration
from top universities in India. She has 6+years of work experience and is a
seasoned software developer with experience working in several application as
well as research-based projects in AI.
---|---
|  Kirtan Shah is a student pursuing Masters in the field of Computing Science
at University of Alberta. He completed his Bachelor’s in the field of Computer
Engineering in India. He has worked on projects that integrate IOT application
for the real-world needs during his bachelor’s. His interest is to apply
various techniques to solve real world problems. He likes to play chess as
well.
---|---
|  Dhara Pancholi is currently a MSc. in Computing Science specializing in
Multimedia at University of Alberta. She completed her bachelor’s degree in
Computer Science from India. Apart from different subjects that she learned,
her interest lies in computer graphics and animation. She has also worked on
different IOT projects in her bachelors and currently pursuing 3 projects at
university of Alberta as course projects.
---|---

##  IX Appendix

Figure 14: Sample of test set 2 after initial preprocessing for
features/feature combinations

Figure 15: Sample of test set 2 after initial preprocessing for RGB enhanced
with features/feature combinations


# Scaling Relations of Spectrum Form Factor and Krylov Complexity at Finite
Temperature

Chengming Tan  Hefei National Research Center for Physical Sciences at the
Microscale and School of Physical Sciences,University of Science and
Technology of China, Hefei 230026, China  Zhiyang Wei  MOE Key Laboratory for
Nonequilibrium Synthesis and Modulation of Condensed Matter, Shaanxi Province
Key Laboratory of Quantum Information and Quantum Optoelectronic Devices,
School of Physics, Xi’an Jiaotong University, Xi’an 710049, China  Ren Zhang
<EMAIL_ADDRESS>MOE Key Laboratory for Nonequilibrium Synthesis and
Modulation of Condensed Matter, Shaanxi Province Key Laboratory of Quantum
Information and Quantum Optoelectronic Devices, School of Physics, Xi’an
Jiaotong University, Xi’an 710049, China  Hefei National Laboratory, Hefei,
230088, China

###### Abstract

In the study of quantum chaos diagnostics, considerable attention has been
attributed to the Krylov complexity and spectrum form factor (SFF) for systems
at infinite temperature. These investigations have unveiled universal
properties of quantum chaotic systems. By extending the analysis to include
the finite temperature effects on the Krylov complexity and SFF, we
demonstrate that the Lanczos coefficients $b_{n}$, which are associated with
the Wightman inner product, display consistency with the universal hypothesis
presented in PRX 9, 041017 (2019). This result contrasts with the behavior of
Lanczos coefficients associated with the standard inner product. Our results
indicate that the slope $\alpha$ of the $b_{n}$ is bounded by $\pi k_{B}T$,
where $k_{B}$ is the Boltzmann constant and $T$ the temperature. We also
investigate the SFF, which characterizes the two-point correlation of the
spectrum and encapsulates an indicator of ergodicity denoted by $g$ in chaotic
systems. Our analysis demonstrates that as the temperature decreases, the
value of $g$ decreases as well. Considering that $\alpha$ also represents the
operator growth rate, we establish a quantitative relationship between
ergodicity indicator and Lanczos coefficients slope. To support our findings,
we provide evidence using the Gaussian orthogonal ensemble and a random spin
model. Our work deepens the understanding of the finite temperature effects on
Krylov complexity, SFF, and the connection between ergodicity and operator
growth.

##  I introduction

Unveiling the scaling relations of physical quantities is an important and
fundamental topic in the study of classical and quantum physics Stanley
(1999); Huckestein (1995); Osterloh  _et al._ (2002); García-García and Wang
(2008); Serra  _et al._ (1998); Pal and Huse (2010); Fisher and Berker (1982).
In the context of equilibrium systems, significant progress has been made in
this direction, leading to a deeper understanding of critical phenomena, phase
transitions, and universal behaviors Hung  _et al._ (2011); Puebla  _et al._
(2019); Chen  _et al._ (2019); Chauhan  _et al._ (2022). The investigation of
scaling relations in numerous systems are still attracting a growing
interests, especially in the context of non-equilibrium systems and quantum
dynamics of chaotic systems Sierant  _et al._ (2020); Šuntajs  _et al._
(2020).

Quantum chaos has been an attractive topic across diverse disciplines,
spanning from condensed matter physics to black hole physics and high-energy
physics Cotler  _et al._ (2017a); García-García and Verbaarschot (2016);
Cotler  _et al._ (2017b). This field holds the potential to offer deeper
insights into phenomena like thermalization Deutsch (2018, 1991); Srednicki
(1994); Rigol  _et al._ (2008); Kim  _et al._ (2014); Abanin  _et al._ (2019),
information scrambling Yan  _et al._ (2020); Shen  _et al._ (2020); et al.
(2021); Landsman  _et al._ (2019), and quantum entanglement Abanin  _et al._
(2019); Žnidarič  _et al._ (2008); Bardarson  _et al._ (2012). However,
comprehensively diagnosing quantum chaos remains a formidable challenge. To
date, several methods have emerged for characterizing the nature of quantum
chaos. These include spectrum statistics Shklovskii  _et al._ (1993); Frisch
_et al._ (2014); Liu (2018), out-of-time-correlation von Keyserlingk  _et al._
(2018); Rakovszky  _et al._ (2018), Loschmidt echo Yan  _et al._ (2020), and
complexity Prosen (2007). Each diagnostic tool captures different aspects of
quantum chaos and provides valuable insights individually. Recently, spectrum
form factor (SFF) Cotler  _et al._ (2017a, b); Gharibyan  _et al._ (2018);
Chen and Ludwig (2018); Chan  _et al._ (2018a) and Krylov complexity
(K-complexity) Parker  _et al._ (2019) stand out as prominent examples,
offering perceptions into the energy spectrum and operator growth in quantum
chaotic systems.

Typically, the characteristics of quantum chaos manifest across the entire
spectrum. To quantify the chaos strength, the statistical properties are
conventionally studied. For example, the Wigner-Dyson distribution describes
the statistical properties of a quantum chaotic system at infinite
temperature, where a level repulsion presents in the energy spectrum Guhr  _et
al._ (1998); Mehta (1991); Haake (2010). The statistical properties of the
spectrum provide information solely about the nearest level spacing within the
spectrum. In contrast, the SFF measures energy correlations between any two
states, which allows it to go beyond the constraints of spectrum statistics
and capture richer information. SFF has found widespread use in diverse
contexts, including random matrix theory (RMT) Guhr  _et al._ (1998); Brody
_et al._ (1981); Mehta (1991); Haake (2010); Kos  _et al._ (2018), the
Sachdev-Ye-Kitaev model Cotler  _et al._ (2017b, a), and the Hubbard model
Zhou  _et al._ (2023); Wittmann W.  _et al._ (2022); Dağ  _et al._ (2023). The
intriguing dip-ramp-plateau structure observed in many chaotic systems
indicats the universality of SFF Gharibyan  _et al._ (2018); Saad  _et al._
(2019); Winer  _et al._ (2020). The SFF can also be generalized to encapsulate
the correlation between multi energy levels Liu (2018); Wei  _et al._ (2024);
Cardella (2021); Gross and Rosenhaus (2017).

In quantum systems, the chaotic feature can also be manifested through
operator growth. Specifically, the evolution of an initially simple operator
$\hat{\cal O}(0)$ gradually becomes more complex under the Heisenberg
evolution. This occurs because the operator $\hat{\cal O}(0)$ does not commute
with the Hamiltonian $\hat{H}$ of an interacting quantum system. Several
approaches have been proposed to quantify the operator growth Nahum  _et al._
(2018); Khemani  _et al._ (2018); Chan  _et al._ (2018b); Roberts  _et al._
(2018). Recently, the concept of K-complexity was introduced to characterize
operator growth Parker  _et al._ (2019) and has been applied to various
systems, including strongly correlated systems, quantum circuits, and black
hole physics Barbón  _et al._ (2020); Magán and Simón (2020); Jian  _et al._
(2021); Kar  _et al._ (2022); Liu  _et al._ (2022); Barbón  _et al._ (2019);
Dymarsky and Gorsky (2020); Xu  _et al._ (2020); Avdoshkin and Dymarsky
(2020); Rabinovici  _et al._ (2021); Dymarsky and Smolkin (2021); Trigueros
and Lin (2022); Kim  _et al._ (2022); Hörnedal  _et al._ (2022); Rabinovici
_et al._ (2022a); Bhattacharjee  _et al._ (2022); Balasubramanian  _et al._
(2022); Heveling  _et al._ (2022); Adhikari and Choudhury (2022); Adhikari
_et al._ (2022); Caputa and Liu (2022); Mück and Yang (2022); Banerjee  _et
al._ (2022); Fan (2022a, b); Rabinovici  _et al._ (2022b); Bhattacharyya  _et
al._ (2023). It was found that operator growth can be quantified by the mean
value of wave packet position on a half-infinite chain. Motivated by this
idea, a universal hypothesis of operator growth has also been proposed, where
the Lanczos coefficients can be used to diagnose the presence of quantum
chaos. Meanwhile, the autocorrelation function ${\cal C}(t)={\rm Tr}(\hat{\cal
O}(t)\hat{\cal O}(0))$ defined at infinite temperature can be readily obtained
and has been intensively studied in various contexts Kubo (1957); Alhambra
_et al._ (2020); Carabba  _et al._ (2022). It is surprising to find that the
universal behavior of the autocorrelation function is determined by the first
few Lanczos coefficients Zhang and Zhai (2023). In addition, some studies have
explored the relations between K-complexity and other complexity measures,
such as circuit complexity, leading to a geometric interpretation of
K-complexity Caputa  _et al._ (2022); Lv  _et al._ (2023); Hörnedal  _et al._
(2023).

In this work, we establish a connection between K-complexity and SFF at finite
temperature. Our main conclusions can be summarized as follows: (1) The use of
the Wightman inner product and the standard inner product results in distinct
Lanczos coefficients $b_{n}$, and the slope $\alpha$ of $b_{n}$ in the former
is limited by $\pi k_{B}T$; (2) The ergodicity indicator $g$ diminishes with
decreasing temperature; (3) We establish a quantitative relationship between
$g$ and $\alpha$ after proper scaling.

##  II Lanczos coefficients at finite temperature

Let us begin with a brief review of the derivation of Lanczos coefficients,
and highlight the finite temperature effect of operator growth. Consider a
scenario where we have a simple operator $\hat{\cal O}$ and a Hamiltonian
$\hat{H}$. Typically, in the presence of interactions, they do not commute.
The evolution of the operator $\hat{\cal O}$ follows the Heisenberg equation
$-i\partial_{t}\hat{\cal O}(t)=[\hat{H},\hat{\cal O}(t)]$, and the formal
solution reads $\hat{\cal O}(t)={\rm e}^{i\hat{H}t}\hat{\cal O}{\rm
e}^{-i\hat{H}t}$. Henceforth, we set $\hbar=1$. Therefore, the initial simple
operator becomes progressively more complex due to the noncommutative relation
between $\hat{\cal O}(t)$ and $\hat{H}$, which is also termed as operator
growth. In order to quantify the complexity of $\hat{\cal O}(t)$, the concept
of K-complexity was introduced by Parker, et.al. Parker  _et al._ (2019).
Given a Hilbert space spanned by ${\ket{i}}$, the operator $\hat{\cal
O}=\sum_{i,j}{\cal O}_{ij}\ket{i}\bra{j}$ can be mapped to an “operator state”
$|{\hat{\cal O}}\rangle=\sum_{i,j}{\cal O}_{ij}\ket{i}\otimes\ket{j}$ living
in the double Hilbert space spanned by ${\ket{i}\otimes\ket{j}}$. By defining
a Liouvillian superoperator $\hat{\cal L}=[\hat{H},\cdots]$, the Heisenberg
equation can be rewritten as $\partial_{t}|{\hat{\cal O}(t)}\rangle=i\hat{\cal
L}|{\hat{\cal O}(t)}\rangle$. In the double Hilbert space, the Liouvillian
superoperator appears as a conventional operator, and can be written as
$\hat{\cal
L}=\hat{H}\otimes\hat{\mathbb{I}}-\hat{\mathbb{I}}\otimes\hat{H}^{T}$ where
$\hat{\mathbb{I}}$ denotes the identity operator with the same dimension of
$\hat{H}$. The superscript $T$ denotes transpose. The time-dependent operator
state can be expressed as:

$\displaystyle|\hat{\cal O}(t)\rangle={\rm e}^{i\hat{\cal L}t}|{\hat{\cal
O}}\rangle=\sum_{n=0}^{\infty}\frac{(it)^{n}}{n!}|{\hat{\cal L}^{n}\hat{\cal
O}}\rangle.$ (1)

This means that $|{\hat{\cal O}(t)}\rangle$ can be expanded under the basis
spanned by ${|{\hat{\cal L}^{n}\hat{\cal O}}}\rangle$. It is important to note
that this basis is neither normalized nor orthogonal. To address this issue,
the Krylov basis is defined by the Gram-Schmidt orthogonalization. It is
pointed out that one must properly define the inner product of two operator
states as follows:

$\displaystyle\langle{\hat{A}}|{\hat{B}}\rangle_{\beta}^{g}=\frac{1}{Z}\int_{0}^{\infty}g(\lambda){\rm
Tr}({\rm e}^{-\beta\hat{H}}{\rm e}^{\lambda\hat{H}}\hat{A}^{\dagger}{\rm
e}^{-\lambda\hat{H}}\hat{B})d\lambda.$ (2)

Here, $Z={\rm Tr}({\rm e}^{-\beta\hat{H}})$ denotes the partition function,
$\beta=1/(k_{\rm B}T)$ represents the inverse temperature, and $k_{\rm B}$
indicates the Boltzmann constant. $g(\lambda)$ satisfies the conditions
$g(\lambda)\geq 0$, $g(\beta-\lambda)=g(\lambda)$, and
$\beta^{-1}\int_{0}^{\beta}d\lambda g(\lambda)=1$ Parker  _et al._ (2019).
Different choices of $g(\lambda)$ define distinct inner products. In this
work, we consider two specific choices: (1) The standard inner product is
defined by choosing $g(\lambda)=[\delta(\lambda)+\delta(\lambda-\beta)]/2$,
and can be explicitly written as:

$\displaystyle\langle{\hat{A}}|{\hat{B}}\rangle_{\beta}^{S}=\frac{1}{2Z}{\rm
Tr}\left({\rm e}^{-\beta\hat{H}}\hat{A}^{\dagger}\hat{B}+\hat{A}^{\dagger}{\rm
e}^{-\beta\hat{H}}\hat{B}\right).$ (3)

Here, by “standard” we mean that this kind of inner product is widely adopted
in the linear response theory. (2) Another inner product often considered in
quantum field theory is the Wightman inner product, defined by
$g(\lambda)=\delta(\lambda-\beta/2)$, and can be explicitly written as:

$\displaystyle\langle{\hat{A}}|{\hat{B}}\rangle_{\beta}^{W}=\frac{1}{Z}{\rm
Tr}\left(e^{-\beta\hat{H}/2}\hat{A}^{\dagger}e^{\beta\hat{H}/2}\hat{B}\right).$
(4)

It should be noted that at infinitely high temperature, i.e., $\beta\to 0$,
both the standard and Wightman inner products reduce to the same one:
$\langle{\hat{A}}|{\hat{B}}\rangle={\rm
Tr}\left(\hat{A}^{\dagger}\hat{B}\right)/{\rm Tr}(\hat{\mathbb{I}})$. Equipped
with the inner product, we can define the Krylov basis denoted by
${|{\hat{\cal O}_{n}}}\rangle$ using the Lanczos algorithm as follows: for
$n=1,2$, we have $|{\hat{\cal O}_{0}}\rangle=|{\hat{\cal O}}\rangle$ and
$|{\hat{\cal O}_{1}}\rangle=b_{1}^{-1}\hat{\cal L}|{\hat{\cal O}_{0}}\rangle$,
respectively, where $b_{1}=\langle{\hat{\cal L}\hat{\cal O}_{0}}|{\hat{\cal
L}\hat{\cal O}_{0}}\rangle^{1/2}$; then for $n>2$, $|{\hat{\cal
O}_{n}}\rangle$ can be defined inductively as:

$\displaystyle|{\hat{\cal A}_{n}}\rangle=$ $\displaystyle\hat{\cal
L}|{\hat{\cal O}_{n-1}}\rangle-b_{n-1}|{\hat{\cal O}_{n-2}}\rangle;$
$\displaystyle b_{n}=$ $\displaystyle\left[\langle{\hat{\cal
A}_{n}}|{\hat{\cal A}_{n}}\rangle_{\beta}^{g}\right]^{1/2};$ (5)
$\displaystyle|{\hat{\cal O}_{n}}\rangle=$ $\displaystyle
b_{n}^{-1}|{\hat{\cal A}_{n}}\rangle.$

Here, $b_{n}$’s indicate the Lanczos coefficients, which obviously depends on
temperature. Previous research has demonstrated that the asymptotic behavior
of these coefficients at large $n$ can be utilized as a diagnostic for chaotic
systems Parker  _et al._ (2019). Using the completeness and orthogonality of
the Krylov basis, the Heisenberg equation can be mapped to a discrete
Schrödinger-like equation
$\partial_{t}\varphi_{n}(t)=b_{n}\varphi_{n-1}(t)-b_{n+1}\varphi_{n+1}(t)$
with $\varphi_{n}(t)=i^{-n}\langle{\cal O}_{n}|{\cal O}(t)\rangle$. The
natural initial condition reads $\varphi_{n}(0)=\delta_{n0}$.

Figure 1: Finite temperature Lanczos coefficients $b_{n}$ for Gaussian
orthogonal ensemble (GOE). The operator is chosen as
$\hat{\sigma}_{z}\otimes\hat{\mathbb{I}}_{L/2}$, and the dimension of random
matrix is $8192$. In (a) and (b), Wightman and standard inner product are
used, respectively. (a): At finite temperature, $\\{b_{n}\\}$ defined by
Wightman inner product shows linear behavior, which is consistent with the
universal hypothesis. (b): The finite temperature $b_{n}$ presents oscillatory
behavior which is in contrast to the universal hypothesis. (c): The slope
$\alpha(T)=\pi k_{\rm B}T$ aligns with upper bound of the operator growth
rate, and approach at infinite temperature.

To illustrate the difference in ${b_{n}}$’s defined by the standard and
Wightman inner products, we calculate ${b_{n}}$ for the Gaussian orthogonal
ensemble (GOE), which serves as an ideal chaotic model. The Hamiltonian can be
expressed as $\hat{H}_{\rm GOE}=\mathcal{E}{\rm GOE}(L,\mu,\sigma)$, where the
$\mathcal{E}$ is the characteristic energy and ${\rm GOE}(L,\mu,\sigma)$
represents a $L\times L$ Hermitian matrix of which the elements are sampled
from Gaussian distribution with mean value $\mu$ and standard deviation
$\sigma$. The operator is chosen as
$\hat{\sigma}_{z}\otimes\hat{\mathbb{I}}_{L/2}$, where $\hat{\sigma}_{z}$ and
$\hat{\mathbb{I}}_{L/2}$ denote the Pauli matrix and $L/2\times L/2$ identity
matrix, respectively. Obviously, this is a simple operator. By simple, we mean
it is represented by a sparse matrix. In evolution, this operator becomes
complex progressively. The ${b_{n}}$’s obtained using the Wightman and
standard inner products are shown in Fig. 1(a) and (b), respectively. It is
important to note that at infinite temperature, both inner products yield the
same ${b_{n}}$ values, aligning with our expectations. However, the $b_{n}$’s
at finite temperature showcase quite different behaviors for the above-
mentioned inner products. The Wightman inner product leads to a linear $b_{n}$
at small $n$, i.e., $b_{n}\approx\alpha(T)n$. The slope $\alpha$ increases as
the temperature rises and approaches infinity at infinite temperature. This is
reasonable in the sense that at higher temperatures, the speed of operator
growth is larger. The range of $n$ that supports linear $b_{n}$ also extends
at low temperatures. This indicates that the universal hypothesis is
applicable even at finite temperature when the Wightman inner product is
adopted. The saturation at large $n$ is due to finite-size effects. In
contrast, $b_{n}$ shows oscillatory behavior at finite temperature when the
standard inner product is used, and the oscillatory behavior becomes more
pronounced at lower temperatures, which is not aligned with the universal
hypothesis. In the following analysis, we will focus on the Wightman inner
product. In Fig. 1(c), we present the slope $\alpha(T)$ for various
temperatures. It is clear that for GOE,

$\displaystyle\alpha(T)=\pi k_{B}T,$ (6)

which, on one hand, is consistent with the universal hypothesis on the upper
bound of the Lanczos coefficient growth rate $\alpha(T)\leq\pi k_{B}T$, and on
the other hand, manifests GOE being the most chaotic model. The slight
derivation at high temperature is due to the finite size effect.

##  III Spectrum form factor

The SFF is another emerging diagnostic of quantum chaosCotler  _et al._
(2017b); Kos  _et al._ (2018), which measures the spectrum correlation of
arbitrary two states. Specifically, we consider the following spectrum
correlation,

$\displaystyle{\cal
C}(E)=\frac{1}{L^{2}}\sum_{i,j}\delta\left[E-\left(E_{i}-E_{j}\right)\right],$
(7)

It is evident that as long as the energy difference between two states is $E$,
they contribute to ${\cal C}(E)$, regardless of whether they are the nearest
neighbors or not. The SFF can be obtained by a Fourier transform as follows:

$\displaystyle K(t)=L\int dEe^{-iEt}{\cal C}(E)=\frac{1}{L}\left|{\rm
Tr}\left({\rm e}^{-i\hat{H}t}\right)\right|^{2}.$ (8)

It has been demonstrated that the “dip-ramp-plateau” structure of SFF is a
universal behavior for quantum chaotic systems. In the long time limit,
$K(t)\to 1$ according to our definition. Moreover, the SFF can also be
generalized to finite temperature by the analytical extension $t\to t-i\beta$,
where $\beta=1/(k_{B}T)$ Cotler  _et al._ (2017b). The finite temperature
effect of SFF has been investigated in paradigmatic quantum chaotic systems,
and the “dip-ramp-plateau” structure survives, which is reproduced in Fig. 2.

Two time scales can be extracted from the SFF Edwards and Thouless (1972);
Thouless (1974); Sierant  _et al._ (2020); Šuntajs  _et al._ (2020); one is
the Heisenberg time $t_{\rm H}$; the other one is the Thouless time $t_{\rm
Th}$. The former one is defined as $t_{\rm H}=2\pi/\Delta$ with $\Delta$ being
the mean energy level spacing which exponentially decreases as the system size
increases Sierant  _et al._ (2020); Šuntajs  _et al._ (2020). In SFF, the
Heisenberg time can be pinned down by the presence of plateau. The Thouless
time is the timescale that characterizes how initial local information
diffuses throughout the system, and can be defined as the timescale beyond
which the SFF becomes consistent the universal one of GOE. In contrast to the
Heisenberg time, the Thouless time increases polynomially as the system time
increases Sierant  _et al._ (2020); Šuntajs  _et al._ (2020); D’Alessio  _et
al._ (2016). In RMT, the analytical form of $K(t)$ at infinite temperature is
available. For instance, $K(t)$ of GOE at intermediate and long time can be
written as Vasilyev  _et al._ (2020)

$\displaystyle K_{\rm a}(t)=\left\\{\begin{aligned} &\frac{2t}{t_{\rm
H}}-\frac{t}{t_{\rm H}}{\rm log}_{10}\left(1+\frac{2t}{t_{\rm
H}}\right),0<t<t_{\rm H}\\\ &2-\frac{t}{t_{\rm H}}{\rm
log}_{10}\left(\frac{2t+t_{\rm H}}{2t-t_{\rm H}}\right),t>t_{\rm
H}\end{aligned}\right.,$ (9)

where we have taken the random average with respect to the probability of GOE.
It can be checked that in the long time limit $K_{\rm a}(t)\to 1$, which is
consistent with our definition in Eq.(8).

Figure 2: The spectrum form factor(SFF) at finite temperature. The “dip-ramp-
plateau” structure of SFF survives at finite temperature(colored line). The
Thouless time(colored diamond) is determined as SFF at finite temperature is
close enough to the analytical SFF(black dash line) of GOE at infinite
temperature. The Heisenberg time is shown as the black triangle marker.

With the Heisenberg and Thouless time, a quantity to measure the ergodicity in
quantum systems is defined,

$\displaystyle g=\log_{10}(t_{\rm H}/t_{\rm Th})$ (10)

which is termed as ergodicity indicator Sierant  _et al._ (2020); Šuntajs  _et
al._ (2020). In the thermodynamic limit, the ergodicity indicator $g$
describes a transition from the quantum ergodic regime $g\to\infty$ ($t_{\rm
H}/t_{\rm Th}\to\infty$) to nonergodic regime $g\to-\infty$ ($t_{\rm H}/t_{\rm
Th}\to 0$).

Similar to the generalization of SFF, we extend the ergodicity indicator to
finite temperature since the Heisenberg and Thouless time can still be
extracted from the SFF at finite temperature. In Fig. 2, we illustrate the
extraction of Heisenberg and Thouless time from SFF at finite temperature.
Specifically, we numerically generate 150 samples of GOE at system size of
$L=4096$ for each considered temperature. Following the method in Sierant  _et
al._ (2020); Šuntajs  _et al._ (2020), we apply the unfolding procedure, which
maps each eigenvalue $E_{i}$ to a dimensionless number $\epsilon_{i}$ times
average level spacing $\Delta$ and avoids the global energy dependence.
Meanwhile, the information in local energy fluctuation reserves Haake (2010).
Then the SFF after the unfolding procedure is written as

$\displaystyle K(t)=\frac{1}{L}\left|\sum_{i}{\rm e}^{-iE_{i}t}\right|^{2}\to$
$\displaystyle\frac{1}{L}\left|\sum_{i}{\rm
e}^{-i(\epsilon_{i}\Delta)t}\right|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{L}\left|\sum_{i}{\rm e}^{-i2\pi\epsilon_{i}t/t_{\rm
H}}\right|^{2}.$ (11)

In the last step, we use the definition of the Heisenberg time $t_{\rm
H}=2\pi/\Delta$. Therefore, the Heisenberg time, labelled by the black
triangle in Fig. 2, naturally serves as the time unit. The colored solid lines
represent the average SFF of 150 samples in GOE at various temperature. It is
evident that the SFF at finite temperature adheres to the analytical SFF
$K_{\rm a}(t)$(black dash line) after the Thouless time(colored diamond).
Numerically, the Thouless time is determined when finite temperature SFFs are
sufficiently close to analytical SFF $K_{\rm a}(t)$, the quantitative
criterion of which reads $\left|{\rm log}_{10}\left[\frac{K(t_{\rm
Th})}{K_{\rm a}(t_{\rm Th})}\right]\right|<0.05$ Sierant  _et al._ (2020). It
is evident that a system with lower temperature exhibits a longer Thouless
time, resulting in a smaller ergodicity indicator. This observation aligns
with our intuition, indicating that the reduction in temperature suppresses
the characteristics of chaos, as similarly observed in the previous section
for Lanczos coefficients.

Figure 3: The scaled ergodicity indicator $g^{\rm(GOE)}_{S}(T)$ and its
relation to the slope $\alpha(T)$ of Lanczos coefficients. (a): The initial
ergodicity indicators displayed in the inset are different by a size
dependence. After scaling, the scaled ergodicity indicators of different
system size show collapse to a single curve. (b): The linear relation between
scaled ergodicity indicator and Lanczos coefficients slope. The error bar of
scaled ergodicity indicator is the standard deviation of 10 sets of samples.
The fluctuation of $\alpha(T)$ is so small during the sampling procedure that
the error bar on Krylov complexity side is not shown. The black dotted line is
plotted to show linearity.

##  IV Scaling relation of GOE

As of now, we present two chaos diagnostic tools that depend on temperature. A
natural question arises: what is the relation between them? In this section,
we present a scaling relationship between the ergodicity indicator $g(T)$ and
the slope of Lanczos coefficients $\alpha(T)$ at different temperature, which
is showcased in Fig. 3(b). To this end, we firstly study the temperature
dependence of ergodicity indicator $g(T)$, as illustrated in the inset of Fig.
3(a). The ergodicity indicator increases as the temperature ramps, as
expected, and it also shows size dependency, the same as SFF Cotler  _et al._
(2017b). This size dependence precludes a meaningful and direct quantitative
comparison between $g(T)$ and $\alpha(T)$.

In order to eliminate the size-dependent effects, a scaled ergodicity
indicator denoted as $g^{\rm(GOE)}_{\rm S}(T)$ has been proposed. It remains
invariant with respect to different system size. Notably, for a particular
system that can be simulated by GOE, the spectrum spans approximately from
$-\mathcal{E}\sqrt{L}$ to $\mathcal{E}\sqrt{L}$. We postulate that temperature
should be scaled by a factor of $\sqrt{L}$. Using the definition of Heisenberg
time, we find that it is system size-dependent, i.e., $t_{\rm H}=2\pi
L/(E_{\rm max}-E_{\rm min})\sim 2\pi\sqrt{L}/\mathcal{E}$. This contrasts with
the Thouless time. As such, we can shift the value of the ergodicity indicator
by amount of $\log_{10}(\sqrt{L})$, and define scaled ergodicity indicator as

$g^{\rm(GOE)}_{\rm S}(T)={\rm log}_{10}\left[\frac{t_{\rm H}}{t_{\rm
Th}(T)\sqrt{L}}\right].$ (12)

In Fig. 3(a), we present the scaled ergodicity indicator for different system
size. Our results show that, within a broad temperature range, the scaled
ergodicity indicators, denoted as $g^{\rm(GOE)}_{\rm S}(T)$, collapse to a
single curve across various system sizes. We also would like to mention that
the temperature in Fig. 3(a) has also be scaled by $\sqrt{L}$, which confirms
our previous postulation. The error bars represent the standard deviations of
scaled ergodicity indicators from ten repetitions of sampling procedure.

Then the relation between $g^{\rm(GOE)}_{\rm S}(T)$ and $\alpha(T)$ can be
well defined. It has been demonstrated that the ratio of the time it takes for
the SFF to reach the minimum and the plateau is proportional to
$\sqrt{L}/(1+\beta^{2}/2)$ for GUE Cotler  _et al._ (2017b). Noticing that the
SFF at Thouless time is close to the minimum and the Heisenberg time is pinned
down by the presence of plateau, we conjecture the ratio of Heisenberg time
and Thouless time for GOE has similar form and can be written as

$\displaystyle\frac{t_{\rm H}}{t_{\rm
Th}}\propto\left(\frac{\sqrt{L}}{1+\beta^{2}\mathcal{E}^{2}/2}\right)^{\delta}.$
(13)

Here, $\delta$ is a fitting parameter and $\beta\mathcal{E}$ is dimensionless.
Upon substituting Eq.(13) and (6) into Eq.(12), we obtain a scaling relation
between $g^{\rm(GOE)}_{\rm S}(T)$ and $\alpha(T)$,

$g^{\rm(GOE)}_{\rm S}(T)=\delta{\rm
log}_{10}\left[\frac{2(\alpha(T)/\mathcal{E})^{2}}{2(\alpha(T)/\mathcal{E})^{2}+\pi^{2}}\right]+C,$
(14)

where we have used $\beta=1/k_{\rm B}T$ and $C$ is a fitting parameter. In
Fig. 3(b), we show that Eq.(14) indeed holds. For different system sizes, the
curves collapse to a single one. By fitting Eq.(14) to numerical results, we
find $\delta=0.25$ and $C=-0.21$. The origin of the error bars is the same as
that for scaled ergodicity indicators.

##  V Scaling relation of random spin model

Up to now, we work on the random matrices which describe sufficient
complicated and chaotic systems. In this section, we study a 1D Ising model in
transverse and longitudinal field, TL-Ising model in short. In our
calculation, the longitudinal field takes a random value on each site. This
model is nonintegrable and chaotic at uniform transverse and random
longitudinal field Noh (2021); Nivedita  _et al._ (2020); Trigueros and Lin
(2022). And the energy spectrum follows the Wigner-Dyson distribution, which
is a hallmark of quantum chaos. Besides, we choose the random parameter far
away from the many-body-localization phase Trigueros and Lin (2022); De Tomasi
_et al._ (2021) to reserve the chaotic characteristics. In this system, the
linear scaling relation is tested.

Specifically, the Hamiltonian of TL-Ising model is written as

$\hat{H}_{\rm
spin}=\sum_{i}^{N}(-J\hat{\sigma}^{x}_{i}\hat{\sigma}^{x}_{i+1}+h\hat{\sigma}^{z}_{i}+g_{i}\hat{\sigma}^{x}_{i}),$
(15)

where $J$ is the near-neighbor coupling and taken as the energy dimension in
the following discussion, $h$ represents the uniform transverse field and
$g_{i}$ represents the site-dependent longitudinal field, $N$ is the total
number of spin and $\hat{\sigma}_{i}^{a}$ with $a=x,z$ is the Pauli operator
on $i$-th site. Formally, this operator should be considered in version of
direct product that
$\hat{\sigma}_{i}^{a}=...\otimes\underbrace{\hat{\mathbb{I}}_{2}}_{i-1\text{-th
site}}\otimes\underbrace{\hat{\sigma}^{a}}_{i\text{-th
site}}\otimes\underbrace{\hat{\mathbb{I}}_{2}}_{i+1\text{-th site}}\otimes...$
with $\hat{\mathbb{I}}_{2}$ being the $2\times 2$ identity operator. When the
longitudinal field is turned off ($g_{i}=0$ for each site), this model can be
mapped to a free fermion model and becomes integrable. As shown in Noh (2021)
with sightly applying uniform longitudinal field, the integrability is broken
and the system tends to be chaotic. In our case, we take site-dependent
longitudinal field $g_{i}$, which is sampled from a uniform distribution.

Figure 4: The SFF and logarithmic corrected slope $A(T)$ of spin model in
transverse and random longitudinal field(TL-Ising model) at finite
temperature. (a): The black dash line is the analytical SFF of GOE at infinite
temperature. The colored solid lines represent the average SFF of TL-Ising
model at finite temperature, from which the Thouless time is extracted(colored
diamond). The Heisenberg time is still labelled by black triangle. (b): The
slope $A(T)$ of Lanczos coefficient defined by the Wightman inner product
exhibits a progressive increase with rising temperature and saturates at high
temperature. The error bar is the standard deviation of 10 sets of samples.

The SFF of TL-Ising model at finite temperature is shown in Fig. 4(a), where
$N=8$, $h=J$ and $g_{i}$ is sampled from $[-J,J]$. Each colored curve of SFF
data is taken average from 200 samples. The “dip-ramp-plateau” structure can
be observed, which confirms the existence of chaos. Using the same method
illustrated in GOE, the Thouless time $t_{\rm Th}$ and Heisenberg time $t_{\rm
H}$ are extracted from the SFF data. The Heisenberg time still serves as a
time unit. At lower temperature, the SFF of TL-Ising model takes longer time
to adhere to the analytical SFF of GOE, indicating larger Thouless time. Then
the ergodicity indicator defined by Eq.(10) can be calculated. As expected,
the ergodicity indicator shows a size dependence, seen in the inset of Fig.
5(a). As such, we need to scale the SFF to avoid the size effect.

The scaling method on TL-Ising model is different from the case of random
matrix. For energy of TL-Ising model ranges from $\sim-JN$ to $\sim JN$. The
temperature $k_{B}T/J$ is scaled to $k_{B}T/NJ$. In contrast to GOE, the
Thouless time is proportion to $N^{2}$ Sierant  _et al._ (2020); Šuntajs  _et
al._ (2020); D’Alessio  _et al._ (2016). Therefore, the scaled ergodicity
indicator for TL-Ising model is written as

$g_{\rm S}^{\rm(Ising)}(T)={\rm log}_{10}\left[\frac{t_{\rm H}N^{2}}{t_{\rm
Th}(T)}\right].$ (16)

As shown in Fig. 5(a), the size effect on ergodicity indicator for TL-Ising
model is almost eliminated.

Figure 5: The scaled ergodicity indicator $g^{\rm(Ising)}_{S}(T)$ and the
relation between $g^{\rm(Ising)}_{S}(T)$ and $A(T)$ for TL-Ising model. (a):
The initial ergodicity indicator is displayed in the inset, which also
presents system size dependence. The finite size dependence is eliminated in
the scaled ergodicity indicator. (b): The relation between the scaled
ergodicity indicator and $A(T)$. The linearity can be observed at relatively
low temperature. The black dotted line is plotted to show linearity.

On the Krylov complexity side, we also use the Wightman inner product in the
calculation of Lanczos coefficient $b_{n}$. In our calculation, the operator
is chosen as
$\hat{\sigma}^{(1)}_{z}\otimes\Pi_{i=2}^{N}\hat{\mathbb{I}}_{2}^{(i)}$. As the
system is one dimensional, the linear growth of $b_{n}$ is corrected by
logarithmic correction $b_{n}=A(T)n/{\rm ln}(n)$. Here $A(T)$ is the slope of
$\left\\{b_{n}\right\\}$ sequence after logarithmic correction. Considering
the disordered longitudinal field, we obtain the final $b_{n}$ by the mean
value of 10 samples, and fit the results by a linear function. The temperature
dependence of $A(T)$ is shown in Fig. 4(b). The points in figure represent the
average of $A(T)$ from five repetitions and the error bars represent their
standard deviations. It is obvious that $A(T)$ tends to saturate at high
temperature limit, which indicates the TL-Ising model is chaotic but not
sufficiently chaotic.

Then the scaling relation between the scaled ergodicity indicator and the
Lanczos coefficients slope can be established. Inspired by the similarity of
spectral properties between GOE and TL-Ising model Sierant  _et al._ (2020);
Šuntajs  _et al._ (2020), we expect that the scaling law established in GOE
should also hold for the TL-Ising model. Therefore, the counterpart of Eq.
(14) can be written as

$g_{\rm S}^{\rm(Ising)}(T)=\delta{\rm
log}_{10}\left[\frac{2(A(T)/J)^{2}}{2(A(T)/J)^{2}+\pi^{2}}\right]+C.$ (17)

As shown in Fig.5(b), for different system size, the linear scaling relation
appears at relatively low temperature, which is consistent with the result of
GOE. By fitting the numerical results to Eq.(17), we find $\delta=1.07$ and
$C=2.94$ for TL-Ising model. We suspect that the deviation from linearity can
be attributed to the property that this model is not sufficiently chaotic.

##  VI conclusion

In summary, we numerically investigate the finite-temperature behavior of
K-complexity and SFF in the context of GOE and TL-Ising model. Within the GOE,
the Lanczos sequence $\left\\{b_{n}\right\\}$ defined by the Wightman inner
product exhibits notable linear growth at finite temperatures, which is not
observed in the $\left\\{b_{n}\right\\}$ defined by the standard inner
product. Utilizing the Wightman inner product, the slope $\alpha$ of $b_{n}$
saturates the universal upper bound $\pi k_{B}T$, indicating that the random
matrix effectively simulates highly complex or chaotic systems. In the case of
TL-Ising model, the behavior of $b_{n}$ defined by the Wightman inner product
displays linear growth after a logarithmic correction. The slope $A$ converges
to a specific value at infinite temperature, suggesting that the chaotic
nature of this spin system is comparatively subdued compared to the GOE.

Furthermore, we extend the concept of the ergodicity indicator $g$ to finite
temperatures. Through a scaling procedure, which relies on system
characteristics, we eliminate the size-dependence and preserve information
reflective of the chaotic features inherent in the system. Then, we establish
a linear relationship between the scaled ergodicity indicator $g_{\rm S}(T)$
and the Lanczos coefficients slope $\alpha(T)$ or $A(T)$. Our results build a
connection between these two aspects of quantum chaos diagnostic, despite
their disparate origins.

Acknowledgement. This work is supported by Innovation Program for Quantum
Science and Technology (Grants No.2021ZD0302001), NSFC (Grants No.12174300),
the Fundamental Research Funds for the Central Universities (Grant No.
71211819000001) and Tang Scholar.

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