Functions, including composition of … ... To implement the inverse Fourier transform to obtain $ f(x,y) $, we need data points on a rectangular grid. from filterpy.kalman import KalmanFilter f = KalmanFilter (dim_x=2, dim_z=1) Assign the initial value for the state (position and velocity). The Radon transform can also be used for line detection, which will be the focus of this report. Each column of the image corresponds to a projection along a different angle. REST Credit cannot also be received for 18.06, 18.700. The Machine Learning Domain IEEE Project Ideas for CSE offers students an opportunity to increase their Machine Learning knowledge while gaining Python experience. The main application of the Radon transform is CAT scans, Figure 1.1, where the inverse Radon transform is applied. I need to get the sinogram this code outputs without using skimage. Götz, Druckmüller, and, independently, Brady have defined a discrete Radon transform (DRT) that sums an image's pixel values along a set of aptly chosen discrete lines, complete in slope and intercept. Figure 2: the sinogram (OK OK, it's the Radon transform) of the target. Installation The package can be installed from the Python package index: pip install radontea Testing Inverse Problems in Science and Engineering 29:8, 1108-1126. binary_blobs¶ skimage.data. We can overcome this restriction by replacing the colored crystal shape by a colored ball. Proceedings of the 2012 SEG Annual Meeting. radon_transform.py. COMP0114 Inverse Problems in Imaging. The CACAO project has been proposed to enhance the quality of SPECT images. radon data.dat which is a dat le with the samples. It is well-known that convolutions can be implemented in the Fourier Transform domain. torch_radon, Release 0.0.1 Torch Radon is a fast CUDA implementation of transforms needed for working with computed tomography data in Pytorch. There are several techniques by which the inverse transform can be calculated but the most common is Filtered Back Projection. Inequalities. Here is a filter that tracks position and velocity using a sensor that only reads position. CACAO is a short hand notation for computer aided collimation tomography. The transform was introduced in 1917 by Johann Radon, who also provided a formula for the inverse transform. In mathematics, the Radon transform is the integral transform which takes a function f defined on the plane to a function Rf defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the line integral of the function over that line. Tomographic reconstruction is a type of multidimensional inverse problem where the challenge is to yield an estimate of a specific system from a finite number of projections.The mathematical basis for tomographic imaging was laid down by Johann Radon.A notable example of applications is the reconstruction of computed tomography (CT) where cross-sectional images … Inverse Radon transform collapse all in page Syntax I = iradon (R,theta) I = iradon (R,theta,interp,filter,frequency_scaling,output_size) [I,H] = iradon ( ___) Description example I = iradon (R,theta) reconstructs the image I from projection data … In this paper, we propose a pipeline to learn a filter and additional weights to invert the weighted forward projection. The Python code for reproducing our results is available at Shifat-E-Rabbi et al. The underlying object is independent of the representation used for initialization. This involves a Fourier transform, followed by multiplication by the (absolute value of) frequency, followed by an inverse Fourier transform. Parameters length int, optional. Real and Complex roots. The Department of Mathematics offers training at the undergraduate, graduate, and postgraduate levels. There are three possible methods of reconstructing an object from a complete set of its parallel projections: (i) back-projection and deconvolution, (ii) filtered back-projection and (iii) reconstruction via the central slice theorem. Thus Computed Tomography is an attempt to achieve an Inverse Radon Transform. MiniProject - Python Hints 1. Take A Sneak Peak At The Movies Coming Out This Week (8/12) Why Your New Year’s Resolution Should Be To Go To The Movies More; Minneapolis-St. Paul Movie Theaters: A Complete Guide Introductory course in linear algebra and optimization, assuming no prior exposure to linear algebra and starting from the basics, including vectors, matrices, eigenvalues, singular values, and least squares. The idea of inverse pole figure color coding of orientation maps is to visualize the orientation of a grain by the color of the crystal face pointing towards you. Python NumPy module deals with creation and manipulation of array data elements. This thesis describes one solution to the inverse problem of solving for the attenuation coe cient in medical imaging. Words - Free ebook download as Text File (.txt), PDF File (.pdf) or read book online for free. However, first, I need to know the exact angle of the object at each step. (Python code implementing the Radon cumulative distribution transform subspace model for image classification. A practical, exact implementation of the inverse Radon transform does not exist, but there are several good approximate algorithms available. In the section Denoising and Filling Missing Data we have discussed how to work with noisy EBSD data the contained non indexed pixels. Computes the inverse of the fan-beam Radon transform using interpolation of the linogram and one of the inverse algorithms for tomography with the Fourier slice theorem. Absolute values. Examples. can take the exact inverse transform from the regularly sampled domain to an irregularly sampled domain and use this as a forward model in an inverse problem. This is the basic approach for the hyperbolic Radon transform and the linear Radon transform by Thorson and Clarebout (1985). Finally, we provide code in Python and MATLAB. But I am unable to find any implementation in python. To reconstruct the image from the sinogram, the inverse Radon transform is applied to the image. Journal of Geophysical Research 112, B10303. The fanbeam function computes the line integrals along paths that radiate from a single source, forming a fan shape. 2 Notation, the NDFT, and the NFFT This section summarises the mathematical theory and ideas behind the NFFT. (We focus on the 2D case throughoutmost of … The figures below show a y-slice of a 3D volume with two linear events, FK transforms of the data, and and Py slices of 3D Tau-Px-Py transforms of the data. Compute the N-dimensional discrete Fourier transform of A using a Fast Fourier Transform (FFT) algorithm. From 2002, the BE Liberty B4 was offered with an automatic transmission for which the EJ208 engine was … Python实现逆Radon变换——直接反投影和滤波反投影. InverseRadonTransform computes a radial Fourier transform, followed by a two-dimensional inverse Fourier transform, to accomplish the above inversion. It allows the training of end-to-end models that takes sinograms as inputs and produce images as output. The mathematical foundation for these reconstruction methods are the Radon transform, the inverse Radon transform, and the projection slice theorem. I should be able to tomographically reconstruct the object using an inverse Radon transform. I focused on code readability and thorough comments. Academia.edu is a platform for academics to share research papers. The fanbeam function computes projections of an image matrix along specified directions. The “filtered back projection” then becomes. Stack Exchange network consists of 178 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.. Visit Stack Exchange s2let_radon_inverse Compute inverse (spin) radon transform in harmonic space. from the slices, and an inverse 2-D Fourier transform recovers f. Practical applications inevitably deal with images f(x,y) and sinograms p(ρ,θ) that are represented discretely, usually as 2-D arrays of values. #include Finds the camera intrinsic and extrinsic parameters from several views of a calibration pattern. As described in Radon Transform, given an image I and a set of angles theta, the radon function can be used to calculate the Radon transform. The function iradon can then be called to reconstruct the image I from projection data. In the example above, projections are calculated from the original image I. There is a large, if scattered, literature concerning approximations of the continuous Radon transform, and its inverse, in such cases. This example shows how to use the pylops.signalprocessing.ChirpRadon2D and pylops.signalprocessing.ChirpRadon3D operators to apply the linear Radon Transform to 2-dimensional or 3-dimensional signals, respectively.. Project description Algorithms to compute the inverse Radon transform (e.g. A practical, exact implementation of the inverse Radon transform does not exist, but there are several good approximate algorithms available. The Radon transform is a process that can be used to simulate the data received by a CT scan. Define the Hilbert transform as Radon Transform of Head Phantom Using 90 Projections Note how some of the features of the input image appear in this image of the transform. The background: I conducted an experiment where I rotated an object (with three-fold symmetry) many times and recorded the projection of the object. Show activity on this post. Several projection geometries are commonly used, including parallel beam, fan beam, and cone beam. Typical linear size of blob, as a fraction of length, should be … The result is a collection of algorithms that are suitable for teaching the basics of computerized tomography. Stack Exchange network consists of 178 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.. Visit Stack Exchange The filtered back projection algorithm is split into two phases, filtration and projection. Author: R. Schultz, 2012 Translated to Python by: S. Schneider, 2016 """ # Check for Data type of variables. 6. I used python as the handling of .fits file is readily available through the astropy.io module and radon/inverse radon transform are available through the skimage module. The R-CDT ˆ s (t, θ) is then obtained by applying the CDT over the t dimension of ˜ s (t, θ), ∀ θ. Professional academic writers. Also, there are two general ways to implement the chosen Radon transform. It can either be a function of θand ρfor which all matching pixels are calculated, or it could be implemented as a function of the image pixels. The latter is the easiest but is, however, the Hough transform. It is implemented in MATLAB. You just need to load this with numpy.load(’SLphan.npy’). The Radon transform yields a better seismic image than the bandpass and inverse Q filters, since it removes the noise and the multiples from the data quite efficiently. Download this file. The Radon transform fits badly Single Photon Emission Tomography (SPECT). (a friendly graphical language) and Python, and will design and implement two projects of their choice. Hoss Belyadi, Alireza Haghighat, in Machine Learning Guide for Oil and Gas Using Python, 2021. A few caveats though:-you would need python and the above modules to run it-it is slow (one night to process one image) The object is then reconstructed using a 2-D inverse Fourier Transform. With a sampled discrete system, the inverse Radon Transform is Which is the back projection method for CT scanner? Relaization of the filtered backprojection. Radon Transform for P projection of image f: Hough transform. Raw. Consider a counter-clockwise rotation of 90 degrees about the z-axis. (a) Write an algorithm to compute the projections (do not use PYTHON’s radon function). In doing so perhaps it can be clearer to the user what operations are actually taking place in the reconstruction process, as the acquisition of projections and the backprojection process … A fast approach to the Radon transform and its inverse, as well as expansions to using other operators besides summation, is presented in Press (2006). Back-projection Radon transform: If image is rotated by the inverse of the output, the lines will be: horizontal (though they may be upside-down depending on the original image) It doesn't work with black borders """ from __future__ import division, print_function: from skimage. The transform is fast, O ( N 2log N ) for an N × N image; it uses only addition, not multiplication or interpolation, and it admits a fast, exact algorithm for the … The inverse Radon transform is used in computed tomography to reconstruct a 2D image from the measured projections (the sinogram). Here is the recipe: given the Radon transform , a function of polar coordinates, model aplikasi bahasa python perancangan program hidrograf satuan sintetis metode nakayasu, snyder, scs, dan gama i Andhy, Setyo Raharjo (, 2021-08-16) Hidrograf Satuan Sintetis (HSS) merupakan salah satu analisis yang digunakan untuk mengubah hujan 1 mm menjadi debit aliran langsung (direct run off) suatu Daerah Aliran Sungai (DAS). 2.2 Radon Transform . This means that if we have a bunch of lines confusing our image, we could do a radon transform and erase the resuling points (circles in practice), then do an inverse transform to go back to an image with no lines in it. Take A Sneak Peak At The Movies Coming Out This Week (8/12) Why Your New Year’s Resolution Should Be To Go To The Movies More; Minneapolis-St. Paul Movie Theaters: A Complete Guide The notation 1j is Python's code for the famous imaginary number sqrt(-1). Radon Transform and Inverse Problems. To represent an image, the fanbeam function takes multiple projections of the image from different … Lets take a part of scanned image and see how to correct skew. The module radonteaimplements some of them. SART, backprojection, Fourier interpolation). Deep convolutional neural networks (CNNs) are broadly considered to be state-of-the-art generic end-to-end image classification systems. The collection of these g(phi,s) at all phi is called the Radon Transform of image f(x,y). The respective data volumes were inverted The Python code for reproducing our results is available at Shifat-E-Rabbi et al. Updated on Nov 1, 2018. As the inverse Radon transform reconstructs the object from a set of projections, the (forward) Radon transform can be used to simulate a tomography experiment. This grid does not match the radial grid. Example. weixin_42699499: 拉东变换：skimage.transform.radon 反变换skimage.transform.iradon. Parameters. Furthermore, using the Fourier Slice Theorem [34, 35], the inverse Radon transform \(s=\mathscr {R}^{-1}(\widetilde{s})\) is defined as Radon transform: If image is rotated by the inverse of the output, the lines will be: horizontal (though they may be upside-down depending on the original image) It doesn't work with black borders """ from __future__ import division, print_function: from skimage. 8.6.2 Inverse Radon Transform. Stack Exchange network consists of 178 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.. Visit Stack Exchange Abstract. F H F = I ). 57248 lines (57247 with data), 623.3 kB IEEE Geoscience and Remote Sensing Letters 19 , 1-5. (Python code implementing the Radon cumulative distribution transform subspace model … When working with the linear Radon transform, this is a faster implementation compared to in … Fig 3: Inverse Radon Transform. The inverse Radon transform is used in computed tomography to reconstruct a 2D image from the measured projections (the sinogram). As described in Radon Transform, given an image I and a set of angles theta, the radon function can be used to calculate the Radon transform. GitHub | Report. Type or paste a DOI name into the text box. By this way, the GAC-based forward blur models only require several FFTs Image containing radon transform (sinogram). Topline method. This is the required formula for inversion of the Radon transform. Python implementation of the Radon Transform. 前几天我学习了Radon变换并用Python做了一个简单的程序（见上一篇博文），昨天看了一下逆Radon变换，尝试了一下简单的实现。Radon变换存在逆变换，我们可以通过Sinogram图来还原原始的图像，一种方法是直接反投影，另一种方法是滤波反投影（这个方法是大部分商用CT机器使用的算法）。 The main idea of this project is to use collimators with much larger holes … As we can see in the bottom figures, the adjoint Radon transform is far from being close to the inverse Radon transform, i.e. These are the top rated real world Python examples of skimagetransform.radon extracted from open source projects. Bandpass filter, Inverse Q, and Radon transform are applied to the migrated prestack data, over the time range of 0-6000 ms. Reconstruct an image from the radon transform, using a single iteration of the Simultaneous Algebraic Reconstruction Technique (SART) algorithm. s,tomo,radon 3.2 Radon transform in 2D (s,tomo,radon) The foundation of analytical reconstruction methods is the Radon transform, which relates a 2D function f(x,y) to the collection of line integrals of that function. This makes this denoise module very concise, if not simple. Radon depends on as few packages as possible. ... Compute the inverse Fourier transform of log spectrum. R … [radon transform] stability [stability inverse nfft] Figure 1.1: Directory structure of the NFFT 3.0 package 4. blob_size_fraction float, optional. A projection of a two-dimensional function f(x,y) is a set of line integrals. In mathematics, the Abel transform, named for Niels Henrik Abel, is an integral transform often used in the analysis of spherically symmetric or axially symmetric functions. A new tool to compute the 3D linear forward and inverse Radon transform. In this method, we will convert image to black (absence of pixel) & white (presence of pixel). (Python code implementing the Radon cumulative distribution transform subspace model … ». The Python code for reproducing our results is available at Shifat-E-Rabbi et al. This code does not explicitly use Radon transform functions. Insertion sorts ... Multiplicative inverse Algorithms: for computing a number's multiplicative inverse (reciprocal). This chapter covers the theory, step-by-step codes, and applications of various supervised learning algorithms including multilinear regression, logistic regression, k-nearest neighbor (KNN), support vector machine (SVM), decision tree, random forest, extra trees, … Our global writing staff includes experienced ENL & ESL academic writers in a variety of disciplines. lds – Distance in pixels between source and detector. Hereby, we made the assumption that the grid before and after the operations is the same. This lets us find the … Without access to a CT scanner, the data needs to be computer-generated in order to run a test. The transform was introduced in 1917 by Johann Radon, who also provided a formula for the inverse transform. Scanline method. linogram (2d ndarray of shape (D, A)) – Input linogram from the synthetic aperture measurement. Chirp Radon Transform¶. Each column of the image corresponds to a projection along a different angle. As an exercise, you may try to implement the inverse radon transform of an image using the projection-slice theorem and the ifft. 宋体的微软雅黑: 可以吧，直线检测就是用的Radon变换啊。 Let the torus Td:= In this paper, we propose to use binary block Walsh-Hadamard transform (WHT) instead of … Radon will run from Python 2.7 to Python 3.8 (except Python versions from 3.0 to 3.3) with a single code base and without the need of tools like 2to3 or six. (Python code implementing the Radon cumulative distribution transform subspace model … Inverse radon transform. Parameters: radon_image: 2D array, dtype=float. Parameters radon_image 2D array. There are several methods to compute the inverse Radon transform. Figure 4: the FFT-ed sinogram transformed to 2D FFT space (DC at center). model aplikasi bahasa python perancangan program hidrograf satuan sintetis metode nakayasu, snyder, scs, dan gama i Andhy, Setyo Raharjo (, 2021-08-16) Hidrograf Satuan Sintetis (HSS) merupakan salah satu analisis yang digunakan untuk mengubah hujan 1 mm menjadi debit aliran langsung (direct run off) suatu Daerah Aliran Sungai (DAS). Jingwei Hu, Sergey Fomel, Laurent Demanet, and Lexing Ying, A fast butterfly algorithm for the hyperbolic Radon transform. You can get the data from any of the two following les in Canvas (both containing the same data in di erent formats): radon data.mat which is a Matlab le with the array Y0 containing the samples. 0 the matrix obtained by sampling the Radon transform of the analytical phantom on this grid. the , . To be able to study different reconstruction techniques, we first needed to write a (MATLAB) program that took projections of a known image. We would like to show you a description here but the site won’t allow us. This script performs the Radon transform to simulate a tomography experiment and reconstructs the input image based on the resulting sinogram formed by the simulation. Most of these packages are built on the Python programming language, but experience with another common programming language is … Radon transform and steerable filter methods of motion blur angle estimation in that order. The proposed radar uses rotation movement and applies inverse Radon transform to resolve a 2D image of each range bin. The Radon transform is a valuable tool in inverse problems such as the ones present in electromagnetic imaging. provides a method for performing an inverse Radon transform). Calculate the Radon transform of an image and test the back-projection method The le SLphan.npycontains the image. However, projection profile method is the simplest and easiest way to determine skew in documents in the range ±5°. ... Filtered back-projection: efficiently computes the inverse 2-dimensional Radon transform. This function is an extension of calibrateCamera with the method of releasing object which was proposed in .In many common cases with inaccurate, unmeasured, roughly planar targets (calibration plates), this method can dramatically improve the precision … Linear size of output image. Click Go. Its expertise covers a broad spectrum of fields ranging from the traditional areas of "pure" mathematics, such as analysis, algebra, geometry, and topology, to applied mathematics areas such as combinatorics, computational biology, fluid dynamics, theoretical computer science, … Geometrically, the inversion procedure recovers an image from the values of its Radon transform along different projections of the image for fixed angles and varying . binary_blobs (length = 512, blob_size_fraction = 0.1, n_dim = 2, volume_fraction = 0.5, seed = None) [source] ¶ Generate synthetic binary image with several rounded blob-like objects. 18.061 Linear Algebra and Optimization (New) Prereq: Calculus II (GIR) U (Fall) 5-0-7 units. For my mini-project in Inverse Problems in Imaging, I investigated the Radon transform and its inverse using MATLAB. In mathematics, the Radon transform is the integral transform which takes a function f defined on the plane to a function Rf defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the line integral of the function over that line. Figure 3: the FFT-ed rows of the sinogram (plotted with DC at center). (2021) Image ... because the Radon transform of an off-center point source is a sinusoid. For large objects, this affects the tomographic reconstruction and leads to artifacts and perturbed phase values. Fig. Ramachandran & Lakshminarayanan (1971) proposed the back projection method to overcome the blurring image which is … Code faster with the Kite plugin for your code editor, featuring Line-of-Code Completions and cloudless processing. Reconstruct an image from the radon transform, using a single iteration of the Simultaneous Algebraic Reconstruction Technique (SART) algorithm. Use the Shep-Logan CT phantom which is attached below to show the effect of a Rayleigh, Exponential and Gaussian noise on the quality images computed using the inverse radon transform function. The reconstruction algorithm of FBP is a direct implementation of the Inverse Radon transform formula with measured projections being filtered before they are back-projected . The first column in the Radon transform corresponds to a projection at 0º that is integrating in the vertical direction. model, a homography is the inverse polar transform of the convolution of a kernel with the transformed sharp image (Please refer to Section 3.2 for detailed proof), in which the convolution operation can be e ciently computed with the help of FFT. 1.1Recommended literature •Aninash C. Kak and Malcom Slaney. ... streaks in the same image. In Australia, the EJ208 engine was introduced in the 2001 Subaru BE Liberty B4 which, with a manual transmission, produced peak outputs of 190 kW and 320 Nm. Python radon - 30 examples found. Image containing radon transform (sinogram). Inverse Radon Transform Definition The iradon function inverts the Radon transform and can therefore be used to reconstruct images. If desired, data estimated in the Fourier domain can be transform Statistiques et évolution des crimes et délits enregistrés auprès des services de police et gendarmerie en France entre 2012 à 2019 We train and validate the method with a … Up to now the inversion of the multiscale discrete Radon transform has been only possible by iterative numerical methods while the continuous Radon transform is usually tackled with the filtered backprojection approach. Documentation The documentation, including the reference and examples, is available at radontea.readthedocs.io. Used in Python 2.3 and up, and Java SE 7. 3 A hybrid Linear-Hyperbolic Radon transform Daniel Trad 1, Mauricio D. Sacchi 2, and Tadeusz J. Ulrych1 1The university of British Columbia, 2University of Alberta Summary The main application of the Radon transform (RT) is to map events with different curvature in the data space to different areas of a model Color is a function of absolute value. The Inverse Radon transform can be expressed as the composition of two operators, the back-projection operator and the derivative filtering operator. Jack Poulson, Bjorn Engquist, S. Li, and Lexing Ying, A parallel sweeping preconditioner for frequency-domain seismic wave propagation. The inverse Radon transform is used in computed tomography to reconstruct a 2D image from the measured projections (the sinogram). 17 Filtered Back-projection Python version None Upload date Dec 8, 2021 Hashes View Filename, size deskew-0.10.39-py2.py3-none-any.whl (4.6 kB) File type Wheel Python version py2.py3 Upload date Dec 8, 2021 Hashes View Close. During the process of image reconstruction, the CT data is converted “back” into the image using the inverse Radon transform [9]. Kite is a free autocomplete for Python developers. 3: The process of calculating the Radon cumulative distribution transform (R-CDT) of an image s (x) (defined as a 2-dimensional probability density function). Applications include interpolation, denoising and deghosting. (2022) Multitarget Time-Reversal Radar Imaging Method Based on High-Resolution Hyperbolic Radon Transform. We would like to show you a description here but the site won’t allow us. Computational techniques include filtered backprojection and a variety of iterative methods. In Section 2 we o er a mathematical model for X-ray tomography and in Sections 3 and 4 we discuss the Radon transform and a parametrization method which presents us with a link between our measured data and the attenuation The latest Lifestyle | Daily Life news, tips, opinion and advice from The Sydney Morning Herald covering life and relationships, beauty, fashion, health & wellbeing In this study, we will show, for the first time, … It can also run on PyPy without any problems (currently PyPy 3.5 v7.3.1 is used in tests). Th Radon transform of a line \(mx +b \) is the point \( (m,b)\) in slope-intercept space. The Radon transform is named after the Austrian mathematician Johann Karl August Radon (December 16, 1887 – May 25, 1956) [4]. A. Hough Transform Method . of and to in a is that for on ##AT##-##AT## with The are be I this as it we by have not you which will from ( at ) or has an can our European was all : also " - 's your We ... (i.e. The Hough transform [17] can be applied to find global patterns such as lines, circles, and ellipses in an image in a parameter space. s2let_transform_synthesis_mw Compute inverse (spin) directional wavelet transform, output in pixel space (MW sampling). Radon transform and Ridgelet transform s2let_radon_transform Compute forward (spin) radon transform in harmonic space. Reconstruction d'une image médicale par l'usage de la transformée inverse de Radon sur les données d'atténuation des radiations traversant le tissu biologique. Imaging mantle discontinuities using least-squares Radon transform. c matlab image-processing filters medical-image-processing denoising radon-transform reconstructed-images. 357463527-Password-List.pdf - Free ebook download as PDF File (.pdf), Text File (.txt) or read book online for free. Consequently, 3D imaging can … The tool is high resolution even with highly aliased data. However, they are known to underperform wh The Radon Transform mathematically describes a one-dimensional projection P of a two-dimensional image as a function of viewing angle θ. The Python code for reproducing our results is available at Shifat-E-Rabbi et al. MATH 0005 [0.5 credit] Precalculus: Functions and Graphs Review of algebraic manipulations. To review, open the file in an editor that reveals hidden Unicode characters. Consider the synthetic CMP gather in Figure 6.4-7a and the velocity-stack gather (Figure 6.4-7b) estimated from it using the discrete Radon transform.By including the entire velocity-stack gather in the summation in equation (), we get the fully reconstructed CMP gather shown in Figure 6.4-7c.That result is reproduced in Figure 6.4-19a.
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