Image Processing: Stochastic Model Based Approach
Image Processing:Stochastic Model BasedApproachK. SEETHARAMANDepartment ofComputer Science and Engineering,Annamalai University1
ContentsPrefacevAcknowledgementsChapter 1viiIntroduction1.1 Background11.2 Problems in Selecting Models71.3 Types of Models71.4 SummaryBibliographyChapter 21112Randomness, Sampling and Quantization2.1 Introduction142.2 Random Process152.3 Randomness, Sampling and Quantization 162.4 Parameter Estimation182.5 Summary21BibliographyChapter 322Image Model and Parameter Estimation3.1 Introduction233.2 The FRGMRF Model293.3 Parameter Estimation of FRGMRFModel323.4 Summary36BibliographyChapter 437Texture Analyses4.1 Introduction414.2 Types of Images452
4.3 Texture Identification484.4 Texture Representation504.5 Texture Classification504.5.1 Supervised Classification514.5.1.1 Bartlett’s Test forHomogeneity of Variances514.5.1.2 Independent Two-Samplet-test4.5.2 Unsupervised Classification4.5.2.1 Methodology54544.6 Experiments and Results554.7 Discussion594.8 Summary59BibliographyChapter 55260Edge Detection5.1 Introduction625.2 Image Smoothing665.3 Edge Magnitude and Direction675.4 Edge Detection Algorithm745.5 Comparison with SomeStandard Techniques5.6 Comparison of Various Edge Maps775.7 Experimental Results and Discussion785.8 Summary79BibliographyChapter 67580Image Compression6.1 Introduction826.2 Image Compression Model896.3 Parameter Estimation916.3.1 Metropolis-Hastings Algorithm916.4 Measure of Performance966.5 Experiments and Results963
6.6 Discussion986.7 Summary99BibliographyChapter 799Image Restoration7.1 Introduction1037.2 Damaged Block and Prior-Information Identification7.3 Restoration Model7.4 Image Restoration7.4.1 Restoration Algorithm1101127.5 Experiments and Results1147.6 Summary121BibliographyChapter 8106109122Conclusion8.1 General Discussion1258.2 Advantages of FRGMRF Model BasedScheme8.3 Scope for Future WorkINDEX1271281294
PrefaceDigital Image analyses at low-level is a complicated task,because it has the responsibility of analysing the images atmicro-level and in-depth. The intention of this book is thathow to effectively and efficiently perform the model basedimage processing tasks at low-level. Attention has beenfocused on the concepts of sampling, quantization,randomness, and how to mathematically characterize andmodel an image, and how to effectively utilize the model toperform various advanced image processing tasks such asTexture Analyses, Edge detection, Compression, Restoration bya single model. This book focuses on construction of astochastic model, which is coined as Full Range GaussianMarkov Random Field model, and also illustrates the efficiencyand effectiveness of the model based work. This bookdiscusses the concept of Bayesian methodology and how toincorporate the prior information in various image processingtopics, in neat and simple way. The way of discussion and theconcept provided are very useful and draw the attention of thepostgraduate students, researchers, scientists and engineersto design image processing systems and perform research atadvanced level in the newly emerging topics. This book alsoillustrates the research concepts with a number of examples.Especially, this book is useful to the researchers, becausesome novel concepts at research level have been introduced,and also more than 250 citations have been incorporated fromvarious scholarly published articles.The advent of the imaging science and technology, and itsapplications help the various fields, namely medicine, defence,remote sensing, robot vision, pattern recognition, traffic,forensic science etc. to flourish with advanced technology andfast growth. Especially in the field of medicine, remote sensingand pattern recognition, image processing plays significantrole. The outcome of the image analysis may be either animage or a set of attributes or parameters related to the image.Image processing is a sub-class of signal processing and also5
it is a discrete space with time dependent, so it is convenientto model the images using mathematical or statistical modelssuch as Wavelet, time series, stochastic models etc.This book discusses the various fundamental imageprocessing methods based on stochastic model, which iscoined as Full Range Gaussian Markov Random Field model.This book explains the concepts, randomness, sampling andquantization, and how the mathematical model is designedand constructed according to the nature of the images, in neatand simple manner. Also, the book discusses the advantagesand disadvantages of the various parameter estimationtechniques in the context of the image processing. Though,this book does not cover a more number of topics in imageprocessing, the covered some topics are very useful to thereaders at advanced level. The book is organised with eightchapters follows.Chapter 1, introduces the basic concepts and variousapplications of the image processing, and also elaborates thedifferent types of image processing models such as causal,semi-causal, non-causal. Chapter 2 discusses the concepts ofrandomness, sampling and quantization in the context ofimage processing.Chapter 3, introduces the stochastic model and itsappropriateness in the image analysis at low-level. Also itnarrates the Bayesian concepts and parameter estimationtechnique. Chapter 4 discusses texture analysis such astexture characterization, identification, representation anddescription. Chapter 5 discusses the identification anddistinguishing the textures from structures and edgedetection. Chapter 6 and 7 discusses the compression andrestoration of the damaged regions in the images, and Chapter8 concludes with conclusion. Each chapter provides asummary and annotated bibliography.6
Chapter 1Introduction1.1BackgroundSince 1960s, the researchers in computer vision aggressivelyconcentrated on image processing and their applications invarious fields such as video conferencing, telemedicine, remotesensing via satellite, forensic science, agricultural, education,defence, news and current affairs etc. To fulfil all theserequirements, it is required to analyse the content of an imagein depth. With the advent of the advanced technologies incomputer vision, advancements in the storage media andelectronic communication channels; above all these things,the invention and discovery of the various mathematical andstatistical techniques to effectively process or analyse theimages becomes more effective for the evolution of the imageprocessing. Digital Image processing enables the reversible,virtually noise-free modification of an image in the form of alattice of integers instead of the classical darkroommanipulations or filtration of time dependent voltagesnecessary for analog images and video signals. In order tounderstand the contents of an image, various tasks areperformed on it such as image understanding, imageprocessing and image analysis.Image understanding is concerned with symbolicdescriptions and structure, namely, image formation, surfaceorientation, image intensity, gradient space and reflectivityfunction. In the context of digital image processing, textureplays noteworthy role in image formation, even if it is textureimage or structure image it is formed by the textures. Simply,we can say that the images cannot be formed without7
textures. The Nature of the images is determined by thestructure and orientation of the textures. Based on thetextures, an image can be analysed, characterized, identified,described, and represented. Image intensities and the texturefeatures are considered for segmentation of an image intovarious regions or edge-fragments. The gradient space isexploited to extract the edges, curves, boundaries etc. Thesetypes of image understanding tools are effectively utilised foranalyses and processing of an image. In order to performproperly the image processing and image analysis tasks, it isnecessary to understand how the images are formed, whatdetermines the observed intensity in the image and thestructure of the images.Image processing deals deterministic and stochasticrepresentation of images, i.e. is image transforms and imagemodels. It also concerned with image data compression andimproving the quality of the image by filtering and by removingany degradations present in the image, viz. imageenhancement and image restoration. On the other hand, theimage analysis deals with the tasks like extraction of lines,curves and regions in images, classification and segmentationof objects in the image using boundary information, textureanalysis, analysis of a sequence of images with the interest ofestimating the motion of objects and scene analysis. Hence, ingeneral, in image processing, the inputs and outputs areimages while in image analysis, the outputs are a list ofobjects present in the image or a set of features such as edges,curves and boundaries.The image processing and image analysis are conductedin two stages, namely, low-level and high-level. The low-level segmentation, object recognition, compression, edge detectionand texture analysis, whereas in the high-level stage, weregard it as an interpretation of the results obtained at thelow-level stage. The low-level stage involved with models thatcontain classical knowledge about image formation and objectrecognition that are independent of the class of images underanalysis. The high-level stage is involved with interpretationsof applications of specific knowledge about the concept of the8
scene analysis. The images are analysed in low-level stage byvarious methods, which are obtained by using either edgebased or region-based approach. Though the region-basedapproach and edge-based approaches are complementary toeach other the edge-based approach has been used widely.Using the edge-based approach, a number of methods havebeen proposed for low-level analysis viz. image compression,classification, segmentation and pattern recognition. Thesemethods can be grouped into two classes. In one class, themethods are proposed based on the underlying assumptionabout the uniformity of intensities in local image regions. Theimages, which are assumed to be consisting of local regionswith uniform intensities are called untextured images. Theimages of real objects often do not exhibit local regions ofuniform intensities due to differences in object surfaceproperties like roughness, orientation, reflectance levels, etc.The former methods of classification and segmentation are notapplicable in the latter class of images. The latter class ofimages is called texture images. Another class of methods,which are usually known as texture analysis methods havebeen proposed for low-level classification and segmentation oftextured images.In general, the mathematical and statistical modelbased techniques play a significant role in image processing.Generally, the fundamental image processing methods can bebroadly classified into image acquisition, representation anddescription, pre-processing, enhancement and filtering, ion,morphological analysis, compression, and object recognition.To perform these fundamental image processing methods, somany mathematical and statistical techniques are available.This book discusses only the model based image processingand analyses.Almost all the image processing and analysis have beenperformed under any one of the following techniques: (i)Transform based and (ii) model based.The transform based techniques are widely used inimage processing such as image coding and restoration.9
Various transform techniques and their efficiency have beenreported in the literature. Among the existing transformtechniques, the most widely used are Discrete FourierTransform (DFT), Discrete Cosine Transform (DCT), DiscreteSine Transform (DST), Karhunen-Loeve Transform (KLT),Walsh-Hadamard Transform, Harr Transform and OrthogonalPolynomial Transform. Each transform technique has its ownspecific features, which are discussed in detail in section 5.1of Chapter 5. In recent years, the last one or two decades, thetransform based Wavelets and Fractal image compressiontechniques becoming popular due to its efficiency. Most of thetransform based techniques demand more computations andsome of them require large amount of memory. Also, theapplications of the transform based techniques are limited atlow-level image processing and analysis when compared tothat of model based techniques.The model based techniques are most appropriate toeffectively handle the problems involved with large amount ofdata like image filtering, object recognition, etc. To handle thisvolume of data, it would be preferable to have an underlyingmodel that explains the dominant statistical characteristics ofthe given data. The different classes of models have beensuggested in the literature, which exploit the statisticalproperties among the neighbouring pixels for low-level imageprocessing or analysis. The statistical models attracted manyresearchers, due to its wide range of applications at low-levelimage processing and analysis such as texture analysis,smoothing, enhancement, restoration, segmentation, edgedetection, image data compression, etc.The statistical models are becoming increasinglyimportant because of their role in the development of usefulalgorithms for image processing and analysis. It is observedthat most of the applications of image processing use somesort of statistical models. Generally, the models are notusually made explicit, but are made implicit by the adoption ofassumptions that incorporate certain model assumptionswithin them. Most of the algorithms, which use theassumption that the image can be treated as a randomprocess with wide sense of stationary properties, linear10
dependency, white noise are uncorrelated. In that sense,Markov Random Field (MRF) and Autoregressive (AR) modelsare most appropriate for almost all the low-level imageprocessing. Many researchers have explored the efficiency ofthe MRF and AR models for low-level image processing suchas image smoothing, object recognition, classification,segmentation, texture representation, texture synthesis,compression and reconstruction, etc. Most of the imagessatisfy the properties of MRF and AR models. The pixels in atwo-dimensional image are spatially equal interval of distancein row wise and column wise as in time series (equal intervalof time) and the pixels in the images satisfy the samplingproperties and it satisfies stationarity, linear dependency,white noise uncorrelated. Hence the MRF and AR models havedrawn the attention of the many researchers in different fieldsof image processing and analysis. As discussed in Chapter 1,different types of stochastic models used for image processingand analysis are reported by many authors, that include,Autoregressive (AR), Moving Average (MA), AutoregressiveMoving Average (ARMA) and Autoregressive Integrated MovingAverage (ARIMA) with various assumptions about the image.The assumptions are Random Field, Markov Random Field,Gibbs Field and -Field etc. are made on the basis of nature ofthe images. A brief review of the related literature, under thedifferent approaches, is given below.Generally, the main advantage of the AR model over theother models is that it is regenerative, that is, it represents allthe information in an N N image by two sets of parameters,one set containing a minimum number of parameters havingmost of the information while another set containing N 2parameters, the so-called residuals, having the remaininginformation. Kashyap suggested that the residuals can bestored with minimum number of bits than the original imagepixels without sacrificing any accuracy. With the use of storedparameters and the residual values, the original image can bereconstructed with good quality, whereas the textured imagescan be generated with the use of stored parameters of themodel only and without any compromise in the quality of theimage.11
Several authors have shown a considerable attention onMRF and AR models, due to its simplicity, i.e. lesscomputational complexity, finite memory or memory less andwide range of applications especially in texture analysis,segmentation, inpainting, reconstruction and in data mining,which searches and retrieve the images from the volume ofdatabase that contains images.1.2 Problems in Selecting ModelsThere are different types of mathematical / statisticalmodels. They can be broadly classified into Probabilisticmodels and Transformable model. The probabilistic modelcompletely deals with the probability, random process,sampling, etc., whereas the transformable models deal withthe quantization, i.e. the data in the spatial domain tofrequency domain. Generally, the actual images are in thespatial domain. By transforming the images into frequencydomain, the features can be extracted and it may beconvenient for any other type of image processing.The mathematical or statistical models can be classifiedinto either linear or non-linear models. The linear model leadsto a better results, if it is adopted for the homogeneous data.In terms of image, it may be texture image. The non-linearmodel is appropriate for the inhomogeneous data. In terms ofimage, it may be structure image. Hence, it is the importantone that the models should be applied carefully; otherwise, itcan lead to wrong results. If we employ a non-linear model onhomogeneous data or texture image, surely it does not yieldbetter results compared to that of linear models.1.3 Types of ModelsThere are a number of probabilistic models. Broadly,they are categorized into three types of stochastic models suchas Causal, Semi-causal and Non-causal.12
X(-1,1) . .X(-1,-1)X(-1,0)X(0,1) . .X(0,-1)X(0,0) . .X(1,-1)X(1,0)X(1,1)X(2,2)X(2,-2)X(2,0) X(-2,-2)X(-1,-1)X(-1,0) .X(0,-1)X(0,0) .X(1,-1)X(1,0)X(1,1)X(2,0) X(-3,3)X(-2,2) .X(2,-2)X(-3,-3)X(3,-3)X(3,3) X(-2,0) X(-2,-2) X(-3,3) X(-3,-3)X(-2,0)X(-1,1) .X(0,1) .X(3,-3)(a) CausalX(-2,2) .X(2,2)X(3,3)(b) SemicausalX(-2,0)X(-1,-1)X(-1,0) .X(0,-1)X(0,0) .X(1,-1)X(1,0) .X(2,-2)X(2,0)X(-2,2)X(-1,1) .X(0,1) .X(1,1) X(-2,-2) X(-3,3) X(-3,-3)X(3,-3) .X(2,2)X(3,3)(c) NoncausalFigure 1.1 The canonical prediction regions.The probabilistic models also can be classified intoTime Series models and Stochastic models. The time seriesmodels deal with the forecasting and prediction, whereas thestochastic models deal with the Markov properties and theconditional probability.The time series models are,1. Autoregressive (AR)2. Moving Average (MA)3. Autoregressive Moving Average (ARMA)13
4. Autoregressive Integrated Moving Average (ARIMA)The stochastic models are,1. Gaussian Random Field (GRF) models2. Mixture Gaussian Random Field (MGRF) models3. Markov Random Field (MRF) models4. Gaussian Markov Random Field (GMRF) models5. Hidden Markov Random Field (HMM) models6. Gibbs Field (GF) modelsMost of the above listed models deal with thehomogeneous data or the texture type of images, while theGibbs Field models are appropriate for analysing theheterogeneous data or structure type of images.The transformation based models play a significant rolein image processing. They yield better results for imagecompression, enhancement, etc. They are,1. Fourier Transform (FT) model2. Discrete Cosine Transform (DCT) model3. Orthogonal Polynomial Transformation (OPT) model4. Wavelet Transform modelIt is the interesting topics in image processing that howto perform different types of image processing steps through asingle statistical model. It is very useful to the researchers,scientist, designer of the image processing oriented softwaresand post graduate students in Computer Science, especiallythose are working in the area of computer vision, signalprocessing, machine learning, data and knowledge discovery,and pattern recognition etc.Throughout this book, it is discussed that the efficiencyand effectiveness of applying the Gaussian Ma
based or region-based approach. Though the region-based approach and edge-based approaches are complementary to each other the edge-based approach has been used widely. Using the edge-based approach, a number of methods have been proposed for low-level analysis viz. image compressi
Jul 09, 2010 · Stochastic Calculus of Heston’s Stochastic–Volatility Model Floyd B. Hanson Abstract—The Heston (1993) stochastic–volatility model is a square–root diffusion model for the stochastic–variance. It gives rise to a singular diffusion for the distribution according to Fell
are times when the fast stochastic lines either cross above 80 or below 20, while the slow stochastic lines do not. By slowing the lines, the slow stochastic generates fewer trading signals. INTERPRETATION You can see in the figures that the stochastic oscillator fluctuates between zero and 100. A stochastic value of 50 indicates that the closing
The input for image processing is an image, such as a photograph or frame of video. The output can be an image or a set of characteristics or parameters related to the image. Most of the image processing techniques treat the image as a two-dimensional signal and applies the standard signal processing techniques to it. Image processing usually .
(e.g. bu er stocks, schedule slack). This approach has been criticized for its use of a deterministic approximation of a stochastic problem, which is the major motivation for stochastic programming. This dissertation recasts this debate by identifying both deterministic and stochastic approaches as policies for solving a stochastic base model,
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Digital image processing is the use of computer algorithms to perform image processing on digital images. As a . Digital cameras generally include dedicated digital image processing chips to convert the raw data from the image sensor into a color-corrected image in a standard image file format. I
What is Digital Image Processing? Digital image processing focuses on two major tasks -Improvement of pictorial information for human interpretation -Processing of image data for storage, transmission and representation for autonomous machine perception Some argument about where image processing ends and fields such as image
Digital image processing is the use of computer algorithms to perform image processing on digital images. As a subfield of digital signal processing, digital image processing has many advantages over analog image processing; it allows a much wider range of algorithms to be applied to the in
