Patent Application
20120004849
The Radon transform has proven to be a useful technique for tasks such as finding lines in images. For example, to find lines in an image, a conventional windowed Radon transform technique rotates a line a number of degrees about each pixel to determine the angle that includes pixels most representative of a line. Such a conventional brute force approach for an image dimensioned M×N has a computational complexity on the order of l*A*M*N, which l is the length of the line in pixels and A is the number of angles. As described herein, various approaches can reduce computational requirements for application of the Radon transform.
One or more computer-readable media including computer-executable instructions to instruct a computing system to define a Radon transform convolution mask; specify an angle that defines a line extending at least partially across a pixel image; and apply the mask successively to target pixels on the line to compute a statistical value for each of the target pixels where application of the mask identifies a set of pixels for computing the statistical value and where each successive application of the mask identifies a set of pixels that includes at least one pixel of a prior set and at least one pixel not included in the prior set to thereby reduce requirements for computing the statistical values. Various other apparatuses, systems, methods, etc., are also disclosed.
This summary is provided to introduce a selection of concepts that are further described below in the detailed description. This summary is not intended to identify key or essential features of the claimed subject matter, nor is it intended to be used as an aid in limiting the scope of the claimed subject matter.
Features and advantages of the described implementations can be more readily understood by reference to the following description taken in conjunction with the accompanying drawings.
The following description includes the best mode presently contemplated for practicing the described implementations. This description is not to be taken in a limiting sense, but rather is made merely for the purpose of describing the general principles of the implementations. The scope of the described implementations should be ascertained with reference to the issued claims.
In the example of
The simulation component 120 may process information to conform to one or more attributes, for example, as specified by the attribute component 130, which may be a library of attributes. Such processing may occur prior to input to the simulation component 120 (e.g., per the processing component 116). Alternatively, or in addition to, the simulation component 120 may perform operations on input information based on one or more attributes specified by the attribute component 130. As described herein, the simulation component 120 may construct one or more models of the geologic environment 150, which may be relied on to simulate behavior of the geologic environment 150 (e.g., responsive to one or more acts, whether natural or artificial). In the example of
As described herein, the management components 110 may include features of a commercially available simulation framework such as the PETREL® seismic to simulation software framework (Schlumberger Limited, Houston, Tex.). The PETREL® framework provides components that allow for optimization of exploration and development operations. The PETREL® framework includes seismic to simulation software components that can output information for use in increasing reservoir performance, for example, by improving asset team productivity. Through use of such a framework, various professionals (e.g., geophysicists, geologists, and reservoir engineers) can develop collaborative workflows and integrate operations to streamline processes.
As described herein, the management components 110 may include features for geology and geological modeling to generate high-resolution geological models of reservoir structure and stratigraphy (e.g., classification and estimation, facies modeling, well correlation, surface imaging, structural and fault analysis, well path design, data analysis, fracture modeling, workflow editing, uncertainty and optimization modeling, petrophysical modeling, etc.). Particular features may allow for performance of rapid 2D and 3D seismic interpretation, optionally for integration with geological and engineering tools (e.g., classification and estimation, well path design, seismic interpretation, seismic attribute analysis, seismic sampling, seismic volume rendering, geobody extraction, domain conversion, etc.). As to reservoir engineering, for a generated model, one or more features may allow for simulation workflow to perform streamline simulation, reduce uncertainty and assist in future well planning (e.g., uncertainty analysis and optimization workflow, well path design, advanced gridding and upscaling, history match analysis, etc.). The management components 110 may include features for drilling workflows including well path design, drilling visualization, and real-time model updates (e.g., via real-time data links).
As described herein, various aspects of the management components 110 may be add-ons or plug-ins that operate according to specifications of a framework environment. For example, a commercially available framework environment marketed as the OCEAN® framework environment (Schlumberger Limited) allows for seamless integration of add-ons (or plug-ins) into a PETREL® framework workflow. The OCEAN® framework environment leverages .NET® tools (Microsoft Corporation, Redmond, Wash.) and offers stable, user-friendly interfaces for efficient development. As described herein, various components may be implemented as add-ons (or plug-ins) that conform to and operate according to specifications of a framework environment (e.g., according to application programming interface (API) specifications, etc.). Various technologies described herein may be optionally implemented as components in an attribute library.
In the field of seismic analysis, aspects of a geologic environment may be defined as attributes. In general, seismic attributes help to condition conventional amplitude seismic data for improved structural interpretation tasks, such as determining the exact location of lithological terminations and helping isolate hidden seismic stratigraphic features of a geologic environment. Attribute analysis can be quite helpful to defining a trap in exploration or delineating and characterizing a reservoir at the appraisal and development phase. An attribute generation process (e.g., in the PETREL® framework or other framework) may rely on a library of various seismic attributes (e.g., for display and use with seismic interpretation and reservoir characterization workflows). At times, a need or desire may exist for generation of attributes on the fly for rapid analysis. At other times, attribute generation may occur as a background process (e.g., a lower priority thread in a multithreaded computing environment), which can allow for one or more foreground processes (e.g., to enable a user to continue using various components).
Attributes can help extract the maximum amount of value from seismic and other data, for example, by providing more detail on subtle lithological variations of a geologic environment (e.g., an environment that includes one or more reservoirs).
In general, an accurate reconstruction of paleostress can be difficult to achieve for a geologic environment. In particular, stress magnitudes can be difficult to reconstruct based on borehole data (e.g., as acquired over a field grid). Stress magnitudes are helpful to understand and exploit resources in reserves such as carbonate reserves, which are estimated to hold more than 60% of the world's oil and 40% of the world's gas reserves. For example, consider that the Middle East has an estimated 62% of the world's proved conventional oil reserves where more than 70% of these reserves are in carbonate reservoirs and that the Middle East has an estimated 40% of the world's proved gas reserves where 90% of these gas reserves lie in carbonate reservoirs.
Unlike sandstones, with their well-characterized correlations of porosity, permeability, and other reservoir properties, heterogeneous pore systems of carbonate rocks can defy routine petrophysical analysis. Carbonates are deposited primarily through biological activity where the resulting rock composition (e.g., of fossil fragments and other grains of widely varying morphology) produces highly complex pore shapes and sizes. Carbonate mineral species are also comparatively unstable and are subjected to multiple stages of dissolution, precipitation, and recrystallization, adding further complexity to the porosity and permeability of the rocks. Further, comparatively simple relationships that might have existed between depositional attributes, porosity, and permeability can be obscured by such physical, biological, and chemical influences, operating at different scales, during and continuing after deposition. One challenge for accurate evaluation of carbonate formations is accounting for reservoir heterogeneity on a multiplicity of scales (e.g., of grains, pores, and textures).
In the oil and gas industry, existing approaches for detection of faults, fractures and estimation of possible stress in layers close to the surface sometimes include analysis of attributes based on local dip angle for the surface, attributes based on local azimuth angle for the surface and attributes based on curvature of a single surface. As described herein, various techniques that rely on the Radon transform can enhance identification of cracks, faults, discontinuities, etc., which, in turn, can provide for a more comprehensive understanding of a reservoir environment. While various examples are described with respect to analysis of seismic data, techniques may be used for other types of data especially where implementation of the Radon transform may benefit from reduction in requirements for computational resources.
The Radon transform may be implemented in the system 100 for tasks such as identifying cracks in the seabed or other geologic environment. Seismic data may be generated by transmitting sound energy in an environment and measuring energy responsive to such transmission, reflections, etc. The measured energy may be used to generate an image (e.g., similar to a technique used for ultrasounds), which may be referred to as a seismic image. A seismic image may be processed using any of a variety of techniques prior to application of the Radon transform. For example, one or more edge enhancement techniques may be applied to removes interface and enhance the irregularities (e.g., subtraction of an offset image).
In various conventional applications of the Radon transform, runtime is dependent on image size, line length searched for and the number of angles. The line length and the number of angles are often about the same size which makes the runtime increase as a quadratic polynomial dependent on line length, if the image size is fixed.
As described herein, the Radon transform is implemented with a moving mask (e.g., a sliding window) that can reuse elements. Such an approach can be referred to as a type of windowed Radon transform. A conventional Radon transform approach may be applied to look for features (e.g., lines, edges, etc.) spanning an entire image while a windowed radon transform approach may be applied to look for features (e.g., lines, edges, etc.) in small windows within an image. As described herein, window or mask size can be determined by a line-length parameter. In a windowed approach, a general Radon transform may be applied to every small window, subset or partition of an image. The term “Radon transform” is used generally herein, for example, as including windowed approaches.
As described herein, a “sliding-window” (or sliding mask) can be implemented as part of a windowed Radon transform approach to data analysis (e.g., edge detection, etc.). Theoretically, such an approach may make the algorithm linearly dependent on line length. Various techniques for optimization are also described herein. In various examples described herein, mask size (or window size) is based on a line length where elements (e.g., pixels, voxels, etc.) are selected to determine a value or values for a single element of interest positioned along the line (e.g., a center element). The selected elements also depend on line angle with respect to the element of interest. Variations of such an approach may be implemented as well (e.g., consider a variation where blocks of pixels values are averaged to create “elements”, etc.).
A particular approach aims to turn mean values into discernable lines via a peak detection process. For example, the result of the transform can be scanned for pixels that are greater in magnitude than their surroundings. Various approaches described herein can optionally decrease computational requirements while keeping noise to an acceptable level.
Various examples refer to fault line detection and more particularly to automatic fault line detection in 2D seismic images. This is a well-known problem in the hydrocarbon energy industry, because the presence of faults in, or around, a reservoir can impact reservoir production performance. Fault line detection is recognized as being a complex problem, where improvements are still actively being sought. A particular approach to fault line detection includes choosing preferred line segments based on both semblance (e.g., mean) and normalized variance, and optionally also based on consistent dip direction between neighboring line segments; improving runtime of such an algorithm by calculating running sums and variances; and calculating running sums in a skewed image, generated through an integer coordinate transform.
The method 210 also includes a process block 226 for processing results of the executed algorithm. Such processing may be optional. Such processing may include global optimization, for example, where a global cost function is applied to all of the results (e.g., to minimize variation of results from pixel to pixel, to maximize mean and variance results for each pixel, etc.). A global cost function may aim to minimize the number of lines or the directions of the lines to ensure that major cracks or faults are well-marked. As indicated in
The method 210 is shown in
As described herein, one or more computer-readable media can include computer-executable instructions to instruct a computing system to: receive data about a geologic environment (see, e.g., blocks 214 and 216); optionally enhance the data using an edge enhancement technique (see, e.g., blocks 218 and 220); execute the Radon transform with a convolution mask that moves along lines defined by angles to generate results (see, e.g., blocks 222 and 224); process results of the Radon transform to identify geological features and artifacts in the data (see, e.g., blocks 226 and 228); and output information sufficient to render the identified geological features to a display (see, e.g., blocks 230 and 232). As described herein, artifacts may be considered noise. As described herein, instructions may be provided to instruct a computing system to compute mean and variance and to identify lines as being characterized by a high mean and a low variance and to identify noise (e.g., speckle noise) as being characterized by a high mean and a high variance.
An approximate graphic in
The method 410 is shown in
As described herein, one or more computer-readable media can include computer-executable instructions to instruct a computing system to: define a Radon transform convolution mask (see, e.g., blocks 414 and 416); specify an angle that defines a line extending at least partially across a pixel image (see, e.g., blocks 418 and 420); and apply the mask successively to target pixels on the line to compute a statistical value for each of the target pixels where application of the mask identifies a set of pixels for computing the statistical value and where each successive application of the mask identifies a set of pixels that includes at least one pixel of a prior set and at least one pixel not included in the prior set to thereby reduce requirements for computing the statistical values (see, e.g., blocks 422 and 424). In the example of
As described herein, information from a Radon transform algorithm, such as statistical values, may be used, at least in part, to identify lines in a pixel image. As described herein, the Radon transform may include a line length parameter, a direction parameter and a location parameter. For example, an angle may specify the direction parameter of the Radon transform and a target pixel may specify the location parameter of the Radon transform.
As described herein, one or more computer-readable media may include computer-executable instructions to instruct a computing system to, for each target pixel, compare its statistical value or values (e.g., optionally via a cost function) to a previously computed value or values for the target pixel where the previously computed value or values correspond to a different angle. Such instructions may be configured to instruct a computing system to select an optimal angle for a target pixel based at least in part on a comparison of values.
As mentioned, instructions may be configured to instruct a computing system to, for each target pixel, compute a cost function where the cost function depends at least in part on a statistical value for a target pixel. For example, a cost function may depend on a mean and a variance for a target pixel.
While various examples pertain to 2D data, various techniques described herein may be applied to higher dimensional data. Further, an image may be a three-dimensional image where the pixels are voxels (volume elements).
The method 510 is shown in
As described herein, a method for identifying lines includes defining a Radon transformation convolution mask (see, e.g., block 514 and 516); specifying angles where each angle defines a line with respect to a set of data (see, e.g., block 518 and 520); moving the mask along the lines to determine cost function values for various data coordinates for each of the angles where, along a line, the cost function value for a data coordinate is determined in part by one or more cost function variable values of an adjacent data coordinate (see, e.g., block 522 and 524); selecting optimal angles for at least some of the various data coordinates based at least in part on the cost function values (see, e.g., block 526 and 528); and outputting information sufficient to render an image to a display where the rendered image includes lines identified at least in part by the selected optimal angles.
As described herein, various techniques allow for implementation of the Radon transform in a more efficient manner, which can reduce requirements for computational resources. Further, various metrics generated via such techniques may be used in manners that can enhance analysis of data (e.g., 2D or 3D data).
The Radon transform (e.g., windowed Radon transform) can be described with respect to three input parameters for a convolution mask (or window): length (l pixels), direction (A angles) and location (the number of pixels in a data set, e.g., in an image M×N). With three parameters, it is possible to fix two and to vary the third. For example, by keeping the length fixed, an implementation may either iterate over possible directions of a line, or locations of a line. A brute force manner of implementing the Radon transform keeps location fixed, which makes it necessary to recompute the transform for all angles A. The resulting computational complexity for an image is on the order of l*A*M*N.
As described herein, where the direction is fixed, a convolution mask (or window) is traversed along that direction. In such an implementation, it becomes possible to reuse computations of pixels overlapping a current and the latest convolution mask since the kernel functions are linear. Accordingly, only 2 computations may be needed for one mask step, which corresponds to a computational complexity for the whole image on the order of 2*A*M*N. In terms of computation time, this may result in a decrease on the order of l/2. Such an approach is referred to herein, at times, as a sliding window approach.
With respect to angles, an algorithm may be configured to narrow down the angle domain of the Radon transform to either: 1) a global angle interval for the entire image or 2) an interval associated to each output pixel [θmin,l, θmax,l] (noting that symmetry of the Radon transform for angles modulo π allows for a smaller interval). Given a priori knowledge of an angle distribution for a data set (e.g., an image), results may be improved. For example, as described herein, an optimization process can be implemented to choose the best angle for each coordinate in a data set (e.g., 2D or 3D), which for image data, may correspond to a pixel.
As described herein, for each pixel at each angle, an implementation of the Radon transform can be used to compute a duple of mean and variance values. For all angles at each pixel, one of the duples is to be selected as the best value, which, for an image, will result in a 2D matrix of best values. One approach chooses the value according to the angle with the largest mean, unless the variance exceeds a cutoff (e.g., a predetermined variance value for a selection process). Such an approach gives strong lines but can have some issues with noise, especially around small areas containing more than one line. To reduce noise, one or more approaches may be taken. One approach considers that variance should be minimized, while the mean peak should be maximized. Accordingly, a local examination of variance and mean may be performed (e.g., within a specified distance from the pixel). Another approach involves determining whether or not a maximum is noise based on angle continuity (e.g., within a specified distance from the pixel). In general, it is unlikely that all three of mean, variance and angle are optimal at the same time. Accordingly, as described herein, an approach can apply weights to one or more of these variables, for example, as part of a cost function.
As described herein, one goal may be to detect edges (e.g., cracks, faults, etc. based on seismic data). To achieve this goal, peak detection may be used to generate lines or planes (e.g., in a 3D data set). In a 2D data set, a final result may be presented that consists of one-element thick lines extracted from the input data. Additionally, where a requirement sets forth that noise in the original image should not be carried over into output results, a peak detection step can be performed in which the optimized results (e.g., a 2D or 3D output array) are scanned for elements that stand out as local peaks. An algorithm may define a peak element, which typically make up, at least in part, output results.
Conventionally, the Radon transform has proven to be a general and helpful tool for finding lines in images. The Radon transform is based on convolution of an image with a kernel that gives high values at output pixels where a line is present and low values where line tendency is low. Often, data is preprocessed (e.g., with an edge enhancing filter to reduce noise affecting the output) prior to implementation of the Radon transform for line detection.
In image processing convolution is a transform that uses regional information when processing pixels. The region used as input for the transform, the convolution mask or kernel mask (or window), is typically defined in relation to a pixel being processed, which may be called the “output” pixel. A transformed value is generated through application of the kernel function or convolution function on the convolution mask.
As described herein, an implementation may consider a line l along the angle α, intersecting an output pixel x, where it is possible to parameterize the pixels p on the line in the Cartesian coordinate system (x1, x2) (e.g., or (x, y)) using a parameter t (e.g., p(x, t)=x+∇l*t) and restrict the pixels p to the set of lines of length l within the image (Img) by L(Img,l,x)={p(x,t): |t|≦[0.5*l],x εImg,p εl∩Img}, which corresponds to a convolution mask shaped as a line with output pixel x in the center.
In various implementations described herein, the Radon transform includes three kernel functions Rμ, Rvar and a cost function over the convolution mask L. These functions correspond to the mean, variance and the cost function rω, an example of which is described further below. In such an approach, the mean is linear in the sum of the pixel intensities (or other data) and the number of elements and the variance is linear in the sum of the squares of the pixel intensities (or other data), the number of elements and the mean.
As described herein a sliding window (or mask) algorithm can be used to process an input array (e.g., 2D or 3D data). The following description provides details of an example of such an algorithm. The example implements the Radon transform with respect to a 2D data set, which is referred to as an image where detection of edges or lines is a goal (e.g., as to identify cracks in a 2D seismic data set).
In the example implementation, angles are selected from an angle interval that covers all of a pixel's angle intervals. The discretization of is accomplished through a uniform distribution of n angles from the smallest to the largest angle in the image, θmin and θmax. This particular approach to discretize the image may miss some angles if n is not large enough. To cover the entire image (or selected portion of an image or data set), starting pixels of a line are selected in a manner that considers the boundary of the image, for example, as shown in Table 1.
For a current output pixel, where the angle of the Radon transform is confirmed to be within the pixel's interval (e.g., −90° to +90° or other selected interval), processing can commence. To compute rμ, and rvar for that pixel, the following information is considered: (i) the number of pixels currently in the convolution mask (e.g., based on line length and angle); (ii) the sum of the pixel intensities; and (iii) the sum of the squares of the intensities. Accordingly, for each output pixel along the line, the implementation notes how many pixels are in the convolution mask and changes the sum of the intensities and the sum of the squares of the intensities based on which pixels are new to the mask, and which are no longer included in the mask.
In this example implementation, also computed is the cost function rw. In such a manner, an optimization is included in implementation of the Radon transform, which can optionally avoid a need to perform a global optimization afterwards (e.g., which may be at the discretion of user, etc.). The foregoing approach that includes an optimization in the implementation of the Radon transform can also reduce memory complexity.
For example, to find the angle with the minimum cost for the output pixel, it can be compared to previous evaluations (e.g., previously determined for other angles of a particular output pixel). To avoid overhead, the cost value can be compared to zero plus an input constant C, initially (e.g., which may have a standard value of 0). If the image has large cost values, the optimum may be found by setting C to a high value; otherwise, they would be thresholded away. The cost results can be written to a final output image; noting that with an angle interval that is the same in the entire image, the code can be shortened substantially. As described herein, where such an approach to optimization is ongoing during implementation of the Radon transform, upon completion of the last angle for a pixel, the optimal result may be readily given for that pixel (e.g., the value for the last angle only needs to be compared to a stored best angle value).
The example implementation is outlined below in pseudo-code and also presented in the form of a flow diagram in
Let rμ, rvar, l, θk, θmin,i, θmax,i and Img be defined as the kernel functions of the Radon transform, a line along an angle, a selected angle between the largest and smallest angles, and an input array or image, respectively.
Let L be a convolution mask for a current output pixel and angle (e.g., not truncated to fit the image)
Let n be the number of pixels inside L
Let pi be the current output pixel Let pfront and pback be the pixel in the front and the back of L
Let Pnot in L be the pixel that was pback back in the latest step
Let s be the sum of the pixel values for the pixels in L
Let s2 be the sum of the squared pixel values for the pixels in L
Let rωk,i be the cost value for the angle θk and the output pixel pi
Let rωi, initialized to zero, be the biggest cost value for pi so far
As mentioned, the example implementation includes an optimization within the Radon transform, which is represented in the method 600 of
As shown in
In the decision block 652, a start pixel is selected, as represented by the data block 654. The decision block 656 decides if the angle is within the interval of the selected pixel. If so, the sliding mask transform block 622 computes information for the local optimization block 626, which applies the cost function 624. In operation, information may flow back and forth from the data store 628 to the local optimization block 628, which may depend on specifics of the cost function 624 and approach to local optimization. For example, if a previously computed result is required for local optimization with respect to a current pixel, then the local optimization block 626 may access the data store 624 (e.g., memory) to acquire one or more values (e.g., a current “best” value for an angle associated with a pixel). Once the local optimization block 626 outputs information for the selected pixel, the method 600 continues at the decision block 658, which decides to select a new pixel along the line (e.g., the line represented by the data block 642). Where pixels are available, the method 600 continues to the decision block 656, which has been described above. A block 657 addresses instances when the angle is not within the interval of the selected pixel (e.g., angle reconciliation, which may optionally skip to another point in the method 600).
In various trials, the aforementioned sliding window Radon transform algorithm was ported to a C program, which was configured to read an edge enhanced input array, perform the Radon transform with optimization and output results to a data file of a data storage device. In this particular trial implementation, memory was allocated and freed manually; noting that the syntax of matrices and arrays differs somewhat from the C# syntax.
As described herein, to pick the best angle a selection procedure must be performed. Various different optimization techniques may be used, such as, a local cost function to pick the optimal value for each pixel, a global weakest link technique or a global full line technique.
For global optimization, mean and variance values must be available. Such an approach was implemented using an additional dimension (e.g., a third dimension for a 2D data set) where the additional dimension was n wide, where n corresponds to the number of angles. In a particular trial, n was 37.
In a trial, a local optimization technique was implemented that ignored the angle continuity. The trial used a cost function rω=w*rvar−rμ, where rω was the cost, w was the weight, rvar was the variance and rμ was the mean). The foregoing example cost function is to be minimized to achieve a small variance and a large mean. A threshold effect approach, as described above, resulted in good noise reducing effects in trial images, although care should be considered in instances where a threshold may act to remove lines in images that have higher variance. In various trials, the constant C was introduced, as already described, to counter such unwanted effects.
As described herein, labeling of pixels may occur based at least in part on mean, variance, mean and variance and optionally with respect to local, global or local and global factors. For example, while line detection can benefit from examination of small variance and large mean, identification of noise such as speckle noise may benefit from examination of large variance and large mean. Accordingly, an approach may include criteria for line detection and criteria for identification of speckle noise. Such criteria may be optionally implemented in parallel or in series (e.g., if not the “best”, is it noise?). As described herein, such criteria may optionally be implemented in the form of one or more cost functions. While speckle noise is mentioned, criteria that act to identify other types of artifacts or features may be included. Accordingly, the Radon transform approach described herein can be adapted for purposes other than line detection. Of course, line detection and one or more other purposes may be accomplished in an essentially simultaneous manner using any of a variety of statistical or other metrics that can be readily computing using the sliding mask (or window) approach described herein.
In the trial implementations, mask coordinates were calculated according to angle. The method then iterated over the mask and calculated the cost for each pixel using the aforementioned cost function: rω=w*rvar−rμ. The method picked the largest cost value on the mask as the weakest link and the angle with the lowest weakest link value was chosen the optimum. The mean and variance values according for the selected angle (lowest weakest link value) was stored in an output matrix.
For a full-line cost function implementation, the mask coordinates were calculated in the same way as in the weakest link approach. However, instead of using the worst value, all cost values from the cost function (rω=w*rvar−rμ) were summed over the mask. The angle with the lowest sum was considered optimal and the mean and variance values that corresponded to the angle with the lowest sum were selected and stored.
An example of a local optimization was implemented that did not require a mask to be calculated. The approach used the aforementioned cost function (rω=w*rvar−rμ) to select the optimal value. The least cost was used without any other alterations of the cost function. In this example of a local optimization, no additional dimension for the mean and variance matrices was needed because the cost could be calculated during the sliding window step, which saved on memory requirements.
As described herein, another approach implemented peak detection algorithms (PDAs). All of the PDAs implemented have a parameter R which denotes the radius of the operation. The radius is interpreted as the number of elements in each direction which are included in the operation. In the examples implemented, the center element was always included and the default radius was set to 3.
A parameter “PixelStep” was introduced to determine how many pixels to step between each peak detection operation. A parameter “SampleStep” was also introduced for how many pixels to step between each sample within the set of data in the current peak detection operation. With a higher PixelStep there will be fewer operations and with a higher SampleStep every operation will be smaller. Both parameters had a default value set to 1.
For a maximum search along a line using the PDAs, one horizontal and one vertical, compares each element in the matrix to every neighbor along a centered line extending R steps in both directions. Accordingly, if the element value is greater than all other along the line, it is considered a peak and, for example, it is copied to a results matrix. This particular approach can be described by the following pseudo-code:
Another approach relies on least-squares fitting to a quadratic curve. In this approach, for every element in a matrix the algorithm uses a least-squares technique to fit a quadratic function of the form f(t)=b0+b1t+b2t2 to the data values along a line (e.g., horizontally or vertically). The line extends a set number of elements away from the current pixel. If the coefficient b2 is negative, the function f has a maximum at tmax=−(0.5)(b1/b2), which is derived by setting the derivative to zero. If this point lies within the current set of data values, the point is classified as a local peak and the data at this point is copied to a results matrix. In general, the algorithm does not consider extreme points on the boundary of the current set because such effects are expected to be handled in passes with origin elements closer to that point. The following pseudo-code describes such a process:
Another approach uses a least-squares fitting technique in a tilted domain. This algorithm can be implemented generally according to the foregoing algorithm (see quadratic curve fitting) with the addition that it selects data values from a domain line that is tilted at an angle, which may be an arbitrary angle. Such an approach can be useful in cases where the line tendencies in an original image change in different areas such that no constant direction would suit the entire image (or portion thereof that is of interest). In the titled domain approach, an expected line direction can be provided per-pixel where the algorithm scans perpendicularly to that direction for each processed pixel. As this approach requires trigonometric computations to resolve the domain at each pixel, it may not be optimal for use with images with slanted but constant line directions. In such instances, a pre-rotation of the image that aligns the expected line direction with the horizontal or vertical direction may prove to be more efficient.
A particular implementation used a least-squares fitting technique as part of a peak detection algorithm (PDA). As mentioned, some PDAs may use least-squares fitting of data to a quadratic function to analyze the data for peaks. In a particular example, such an approach was implemented where fitting was performed by constructing a matrix X which contains a row vector with the values (1, t, t2) for each coordinate of the data. In such an approach, if b is a column vector with the coefficients bn and y is a column vector with all the data values, then Xb=y is an overdetermined system for the vector b. By QR-decomposition of X, Rb=QTy, which is a well determined system of equations that minimizes the two-norm of Xb−y. Since R is an upper-triangular matrix b can be computed directly without elimination. The QR-decomposition of X can be performed, for example, using Gram-Schmidt-orthogonalization. Note that such an implementation solves the transpose of the problem in order to use row-vector operations to improve cache efficiency. Pseudo-code describing the foregoing example follows:
Ingoing system matrix X, coefficient vector b and value vector y
Let Ny be the number of elements in y
Let Nb be the number of elements in b
Define matrix Q with size Ny×Ny
Define matrix R with size Nb×Nb
{Compute QR-decomposition using Gram-Schmidt-orthogonalization}
{Solve overdetermined system using forward substitution}
Trial results were generated for implementation of the Radon transform using a sliding window algorithm. One trial implementation used the same angle span over the whole input image data set, and thus did not need to check for the angle span for each pixel. Another trial implementation used different angle spans for each pixel, which required more resources.
In trial implementations, a script looped the Radon transform over different line lengths and saved the times in a file. The line interval 10-100 was chosen since it covered over three octaves. A step length of 5 pixels gave the interval [10, 15, 20, . . . , 90, 95, 100]. The script ran over 50 intervals. The minimum values for each length were used for comparison. In trials, the fastest values occurred when the runs had the least interference from other running applications on the computing device. The times and the line length were used to create the plot of trial results 800 of
Another approach involved implementation of the Radon transform where only lines along the angle interval specified by each pixel were selected. Trials demonstrated a slower speed than the other approach that depended highly on the distribution of the angle intervals.
In various trials, with respect to optimizations, the weakest link and full line methods executed about 10 times slower than the local method. The weakest link approach made calculations that were superfluous. With an appropriately selected weight, noise was reduced; noting that all methods provided qualitatively good results. For the global techniques, ends of some lines were shortened without any noticeable insufficiencies.
In trial implementations, the local method used simple binary operations, which allowed for more rapid computations. Where a suitable weight value was used, results demonstrated marked lines and low noise. In a trial implementation, global methods were removed and the associated 3D matrix was changed to a 2D matrix; best values were selected during running of the algorithm. From this approach a noticeable increase in speed (decreased computation time) was measured; it also saved significantly on memory.
In various trials, it was noted that the weight parameter had a large effect on the output data. As noted, when the global optimization methods were removed the memory consumption became smaller. For the weighted approach described above (i.e., weight “w”), noise was filtered by a selecting a large value. However, large values for w result in some lines becoming less defined in noisy areas. Accordingly, the weight w should be selected for suitable noise reduction while still producing well-marked lines. Trials implemented values for w between 1 and 10. The standard value was chosen to be 5, but this value could be modified as appropriate (e.g., tuned). The effects of the weight in the mean value images are presented in Table 2.
Various trials were run that were based on applying the horizontal and vertical search operations and the horizontal and vertical least-squares operations to the original image on the left with the detection radius R=10, PixelStep=1 and SampleStep=1.
In a comparison of horizontal versus vertical sample domains, the horizontal search and least-squares algorithms produce discernible lines from the input data while the corresponding vertical algorithms produced more noise. This is in part due to the fact that the line tendencies in the original data were towards vertical and the best results were obtained when using a sample domain perpendicular to the tendencies. If the sample domain is tangential to the lines it is likely to cancel several line elements in one operation step, for example, only selecting one of them as a peak. A more optimal situation is to have only one line element in the sample domain so that it can be correctly selected as the peak element.
In various trial implementations, a criterion was used that stated that detected lines should be no wider than one element (e.g., one pixel). With this criterion, trial results demonstrated that the search-based algorithms comply while the least-squares-based algorithms do not. As described herein, such a criterion was used for a particular purpose and may be optional or one or more other criteria may be used (e.g., where width depends on another factor, depending of whether lines, noise or other features are of interest, etc.).
In various trial implementations, a horizontal least-squares algorithm produced results where, in the gaps between lines, there was no significant noise. This result is at least in part due to the fact that single noise elements do not contribute much to the quadratic function. While some noise exists around detected lines, this is introduced within the algorithm rather than carried over from the original data. In another approach, a horizontal search algorithm did not cancel ingoing noise and tended to pick up short line segments from noise in areas far from real lines. This is at least in part because with no real line element within the sample domain the approach picks up noise elements as peaks instead. It also tends to break off real lines when there are noise elements close by with greater value than the line element. For the data sets used in the trials, the vertical algorithms were more difficult to evaluate because of significant noise.
As described herein, an approach may implement a two-pass peak detection process. For example, to fulfill both the requirement that lines should be one element wide and that original noise should be suppressed a two-pass operation may be used. The foregoing requirements are met by first running a least-squares-based algorithm, then performing a mean-filter on the result and finally running a search-based algorithm on the filtered result. The least-squares algorithm removes noise and concentrates data around the lines. The mean filter smoothes the noise introduced by the least-squares operation. Finally the search-algorithm narrows the smoothed lines to one pixel in width. Accordingly, by using a combination of a search-based and a least-squares-based PDA noise was reduced and lines were kept sharp for particular sets of input data.
As mentioned, various techniques described herein may be applied or adapted for use on data sets having more than two dimensions. The example of
As described herein, one or more computer-readable media may include computer-executable instructions to instruct a computing system to output information for controlling a process. For example, such instructions may provide for output to sensing process, an injection process, drilling process, an extraction process, etc.
As described herein, components may be distributed, such as in the network system 1010. The network system 1010 includes components 1022-1, 1022-2, 1022-3, . . . 1022-N. For example, the components 1022-1 may include the processor(s) 1002 while the component(s) 1022-3 may include memory accessible by the processor(s) 1002. Further, the component(s) 1002-2 may include an I/O device for display and optionally interaction with a method. The network may be or include the Internet, an intranet, a cellular network, a satellite network, etc.
Although various methods, devices, systems, etc., have been described in language specific to structural features and/or methodological acts, it is to be understood that the subject matter defined in the appended claims is not necessarily limited to the specific features or acts described. Rather, the specific features and acts are disclosed as examples of forms of implementing the claimed methods, devices, systems, etc.
This application claims the benefit of U.S. Provisional Application having Ser. No. 61/316,127 entitled “Fault Line Detection,” filed Mar. 22, 2010, which is incorporated by reference herein.
| Number | Date | Country | |
|---|---|---|---|
| 61316127 | Mar 2010 | US |
