\r\n\tBacteriology is subdivision of microbiology which deals with morphology, ecology and biotechnology of bacteria that found in different environmental niches - either inside living organisms, or free living in soil, marine and fresh water. It is also connected to medicine concerning spoilage of foods and bacterial associated diseases (pathogenic bacteriology). On the other hand, good use of friendly bacteria gives protection from other bad microbes causing serious illness. These beneficial bacteria promote absorption of nutrients and aid in healthy digestion.
\r\n
\r\n\tBacteria are key players in bioremediation.They can play a significant role in the mitigation or removal of contaminants in the environment, both organic and inorganic.
\r\n
\r\n\tIn natural environment, bacteria produce nanoparticles as part of their metabolism. Bacteria grab target ions from their environment and then turn the metal ions into the element metal through enzymes generated by the cell activities.The biosynthesized nanoparticles have been used in a variety of applications including drug carriers for targeted delivery, cancer treatment, gene therapy and DNA analysis, antibacterial agents, biosensors and magnetic resonance imaging (MRI).
\r\n
\r\n\tThis book intends to provide the reader with a comprehensive overview of bacterial science and it's applications in different disciplines.
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She received her BSc in Microbiology, MSc and Ph.D. in Marine Microbiology from Faculty of Science, Alexandria University in 1998, 2003 and 2008, respectively. She had 25 published papers in local and international peer-reviewed journals and 2 abstracts conference proceedings in the field of marine and microbial biotechnology. She worked as an Assistant Professor at Faculty of Science and Humanities studies, Shaqra University, Saudi Arabia, from 2010 -2012 where she conducted lectures on General Microbiology, Bacteriology and Pollution. She is a member of numerous local societies and serves as an editorial board of the International Journal of Scientific and Technology Research, International Archive of Medicine, Lawarence Press, International Journal of Natural Resource Ecology and Management. She was also selected as a member at Who\\'s Who in the World for inclusion in the forthcoming 31st Edition 2014. 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1. Introduction
There are fascinating problems with computer vision, such as image classification and object detection, both of which are part of an area called object recognition. For these types of issues, there has been a robust scientific development in the last years, mainly due to the advances of convolutional neural networks, deep learning techniques, and the increase of the parallelism processing power offered by the graphics processing units (GPUs). The image classification problem is the task of assigning to an input image one label from a fixed set of categories. This classification problem is central within computer vision because, despite its simplicity, there are a wide variety of practical applications and has multiple uses, such as labeling skin cancer images [1], use of high-resolution images to detect natural disasters such as floods, volcanoes, and severe droughts, noting the impacts and damage caused [2, 3, 4].
The performance of image classification algorithms crucially relies on the features used to feed them [5]. It means that the progress of image classification techniques using machine learning relied heavily on the engineering of selecting the essential features of the images that make up the database. Thus, obtaining these resources has become a daunting task, resulting in increased complexity and computational cost. Commonly, two independent steps are required for image classification, feature extraction, and learning algorithm choice, and this has been widely developed and enhanced using support vector machines (SVMs).
The SVM algorithm, when considered as part of the supervised learning approach, is often used for tasks as classification, regression, and outlier detection [6]. The most attractive feature of this algorithm is that its learning mechanism for multiple objects is simpler to be analyzed mathematically than traditional neural network architecture, thus allowing to complex alterations with known effects on the core features of the algorithm [7]. In essence, an SVM maps the training data to higher-dimensional feature space and constructs a separation hyperplane with maximum margin, producing a nonlinear separation boundary in the input space [8].
Today, the most robust object classification and detection algorithms use deep learning architectures, with many specialized layers for automating the filtering and feature extraction process. Machine learning algorithms such as linear regression, support vector machines, and decision trees all have its peculiarities in the learning process, but fundamentally they all apply similar steps: make a prediction, receive a correction, and adjust the prediction mechanism based on the correction, at a high level, making it quite similar to how a human learns. Deep learning has appeared bringing a new approach to the problem, which attempted to overcome previous drawbacks by learning abstraction in data following a stratified description paradigm based on a nonlinear transformation [9]. A key advantage of deep learning is its ability to perform semi-supervised or unsupervised feature extraction over massive datasets.
The ability to learn the feature extraction step present in deep learning-based algorithms comes from the extensive use of convolutional neural networks (ConvNet or CNN). In this context, convolution is a specialized type of linear operation and can be seen as the simple application of a filter to a determined input [10]. Repeated application of the same filter to an input results in a map of activations called a feature map, indicating the locations and strength of a detected feature in the input by tweaking the parameters of the convolution. The network can adjust itself to reduce the error and therefore learn the best parameters to extract relevant information on the database.
Many deep neural network (DNN)-based object detectors have been proposed in the last few years [11, 12]. This research investigates the performance of state-of-the-art DNN models of SSD and Faster RCNN applied to a classical detection problem where the algorithms were trained to identify several animals in images; furthermore to exemplify the application in scientific research, the YOLO network was trained to solve the mice tracking problem. The flowing sections describe the DNN models mentioned earlier in more details [13, 14, 15].
2. Object detection techniques
2.1 Single shot multibox detector
The single shot multibox detector [13] is one of the best detectors in terms of speed and accuracy comprising two main steps, feature map extraction and convolutional filter applications, to detect objects.
The SSD architecture builds on the VGG-16 network [16], and this choice was made based on the strong performance in high-quality image classification tasks and the popularity of the network in problems where transfer learning is involved. Instead of the original VGG fully connected layers, a set of auxiliary convolutional layers change the model, thus enabling to extract features at multiple scales and progressively decrease the size of the input to each subsequent layer.
The bounding box generation considers the application of matching pre-computed, fixed-size bounding boxes called priors with the original distribution of ground truth boxes. These priors are selected to keep the intersection over union (IoU) ratio equal to or greater than 0.5.
The overall loss function defined in Eq. (1) is a linear combination of the confidence loss, which measures how confident the network is of the computed bounding box using categorical cross-entropy and location loss, which measures how far away the networks predicted bounding boxes are from the ground truth ones using L2 norm.
Lxclg=1NLconfxc+αLlocxlgE1
where N is the number of matched default boxes and Lconf and Lloc are the confidence and location loss, respectively, as defined in [13]. Figure 1 depicts how to apply the convolutional kernels to an input image in the SSD architecture.
Figure 1.
The SSD network has several feature layers to the end of the base network, which predicts the offsets to default boxes of different scales, aspect ratios, and their associated confidences. Figure based on [13].
2.2 You only look once
You only look once [14] is a state-of-the-art object detection algorithm which targets real-time applications, and unlike some of the competitors, it is not a traditional classifier purposed as an object detector.
YOLO works by dividing the input image into a grid of S×S cells, where each of these cells is responsible for five bounding boxes predictions that describe the rectangle around the object. It also outputs a confidence score, which is a measure of the certainty that an object was enclosed. Therefore the score does not have any relation with the kind of object present in the box, only with the box’s shape.
For each predicted bounding box, a class it’s also predicted working just like a regular classifier giving resulting in a probability distribution over all the possible classes. The confidence score for the bounding box and the class prediction combines into one final score that specifies the probability for each box includes a specific type of object. Given these design choices, most of the boxes will have low confidence scores, so only the boxes whose final score is beyond a threshold are kept.
Eq. (2) states the loss function minimized by the training step in the YOLO algorithm.
where 1iobj indicates if an object appears in cell i and 1ijobj denotes the jth bounding box predictor in cell i responsible for that prediction; x, y, w, h, and C denote the coordinates that represent the center of the box relative to the bounds of the grid cell. The width and height predictions are relative to the whole image. Finally, C denotes the confidence prediction, that is, the IoU between the predicted box and any ground truth box.
Figure 2 describes how the YOLO network process as image. Initially, the input gets passed through a CNN producing the bounding boxes with its perspectives confidences scores and generating the class probability map. Finally, the results of the previous steps are combined to form the final predictions.
Figure 2.
YOLO model detection as a regression problem [17]. Thus the input image is divided into a S×S grid and for each grid cell, B bounding boxes, confidence for those boxes, and C class probabilities are predicted. These encoded predictions are as an S×S×B∗5+C tensor. Figure based on [17].
2.3 Faster region convolutional neural network
The faster region convolutional neural network [15] is another state-of-the-art CNN-based deep learning object detection approach. In this architecture, the network takes the provided input image into a convolutional network which provides a convolutional feature map. Instead of using the selective search algorithm to identify the region proposals made in previous iterations [18, 19], a separate network is used to learn and predict these regions. The predicted region proposals are then reshaped using a region of interest (ROI) pooling layer, which is then used to classify the image within the proposed region and predict the offset values for the bounding boxes.
The strategy behind the region proposal network (RPN) training is to use a binary label for each anchor, so the number one will represent the presence of an object and number zero the absence; with this strategy any IoU over 0.7 determines the object’s presence and below 0.3 indicates the object’s absence.
Thus a multitask loss function shown in Eq. (3) is minimized during the training phase.
Lpiti=1Ncls∑iLclspipi∗+λ1Nreg∑ipi∗Lregtiti∗E3
where i is the index of the anchor in the batch, pi is the predicted probability of being an object, pi∗ is the ground truth probability of the anchor, ti is the predicted bounding box coordinate, ti∗ is the ground truth bounding box coordinate, and Lcls and Lreg are the classification and regression loss, respectively.
Figure 3 depicts the unified network for object detection implemented in the Faster RCNN architecture. Using the recently popular terminology of neural networks with “attention” mechanisms [20], the region proposal network module tells the Fast RCNN module where to look [15].
Figure 3.
Faster RCNN acts as a single, unified network for object detection [15]. The region proposal network module serves as the “attention” of this unified network. Figure based on [15].
3. Datasets
A sample of the PASCAL VOC [21] dataset is used to exemplify the use of SSD and RCNN object detection algorithms. A sample of 6 classes of the 20 available were selected. Table 1 describes the sample size selected for each class.
Class
Number of images
Bird
811
Cat
1128
Dog
1341
Horse
526
Sheep
357
Total
4163
Table 1.
SSD and RCNN network dataset description.
The images presented in the dataset were randomly divided as follows: 1911 for training corresponding to 50%, 1126 for validation corresponding to 25% and test also corresponding to 25%.
To further illustrate the applications of such algorithms in scientific research, the dataset used for the YOLO network presented in [22] was also analyzed. As described in [22], the dataset is composed of images from three researches that involve behavioral experiments with mice:
Ethological evaluation [23]: This research presents new metrics for chronic stress models of social defeat in mice.
Automated home-cage [24]: This study introduces a trainable computer vision system that allows the automated analysis of complex mouse behaviors; they are eat, drink, groom, hang, micromovement, rear, rest, and walk.
Caltech Resident-Intruder Mouse dataset (CRIM13) [25]: It has videos recorded with superior and synchronized lateral visualization of pairs of mice involved in social behavior in 13 different actions.
Table 2 describes the sample size selected from each of the datasets used in this paper. For the ethological evaluation [23], 3707 frames were used, captured in a top view of the arena of social interaction experiments among mice. For the automated home-cage [24], a sample of 3073 frames was selected from a side view of behavioral experiments. For the CRIM13 [25], a sample of 6842 frames was selected, 3492 from a side view and 3350 from a top view.
Description of the dataset for use with the YOLO network as earlier used in [22].
The same dataset division used in [22] was also reproduced resulting in 6811 images for training, 3405 for validation, and 3406 for the test.
4. Material and methods for object detection
In this work, the previously described SDD and Faster RCNN networks are compared in the task of localization and tracking of six species of animals in diversified environments. Having accurate, detailed, and up-to-date information about the location and behavior of animals in the wild would improve our ability to study and conserve ecosystems [26]. Additionally, results from the YOLO network, reproduced from [22], to detect and track mice in videos are recorded during behavioral neuroscience experiments. The task of mice detection consists of determining the location in the image where the animals are present, for each frame acquired.
The computational development here presented was performed on a computer with CPU AMD Athlon II X2 B22 at 2.8GHz, 8GB of RAM, NVIDIA GeForce GTX 10708GB GPU, Ubuntu 18.04 LTS as OS, CUDA 9, and CuDNN 7. Our approach used the convolutional networks described in Section 2.
5. Results and conclusion
The results obtained for the SSD and Faster RCNN networks in the experiments were based on the analysis of 4163 images, organized according to the dataset described in Section 3.
Figure 4(a) depicts the increasing development of the mean average precision values in the epochs of training. Both architectures reached high mean average precision (mAP) while successfully minimizing the values of their respective loss functions. The Faster RCNN network presented higher and better stability in precision, which can be seen by the smoothness in its curve. Figure 4(b) is a box plot of the time spent by each network on the classification of a single image, whereas the SSD came ahead with 17±2 ms as the mean and standard deviation values, and the Faster RCNN translated its higher computational complexity in the execution time with 30±2ms as the mean and standard deviation values, respectively.
Figure 4.
(a) Comparison of the mAP models during the training phase. (b) Time spent to execute each architecture on a single image.
Table 3 presents more results related to object detection performance. First, it shows the mean average precision, which is the mean value of the average precisions for each class, where average precision is the average value of 11 points on the precision-recall curve for each possible threshold, that is, all the probability of detection for the same class (Precision-Recall evaluation according to the terms described in the PASCAL VOC [21]).
Mean average precision results after 100 epochs of training.
Figure 5 shows some selected examples of object detection results on the dataset used. Each output box is associated with a category label and a softmax score in 01. A score threshold of 0.5 is used to display these images.
Figure 5.
Output examples of the networks. (a)–(d) refer to SSD and (e)–(i) to Faster RCNN.
Our approach, as in [22], also used two versions of the YOLO network to detect mice within three different experimental setups. The results obtained were based on the analysis of 13,622 images, organized according to the dataset described in Section 3.
The first version of YOLO being trained was the YOLO Full network which uses the Darknet-53 [14] convolutional architecture that comprises 53 convolutional layers. Such a model was trained as described in [17], starting from an ImageNet [28] pre-trained model. Each model requires 235 MB of storage size. We used a batch of eight images, a momentum of 0.9, and weight decay of 5×10−4. The model took 140 hours to be trained.
A smaller and faster YOLO alternative was also trained and named as YOLO Tiny. To speed up the process, this “tiny” version comprises only a portion of the Darknet-53 [14] resources: 23 convolutional layers. Each model requires only 34 MB of storage size. The network training follows as described in [17], fine-tuning an ImageNet [28] pre-trained model. We used a batch of 64 images, a momentum of 0.9, and weight decay of 5×10−4. The model took 18 hours to be trained.
Figure 6 shows the comparison of the two YOLO models used, YOLO Full and Tiny. Figure 6(a) shows high accuracy of the Full architecture with small oscillations of the accuracy curve during the training. In Figure 6(b), the high accuracy is maintained from the earliest times and remains practically unchanged up to the limit number of epochs. Both architectures reached high mean average precision values while successfully minimizing the values of their loss function. The Tiny version of the YOLO network presented better stability in precision, which can be seen by the smoothness in its curve. The results show that the mean average precision reached by this re-implementation was 90.79 and 90.75% for the Full and Tiny versions of YOLO, respectively. The use of the Tiny version is a good alternative for experimental designs that require real-time response.
Figure 6.
(a) and (b) YOLO architecture evolution in mean average precision and minimization of the loss function during the training phase. (c) GPU time required to obtain the classification of an image in each of the networks.
Figure 6(c) is a bar graph showing the mean time spent on the classification of a single image in both architectures. The smaller size of the Tiny version gets a direct translation in execution time, having 0.08±0.06s as the mean and standard deviation values, whereas the Full version has 0.36±0.16s as the mean and standard deviation values, respectively.
Given the aforementioned small difference between the two versions of the YOLO object detector, the possibility of designing real-time systems for experiments involving animal tracking is closer to reality with the Tiny architecture. Derived from the smaller demand for computing power, systems where actions are taken while the experiment takes place can be designed without the need for human intervention.
Figure 7 shows some examples, resulting from mice tracking performed on the three different datasets used. Thus, it is possible to verify the operation of mouse tracking in different scenarios. In (a)–(c), the black mouse appears over a white background, the video is recorded from a top view camera in a typical configuration in behavioral experiments. For Figures (d)–(f), the camera was positioned on the side of the experimental box; the algorithm performed the tracking correctly for different positions of the animal. Finally, in Figures (g)–(i), the images were recorded by a top-view camera, and it is possible to verify a large amount of information besides the tracked object. However, the algorithm worked very well, even for two animals in the same arena.
Figure 7.
Output examples of the YOLO network. (a)–(c) refer to ethological evaluation [23], (d)–(f) refer to automated home-cage [24], and (g)–(i) refer to CRIM13 [25].
This chapter presented an overview of the machine learning techniques using convolutional neural networks for image object detection. The main algorithms for solving this type of problem were presented: Faster RCNN, YOLO, and SSD. To exemplify the functioning of the algorithms, datasets recognized by the scientific literature and in the field of computer vision were selected, tests were performed, and the results were presented, showing the advantages and differences of each of the techniques. This content is expected to serve as a reference for researchers and those interested in this broadly developing area of knowledge.
At the moment, we are experiencing the era of machine learning applications, and much should be developed in the coming years from the use and improvement of these techniques. Further improvements in the development of even more specific hardware and fundamental changes in related mathematical theory are expected shortly, making artificial intelligence increasingly present and important to the contemporary world.
\n',keywords:"machine learning, convolutional neural network, object detection",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/69955.pdf",chapterXML:"https://mts.intechopen.com/source/xml/69955.xml",downloadPdfUrl:"/chapter/pdf-download/69955",previewPdfUrl:"/chapter/pdf-preview/69955",totalDownloads:637,totalViews:0,totalCrossrefCites:0,totalDimensionsCites:0,hasAltmetrics:0,dateSubmitted:"May 6th 2019",dateReviewed:"September 14th 2019",datePrePublished:"November 7th 2019",datePublished:null,dateFinished:null,readingETA:"0",abstract:"This chapter intends to present the main techniques for detecting objects within images. In recent years there have been remarkable advances in areas such as machine learning and pattern recognition, both using convolutional neural networks (CNNs). It is mainly due to the increased parallel processing power provided by graphics processing units (GPUs). In this chapter, the reader will understand the details of the state-of-the-art algorithms for object detection in images, namely, faster region convolutional neural network (Faster RCNN), you only look once (YOLO), and single shot multibox detector (SSD). We will present the advantages and disadvantages of each technique from a series of comparative tests. For this, we will use metrics such as accuracy, training difficulty, and characteristics to implement the algorithms. In this chapter, we intend to contribute to a better understanding of the state of the art in machine learning and convolutional networks for solving problems involving computational vision and object detection.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/69955",risUrl:"/chapter/ris/69955",book:{slug:"recent-trends-in-artificial-neural-networks-from-training-to-prediction"},signatures:"Richardson Santiago Teles de Menezes, Rafael Marrocos Magalhaes and Helton Maia",authors:null,sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Object detection techniques",level:"1"},{id:"sec_2_2",title:"2.1 Single shot multibox detector",level:"2"},{id:"sec_3_2",title:"2.2 You only look once",level:"2"},{id:"sec_4_2",title:"2.3 Faster region convolutional neural network",level:"2"},{id:"sec_6",title:"3. Datasets",level:"1"},{id:"sec_7",title:"4. Material and methods for object detection",level:"1"},{id:"sec_8",title:"5. Results and conclusion",level:"1"}],chapterReferences:[{id:"B1",body:'Esteva A et al. Dermatologist-level classification of skin cancer with deep neural networks. Nature. 2017;542(7639):115'},{id:"B2",body:'Jayaraman V, Chandrasekhar MG, Rao UR. Managing the natural disasters from space technology inputs. Acta Astronautica. 1997;40(2–8):291-325'},{id:"B3",body:'Leonard M et al. A compound event framework for understanding extreme impacts. Wiley Interdisciplinary Reviews: Climate Change. 2014;5(1):113-128'},{id:"B4",body:'Kogan FN. Global drought watch from space. Bulletin of the American Meteorological Society. 1997;78(4):621-636'},{id:"B5",body:'Srinivas S, Sarvadevabhatla RK, Mopuri RK, Prabhu N, Kruthiventi SSS, Venkatesh Babu R. An introduction to deep convolutional neural nets for computer vision. In: Deep Learning for Medical Image Analysis. Academic Press; 2017. pp. 25-52'},{id:"B6",body:'de Menezes RST, de Azevedo Lima L, Santana O, Henriques-Alves AM, Santa Cruz RM, Maia H. Classification of mice head orientation using support vector machine and histogram of oriented gradients features. In: 2018 International Joint Conference on Neural Networks (IJCNN). IEEE; 2018. pp. 1-6'},{id:"B7",body:'Oskoei MA, Gan JQ, Hu H. Adaptive schemes applied to online SVM for BCI data classification. In: 2009 Annual International Conference of the IEEE Engineering in Medicine and Biology Society. IEEE; 2009. pp. 2600-2603'},{id:"B8",body:'Hearst MA, Dumais ST, Osuna E, Platt J, Scholkopf B. Support vector machines. IEEE Intelligent Systems and their Applications. 1998;13(4):1828'},{id:"B9",body:'Pan WD, Dong Y, Wu D. Classification of malaria-infected cells using deep convolutional neural networks. In: Machine Learning: Advanced Techniques and Emerging Applications. 2018. p. 159'},{id:"B10",body:'Goodfellow I, Bengio Y, Courville A. Deep Learning. MIT Press; 2016'},{id:"B11",body:'Deng L, Hinton G, Kingsbury B. New types of deep neural network learning for speech recognition and related applications: An overview. In: 2013 IEEE International Conference on Acoustics, Speech and Signal Processing. IEEE; 2013. pp. 8599-8603'},{id:"B12",body:'Kriegeskorte N. Deep neural networks: A new framework for modeling biological vision and brain information processing. Annual Review of Vision Science. 2015;1:417-446'},{id:"B13",body:'Liu W, Anguelov D, Erhan D, Szegedy C, Reed S, Fu C-Y, et al. SSD: Single shot multibox detector. In: European Conference on Computer Vision. Cham: Springer; 2016. pp. 21-37'},{id:"B14",body:'Redmon J, Farhadi A. Yolov3: An Incremental Improvement. arXiv; 2018'},{id:"B15",body:'Ren S, He K, Girshick R, Sun J. Faster r-cnn: Towards real-time object detection with region proposal networks. In: Advances in Neural Information Processing Systems. 2015. pp. 91-99'},{id:"B16",body:'Simonyan K, Zisserman A. Very deep convolutional networks for large-scale image recognition. arXiv preprint arXiv:1409.1556; 2014'},{id:"B17",body:'Redmon J, Divvala S, Girshick R, Farhadi A. You only look once: Unified, real-time object detection. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. 2016. pp. 779-788'},{id:"B18",body:'Girshick R, Donahue J, Darrell T, Malik J. Rich feature hierarchies for accurate object detection and semantic segmentation. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. 2014. p. 580587'},{id:"B19",body:'Girshick R. Fast r-cnn. In: Proceedings of the IEEE International Conference on Computer Vision. 2015. pp. 1440-1448'},{id:"B20",body:'Chorowski JK, Bahdanau D, Serdyuk D, Cho K, Bengio Y. Attention-based models for speech recognition. In: Advances in Neural Information Processing Systems. 2015. pp. 577-585'},{id:"B21",body:'Everingham M et al. The Pascal visual object classes (VOC) challenge. International Journal of Computer Vision. 2010;88(2):303-338'},{id:"B22",body:'Peixoto HM, Teles RS, Luiz JVA, Henriques-Alves AM, Santa Cruz RM. Mice Tracking Using the YOLO Algorithm. Vol. 7. PeerJ Preprints; 2019. p. e27880v1'},{id:"B23",body:'Henriques-Alves AM, Queiroz CM. Ethological evaluation of the effects of social defeat stress in mice: Beyond the social interaction ratio. Frontiers in Behavioral Neuroscience. 2016;9:364'},{id:"B24",body:'Jhuang H et al. Automated home-cage behavioural phenotyping of mice. Nature Communications. 2010;1:68'},{id:"B25",body:'Burgos-Artizzu XP, Dollár P, Lin D, Anderson DJ, Perona P. Social behavior recognition in continuous video. In: 2012 IEEE Conference on Computer Vision and Pattern Recognition. IEEE; 2012. pp. 1322-1329'},{id:"B26",body:'Norouzzadeh MS et al. Automatically identifying, counting, and describing wild animals in camera-trap images with deep learning. Proceedings of the National Academy of Sciences of the United States of America. 2018;115(25):E5716-E5725'},{id:"B27",body:'Guo J, He H, He T, Lausen L, Li M, Lin H, et al. GluonCV and GluonNLP: Deep Learning in Computer Vision and Natural Language Processing. arXiv preprint arXiv:1907; 2019. p. 04433'},{id:"B28",body:'Deng J, Dong W, Socher R, Li L-J, Li K, Fei-Fei L. ImageNet: A large-scale hierarchical image database. In: 2009 IEEE conference on computer vision and pattern recognition. IEEE; 2009. pp. 248-255'},{id:"B29",body:'Chen X-L et al. Remote sensing image-based analysis of the relationship between urban heat island and land use/cover changes. Remote Sensing of Environment. 2006;104(2):133-146'}],footnotes:[],contributors:[{corresp:null,contributorFullName:"Richardson Santiago Teles de Menezes",address:null,affiliation:'
'}],corrections:null},book:{id:"7607",title:"Recent Trends in Artificial Neural Networks",subtitle:"from Training to Prediction",fullTitle:"Recent Trends in Artificial Neural Networks - from Training to Prediction",slug:"recent-trends-in-artificial-neural-networks-from-training-to-prediction",publishedDate:"March 4th 2020",bookSignature:"Ali Sadollah and Carlos M. Travieso-Gonzalez",coverURL:"https://cdn.intechopen.com/books/images_new/7607.jpg",licenceType:"CC BY 3.0",editedByType:"Edited by",editors:[{id:"147215",title:"Dr.",name:"Ali",middleName:null,surname:"Sadollah",slug:"ali-sadollah",fullName:"Ali Sadollah"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},chapters:[{id:"69476",title:"Time Series from Clustering: An Approach to Forecast Crime Patterns",slug:"time-series-from-clustering-an-approach-to-forecast-crime-patterns",totalDownloads:289,totalCrossrefCites:0,signatures:"Miguel Melgarejo, Cristian Rodriguez, Diego Mayorga and Nelson Obregón",authors:[null]},{id:"68706",title:"Encountered Problems of Time Series with Neural Networks: Models and Architectures",slug:"encountered-problems-of-time-series-with-neural-networks-models-and-architectures",totalDownloads:277,totalCrossrefCites:3,signatures:"Paola Andrea Sánchez-Sánchez, José Rafael García-González and Leidy Haidy Perez 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1. Potential well
The potential well is the region where the particle is confined in a small region. In general, the potential of the confined region is lower than the surroundings (Figure 1) [1, 2].
Figure 1.
Infinite potential well.
The potential of the system is defined as
V=0,−L<x<L∞,Otherwise
The one dimensional Schrödinger equation in Cartesian coordinate is given as
−ℏ22mΨ′′+VΨ=EΨ⟹Ψ′′+2mℏ2E−VΨ=0E1
In the infinite potential well, the confined particle is present in the well region (Region-II) for an infinitely long time. So the solution of the Schrödinger equation in the Region-II and Region-III can be omitted for our discussion right now. The Schrödinger equation in the Region-II is written as
The integer “n” is the quantum number and it denotes the discrete energy states in the quantum well. We can extract some physical information from the eigen solutions.
The minimum energy state can be calculated by setting n=1, which corresponds to the ground state. The ground state energy is
E1=π2ℏ2/8mL2E11
This is known as zero-point energy in the case of the potential well. The excited state energies are E2=4π2ℏ2/8mL2, E3=9π2ℏ2/8mL2, E4=16π2ℏ2/8mL2, and so on. In general, En=n2×E1.
The energy difference between the successive states is simply the difference between the energy eigen value of the corresponding state. For example, ∆E12=E1∼E2=3E1 and ∆E23=E2∼E3=5E1. Hence the energy difference between any two successive states is not the same.
Though the eigen functions for odd and even values of “n” are different, the energy eigen value remains the same.
If the potential well is chosen in the limit 0<x<2L (width of the well is 2L), the energy eigen value is the same as given in Eqs.(6) and (8). But if the limit is chosen as 0<x<L (width of the well is L), the for all positive integers of “n,” the eigen function is Ψ=2/L1/2sinnπx/L and the energy eigen function is E=n2π2ℏ2/2mL2.
2. Step potential
Step potential is a problem that has two different finite potentials [3]. Classically, the tunneling probability is 1 when the energy of the particle is greater than the height of the barrier. But the result is not true based on wave mechanics (Figure 2).
Figure 2.
Step potential.
The potential of the system
V=0,−∞<x<0V0,0≤x<∞
The Schrödinger equation in the Region-I and Region-II is given, respectively as,
Ψ′′+2mℏ2EΨ=0E12
Ψ′′+2mℏ2E−VΨ=0E13
Case (i): when E<V0, the solutions of the Schrödinger equations in the Region-I and Region-II, respectively, are given as
Ψ1=A1expiαx+B1exp−iαxE14
Ψ2=A2exp−βx+B2expβx
where α2=2mEℏ2 and β2=2mE−V0ℏ2. Here, B2expβx represents the exponentially increasing wave along the x-direction. The wave function Ψ2 must be finite as x→∞. This is possible only by setting B2=0. Hence the eigen function in the Region-II is
Ψ2=A2exp−βxE15
According to admissibility conditions of wave functions, at x=0, Ψ1=Ψ2 and Ψ1′=Ψ2′. It gives us
A1+B1=A2E16
A1−B1=iβαA2E17
From these two equations,
A2=2αα+iβA1
B1=α−iβα+iβA1
The reflection coefficient R is given as
R=B12A12=α−iβα+iβ2=1E18
It is interesting to note that all the particles that encounter the step potential are reflected back. This is due to the fact that the width of the step potential is infinite. The number of particles in the process is conserved, which leads that T=0, since T+R=1.
Case (ii): when E>V0, the solutions are given as
Ψ1=A1expiαx+B1exp−iαx
Ψ2=A2expiβx+B2exp−iβx
where β2=2mE−V0ℏ2. As x→∞, the wave function Ψ2 must be finite. Hence
Ψ2=A2expiβx by setting B2=0. According to the boundary conditions at x=0,
A1+B1=A2E19
A1−B1=βαA2E20
From these equations,
A2=2αα+βA1
B1=α−βα+βA1
The reflection coefficient R and the transmission coefficient T, respectively, are given as
R=B12A12=α−βα+β2E21
T=A22A12=4αβα+β2E22
From these easily one can show that
T+R=4αβα+β2+α−βα+β2=1E23
The results again indicate that the total number of particles which encounters the step potential is conserved.
3. Potential barrier
This problem clearly explains the wave-mechanical tunneling [3, 4]. The potential of the system is given as (Figure 3)
Figure 3.
Potential barrier.
V=V0,0<x<L0,Otherwise
In the Region-I, the Schrödinger equation is Ψ′′+α2Ψ=0. The wave function in this region is given as
Ψ1=A1expiαx+B1exp−iαxwhereα2=2mEℏ2E24
In Region-II, if E<V0, the Schrödinger equation is Ψ′′−β2Ψ=0. The solution of the equation is given as
Ψ2=A2expβx+B2exp−βxwhereβ2=2mE−V0ℏ2E25
The Schrödinger equation in the Region-III is Ψ′′+α2Ψ=0. The corresponding solution is Ψ3=A3expiαx+B3exp−iαx. But in the Region-III, the waves can travel only along positive x-direction and there is no particle coming from the right,B3=0. Hence
Ψ3=A3expiαxE26
At x=0, Ψ1=Ψ2 and Ψ1′=Ψ2′. These give us two equations
A1+B1=A2+B2E27
A1−B1=βiαA2−B2E28
At x=L, Ψ2=Ψ3 and Ψ2′=Ψ3′. These conditions give us another two equations
A2expβL+B2exp−βL=A3expiαLE29
A2expβL−B2exp−βL=A3iαβexpiαLE30
Solving the equations from (27) to (30), one can find the coefficients in the equations. The reflection coefficient is R is found as
R=B12A12=V024EV0−Esinh2βL1+V024EV0−Esinh2βL−1E31
The transmission coefficient T is found as
T=A22A12=1+V024EV0−Esinh2βL−1E32
From Eqs. (31) and (32), one can show that T+R=1. The following are the conclusions obtained from the above mathematical analysis.
When E<V0, though the energy of the incident particles is less than the height of the barrier, the particle can tunnel into the barrier region. This is in contrast to the laws of classical physics. This is known as the tunnel effect.
As V0→∞, the transmission coefficient is zero. Hence the tunneling is not possible only when V0→∞.
When the length of the barrier is an integral multiple of π/β, there is no reflection from the barrier. This is termed as resonance scattering.
The tunneling probability depends on the height and width of the barrier.
Later, Kronig and Penney extended this idea to explain the motion of a charge carrier in a periodic potential which is nothing but the one-dimensional lattices.
4. Delta potential
The Dirac delta potential is infinitesimally narrow potential only at some point (generally at the origin, for convenience) [3]. The potential of the system
V=−λδx,x=00,Otherwise
Here λ is the positive constant, which is the strength of the delta potential. Here, we confine ourselves only to the bound states, hence E<0 (Figure 4).
Figure 4.
Dirac delta potential.
The Schrödinger equation is
Ψ′′+2mℏ2E−VΨ=0⟹Ψ′′+2mℏ2E+λδxΨ=0E33
The solution of the Schrödinger equation is given as
Region−I:Ψ1=A1expβxE34
Region−II:Ψ2=A2exp−βxE35
where β2=−2mEℏ2. At x=0, Ψ1=Ψ2. So the coefficients A1 and A2 are equal. But Ψ1′≠Ψ2′, since the first derivative causes the discontinuity. The first derivatives are related by the following equation
Ψ2′−Ψ1′=−2mλℏ2E36
This gives us
β=mλℏ2E37
Equating the value of β gives the energy eigen value as
E=−mλ22ℏ2E38
The energy eigen value expression does not have any integer like in the case of the potential well. Hence there is only one bound state which is available for a particular value of “m.”
The eigen function can be evaluated as follows: The eigen function is always continuous. At x=0 gives us A1=A2=A. Hence the eigen function is
Ψ=Aexpβx
To normalize Ψ,
∫−∞∞Ψ2dx=1⇒2∫0∞Ψ2dx=1
This gives us A=β=mλℏ.
5. Linear harmonic oscillator
Simple harmonic oscillator, damped harmonic oscillator, and force harmonic oscillator are the few famous problems in classical physics. But if one looks into the atomic world, the atoms are vibrating even at 0 K. Such atomic oscillations need the tool of quantum physics to understand its nature. In all the previous examples, the potential is constant in any particular region. But in this case, the potential is a function of the position coordinate “x.”
5.1 Schrodinger method
The potential of the linear harmonic oscillator as a function of “x” is given as (Figure 5) [4, 5, 6]:
Figure 5.
Potential energy of the linear harmonic oscillator.
V=mω2x22E39
The time-independent Schrödinger equation is given as
Ψ′′+2mℏ2E−mω2x22Ψ=0E40
The potential is not constant since it is a function of “x”; Eq. (40) cannot solve directly as the previous problems. Let
α=mωℏ1/2xandβ=2Eℏω.
Using the new constant β and the variable α, the Schrödinger equation has the form
d2Ψdα2+β−α2Ψ=0E41
The asymptotic Schrödinger equation α→∞ is given as
d2Ψdα2−α2Ψ=0E42
The general solution of the equation is exp±a2/2. As α→∞, exp+a2/2 becomes infinite, hence it cannot be a solution. So the only possible solution is exp−a2/2. Based on the asymptotic solution, the general solution of Eq. (42) is given as
Ψ=Hnαexp−a2/2
The normalized eigen function is
Ψ=mωℏπ1/212n×n!1/2Hnαexp−a2/2E43
The solution given in Eq. (43) is valid if the condition 2n+1−2Eℏω=0 holds. This gives the energy eigen value as
E=n+12ℏωE44
The important results are given as follows:
The integer n=0 represents the ground state, n=1 represents the first excited state, and so on. The ground state energy of the linear harmonic oscillator is E=ℏω/2. This minimum energy is known as ground state energy.
The ground state normalized eigen function is
Ψ0x=mωℏπ1/4exp−mωx22ℏE45
The energy difference between any two successive levels is ℏω. Hence the energy difference between any two successive levels is constant. But this is not true in the case of real oscillators.
5.2 Operator method
The operator method is also one of the convenient methods to solve the exactly solvable problem as well as approximation methods in quantum mechanics [5]. The Hamiltonian of the linear harmonic oscillator is given as,
H=p22m+12mω2x2E46
Let us define the operator “a,” lowering operator, in such a way that
a=2mωℏ−1/2mωx+ipE47
and the corresponding Hermitian adjoint, raising operator, is
Similarly, the energy of the first excited state is found as follows:
1a+a1=E1ℏω−12
11a+0=E1ℏω−12
1.111=E1ℏω−12
1=E1ℏω−12⇒E1=32ℏωE56
In the same way, E2=5ℏω/2, E3=7ℏω/2, and so on. Hence, one can generalize the result as
En=n+12ℏωE57
The uncertainties in position and momentum, respectively, are given as
∆x=x2−x2E58
∆p=p2−p2E59
In order to evaluate the uncertainties x2, x2, p2, and p2 have to be evaluated. From Eqs. (47) and (48) the position and momentum operators are found as
From Eq. (58) and (59), the uncertainty in position and momentum, respectively are given as,
∆x=ℏ2mω2n+11/2E62
∆p=mωℏ22n+11/2E63
∆x.∆p=ℏ22n+1E64
6. Conclusions
The minimum uncertainty state is the ground state. In this state, ∆x=ℏ2mω1/2 and ∆p=mωℏ21/2.
Hence the minimum uncertainty product is ∆x.∆p=ℏ2. Since the other states have higher uncertainty than the ground state, the general uncertainty is ∆x.∆p≥ℏ2. This is the mathematical representation of Heisenberg’s uncertainty relation.
Since Ψ0x corresponds to the low energy state, aΨ0x=0. This gives us the ground state eigen function. This can be done as follows:
aΨ0x=0
2mωℏ−1/2mωx+ipΨ0x=0
mω2ℏ1/2x+i−iℏ∂/∂x2mωℏ1/2Ψ0x=0
ℏmω∂Ψ0x∂x=−xΨ0x
dΨ0xΨ0x=−mωxℏdx
Integrating the above equation gives,
lnΨ0x=−mωℏx22+lnA
Ψ0x=Aexp−mωx22ℏ
The normalized eigen function is given as
Ψ0x=mωℏπ1/4exp−mωx22ℏ
One can see that this result is identical to Eq. (45).
The other eigen states can be evaluated using the equation, Ψnx=a+n/n!Ψ0x.
7. Particle in a 3D box
The confinement of a particle in a three-dimensional potential is discussed in this section [4, 6]. The potential is defined as (Figure 6)
Figure 6.
Three-dimensional potential box.
V=0,0≤x<a;0≤y<b;0≤z<c∞,Otherwise
The three dimensional time-independent Schrödinger equation is given as
∇2Ψxyz−2mℏ2VΨxyz=−EΨxyzE65
Let the eigen functionΨxyz is taken as the product of Ψxx, Ψyy and Ψzz according to the technique of separation of variables. i.e., Ψxyz=ΨxxΨyyΨzz.
In a cubical potential box, a=b=c, then the energy eigen value becomes,
E=π2ℏ22ma2nx2+ny2+nz2.
The minimum energy that corresponds to the ground state is E1=3π2ℏ22ma2. Here nx=ny=nz=1.
Different states with different quantum numbers may have the same energy. This phenomenon is known as degeneracy. For example, the states (i) nx=2;ny=nz=1, (ii) ny=2;nx=nz=1; and (iii) nz=2;nx=ny=1 have the same energy of E=6π2ℏ2ma2. So we can say that the energy 6π2ℏ2ma2has a 3-fold degenerate.
The states (111), (222), (333), (444),…. has no degeneracy.
In this problem, the state may have zero-fold degeneracy, 3-fold degeneracy or 6-fold degeneracy.
\n',keywords:"exactly solvable, Schrödinger equation, eigen function, eigen values",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/73016.pdf",chapterXML:"https://mts.intechopen.com/source/xml/73016.xml",downloadPdfUrl:"/chapter/pdf-download/73016",previewPdfUrl:"/chapter/pdf-preview/73016",totalDownloads:277,totalViews:0,totalCrossrefCites:0,dateSubmitted:"April 29th 2020",dateReviewed:"July 3rd 2020",datePrePublished:"August 18th 2020",datePublished:"October 14th 2020",dateFinished:null,readingETA:"0",abstract:"Some of the problems in quantum mechanics can be exactly solved without any approximation. Some of the exactly solvable problems are discussed in this chapter. Broadly there are two main approaches to solve such problems. They are (i) based on the solution of the Schrödinger equation and (ii) based on operators. The normalized eigen function, eigen values, and the physical significance of some of the selected problems are discussed.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/73016",risUrl:"/chapter/ris/73016",signatures:"Lourdhu Bruno Chandrasekar, Kanagasabapathi Gnanasekar and Marimuthu Karunakaran",book:{id:"10076",title:"Quantum Mechanics",subtitle:null,fullTitle:"Quantum Mechanics",slug:"quantum-mechanics",publishedDate:"October 14th 2020",bookSignature:"Paul Bracken",coverURL:"https://cdn.intechopen.com/books/images_new/10076.jpg",licenceType:"CC BY 3.0",editedByType:"Edited by",editors:[{id:"92883",title:"Prof.",name:"Paul",middleName:null,surname:"Bracken",slug:"paul-bracken",fullName:"Paul Bracken"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}},authors:[{id:"239576",title:"Dr.",name:"Marimuthu",middleName:null,surname:"Karunakaran",fullName:"Marimuthu Karunakaran",slug:"marimuthu-karunakaran",email:"tvdkaruna@gmail.com",position:null,institution:null},{id:"252354",title:"Dr.",name:"Bruno Chandrasekar",middleName:null,surname:"L",fullName:"Bruno Chandrasekar L",slug:"bruno-chandrasekar-l",email:"brunochandrasekar@gmail.com",position:null,institution:{name:"Periyar Maniammai University",institutionURL:null,country:{name:"India"}}},{id:"325784",title:"Dr.",name:"K",middleName:null,surname:"Gnanasekar",fullName:"K Gnanasekar",slug:"k-gnanasekar",email:"kgnnskr@yahoo.com",position:null,institution:{name:"The American College",institutionURL:null,country:{name:"Ireland"}}}],sections:[{id:"sec_1",title:"1. Potential well",level:"1"},{id:"sec_2",title:"2. Step potential",level:"1"},{id:"sec_3",title:"3. Potential barrier",level:"1"},{id:"sec_4",title:"4. Delta potential",level:"1"},{id:"sec_5",title:"5. Linear harmonic oscillator",level:"1"},{id:"sec_5_2",title:"5.1 Schrodinger method",level:"2"},{id:"sec_6_2",title:"5.2 Operator method",level:"2"},{id:"sec_8",title:"6. Conclusions",level:"1"},{id:"sec_9",title:"7. Particle in a 3D box",level:"1"}],chapterReferences:[{id:"B1",body:'Griffiths DJ. Introduction to Quantum Mechanics. 2nd ed. India: Pearson'},{id:"B2",body:'Singh K, Singh SP. Elements of Quantum Mechanics. 1st ed. India: S. Chand & Company Ltd'},{id:"B3",body:'Gasiorowicz S. Quantum Mechanics. 3rd ed. India: Wiley'},{id:"B4",body:'Schiff LI. Quantum Mechanics. 4th ed. India: McGraw Hill International Editions'},{id:"B5",body:'Peleg Y, Pnini R, Zaarur E, Hecht E. Quantum Mechanics. 2nd ed. India: McGraw Hill Editions'},{id:"B6",body:'Aruldhas G. Quantum Mechanics. 2nd ed. India: Prentice-Hall'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Lourdhu Bruno Chandrasekar",address:"brunochandrasekar@gmail.com",affiliation:'
Department of Physics, Periyar Maniammai Institute of Science and Technology, Vallam, India
Department of Physics, Alagappa Government Arts College, India
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