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Open access peer-reviewed chapter

By Sébastien Leclaire, Maud El-Hachem and Marcelo Reggio

Submitted: December 20th 2011Reviewed: March 14th 2012Published: September 26th 2012

DOI: 10.5772/46455

Downloaded: 3143

The design of discrete operators or filters for the calculation of gradients is a classical topic in scientific computing. Typical applications are gradient reconstruction in computational fluid dynamics, edge detection in computer graphics and biomedical imaging, and phase boundary definition in the modeling of multiphase flows.

Edge detection, which is widely performed in image analysis, is an operation that requires gradient calculation. Commonly used edge detection methods are *Canny, Prewitt, Roberts* and *Sobel*, which can be found in MATLAB’s platform. In this field, edge detection techniques rely on the application of convolution masks to provide a filter or kernel to calculate gradients in two perpendicular directions. A threshold is then applied to obtain an edge shape.

For multiphase flows, an edge or contour corresponds to the interface between the fluids. In this respect, traditional gradient calculation methods based on 1D edge detection are not necessarily suited for the underlying physics, because there is no direction in which the gradients of the phase contours tend to evolve over time. As a result, definition of the geometric progress of the interface requires many gradient estimation computations, as is the case in moving and deforming bubbles or droplets, for example. Although it can still be a viable tool, it is clear that the 1D-based method is becoming less useful for simulating these phenomena, which are not, in general, biased toward any particular direction.

To address this issue, we present an efficient computational method for obtaining discrete isotropic gradients that was previously applied to simulate two-phase flows within the lattice Boltzman framework [1, 2]. This "omnidirectional" approach makes it possible to improve the limitations inherent in handling high density ratios between the phases and to significantly reduce spurious currents at the interface. The method is based on a filter which is generally not split along any direction, and there is no need to make the assumption of a continuous filter to reach isotropy, as done by [3]. We also believe that optimal or maximal isotropy can only be reached with a discrete filter when the error terms of Taylor’s series expansion are isotropic, as explained in detail by [1, 2].

Below, we describe isotropic and anisotropic discretizations that will and will not conserve the isotropic property of the differentiated function respectively. This is followed by a description of how convolution can be used to reduce computer time in the gradient calculation. We then present details of the MATLAB implementation of these ideas, along with speedup comparisons of convolution performed on a single core of an Intel® Core i7-970 processor and on an Nvidia® GeForce GTX 580 GPU using the Jacket plugin for MATLAB developed by AccelerEyes®. The GPU with the Jacket plugin for MATLAB speeds up gradient computation by a factor of up to

Let us define the real scalar function

with

This finite difference discretization is very similar to Prewitt’s operator/kernel [4], which is used in image processing. Note that

As in Ref. [2] and without loss of generality, the function

with

To calculate the gradient in the

A similar expression is found for the

In the gradients of Eqs. (5) and (6), the leading

Using the following transformation from a Cartesian to a polar partial derivative operator:

And, by supposing that

Taking into consideration the previous anisotropy problem, it is possible to change the weights of the grid points when computing the gradients to make them isotropic, up to the second order in space, by defining gradients in the

With this new discretization, the dominant differential operator of the second order error term takes the form:

If a gradient has a small dependence on direction, this would imply that the dominant error term has only an axial dependence when the function being derived also only depends on the radius. That is, the operator in Eq. (12) applied on

with

which can be rewritten in the same form as Eq. (13). Similarly, the first differential operator of the fourth order error term in Eqs. (10) and (11) takes the form:

and the associated components in polar coordinates, when applied to a rotationally invariant function

which again meets the rotational invariance requirement, and can be rewritten in the same form as given in Eq. (13). The last differential error operator of the fourth order error term in Eqs. (10) and (11) is:

and can be shown to be anisotropic (i.e.

In this work, we only consider the gradient approximation of scalar functions over a square or cubic lattice of unit spacing, so we need to take

Here, we present the mathematical and computational aspects of convolution. As finite difference discretization and edge detection kernels are very similar, let us return to the mathematical foundations of these techniques. We often give examples involving 2D images, but we could give the same examples and talk about 2D discrete functions. We don’t know the exact coding in the MATLAB and Jacket libraries, as they are under license, but we do have a general idea about function algorithms, which is given in the next section.

The convolution product of two functions

If Eq. (20) defines the convolution of two functions in the space domain, it is also equivalent to the point wise product of these functions in the frequency domain. In mathematics, the convolution theorem for Fourier transforms [5, chap. 4] is formalized by the following equation, where

The Fourier transform of a function generates the frequency spectrum of a function. This representation can easily be extended to two dimensions, which is more suitable to image or spatial analysis if the image is a function of two spatial variables. In our case, the output of the Fourier transform will generate a 2D function in the frequency domain from the 2D spatial domain. High frequencies will correspond to information varying rapidly in the original function, while low frequencies correspond to slow variations.

By applying the inverse Fourier Transform to both sides of Eq. (21), we obtain a method for calculating the convolution of two functions:

Therefore, there are two possible ways of convoluting two functions: the Fourier transform pipeline in three steps (Fourier transform, frequency spectrum filtering, and Fourier inverse transform), or the application of a stencil in the spatial domain. Indeed, edge detection kernels acts as high pass filters, accentuating high frequency contributions to the image or to the 2D function in the form of edges and details.

Knowing that the properties of the Fourier transform also work for a sampled signal, we present the definition of the 2D discrete Fourier transform (2D DFT), where M and N represent the number of samples in each dimension, x and y are the discrete spatial variables, and u and v are the transform or frequency variables:

In addition to having the same properties as the continuous Fourier transform, which are outside the scope of this presentation and can be found in [6], there are two more in 2D DFT that are important: periodicity and separability. Since the discrete signal is sampled from the finite length sequences, its frequency spectrum will be periodic. In the spatial domain, this property allows us to slide the filter from one frontier to the opposite one, and then start again at the frontier where we began. Moreover, its Fourier transform is separable. The 2D DFT can be processed in two steps: applying 1D DFT on the lines, and then applying 1D DFT on the resulting columns. In the spatial domain, a 2D filter represented by

As an example, let us take kernelX, the 2D separable kernel in section (4.2):

We can easily deduce the main functionality of the 2D kernel, the gradient component in x that approximates the derivative in x, by calculating the difference between the first and third rows in the neighborhood of a point, instead of the first and third columns, because of the rotation of our axis system. The separated 1D filters give us further information about the kernel function: a smoothing mask is applied on the differentiating mask, in order to reduce noise that could be exaggerated by the first filter.

Now, to explain how to apply a circular convolution mask in the spatial domain, we go back to the definition of convolution for discrete functions, presented in 2D and 3D in Eqs. (25) and (26) respectively, where A is the image and B is the mask. Processing a convolution filter consists of computing the scalar product of the filter weights with the input values within a window of the filter dimensions surrounding each of the output values, after flipping the mask in each dimension, as described in [7, chap. 4].

In circular convolution, the equation is slightly modified to model periodic shifting. Here we consider a square image N by N:

The values on the image we see in the window of the K-by-K filter are distributed on a periodic surface. The 2D filter is rolled up in the opposite direction and will turn in a clockwise direction. For each output value, the stencil is lined up on the input values, the scalar product is applied, and then the stencil rotates to shift to the next position to compute. Therefore, in the spatial domain, an N-by-N image convoluted with a K-by-K filter requires

In the frequency domain, the 2D discrete Fourier transform and its inverse are computed using the divide-and-conquer algorithm of the Fast Fourier Transform (FFT and IFFT), which reduces the complexity from

In this section, we present a systematic method that can be applied in MATLAB to calculate the gradient of scalar functions on a square lattice, or in 2D, which is simply the gradient of the images. The 3D case is a straightforward extension of the 2D case. First, let us define a function with which it is possible to test the gradient discretization previously defined in section (2.2):

For the purpose of this test and for simplicity, we consider only a square image

Figures 1(a) and 1(b) show the surface shape and contour of this function.

When the gradient of a function needs to be computed, it may be tempting to apply Eqs. (10) and (11) directly. This is quite a simple approach, and in MATLAB can be applied as follows:

However, this is rather a naive and slow implementation for MATLAB. It is far more efficient to evaluate the gradient by using a convolution product, as presented in section (4.2).

Instead of calculating the gradient as previously shown, the use of a convolution product between the functions to differentiate them, and a kernel representing the weights and stencils of the finite difference approximation is more beneficial. For the isotropic discretization of Eqs. (10) and (11), this computation can be performed in MATLAB as follows:

A useful list of 2D and 3D kernels for calculating isotropic gradients is available in sections (6.1) and (6.2).

Sometimes the kernel is separable, which means that, instead of applying an nD convolution, we can apply an 1D convolution

Note that in the work of [1], one of the most important 3D isotropic kernels, which is the one that uses the nearest neighbors only, was not presented. In this work, two variants of this kernel were obtained, one using only 10 nearest neighbors and the other using only 18 nearest neighbors, and their MATLAB form are given in section (6.2). Moreover, the 3D kernel using 18 nearest neighbors has the advantage of being separable, which means that we should expect to be able to rapidly compute a 3D gradient of low isotropic order using the knowledge presented in this chapter. Also note that the higher isotropic order (>2nd) kernels given in sections (6.1) and (6.2) are not separable.

In this work, the order accuracy is defined by

The main point about the kernel that we make in this study is that, as its size increases, it becomes possible to set isotropic the higher order error term in the Taylor series expansion at fixed space second order accuracy. It is therefore important to be careful not to confuse space order accuracy and isotropic order accuracy. We believe that, based on this result, future research could provide other kernels (perhaps of similar size) for which the leading space order accuracy would be the same as the leading isotropic order accuracy. For some applications, finding such kernels could be a significant step forward. In fact, achieving leading higher space order accuracy with equal leading isotropic order accuracy might have greater impact than achieving leading low space order with very high isotropic order accuracy, as is currently the case. However, higher space order gradient discretizations may suffer from another non physical numerical artifacts, known as spurious oscillations.

As previously indicated, computation of the gradient by applying convolution is faster than using a simpler, more straightforward method, but a naive one. We present some numerical results in this section that will show that this is indeed true. We also show that using a GPU instead of a CPU significantly reduces computation time.

First, all performance testing consists in evaluating gradients of random 2D and 3D double precision images. Note that we suppose a periodic padding for these images. All these image gradients are computed using MATLAB with the -singleCompThread startup option. This is done for benchmarking purposes, because the reference case should be computed using a sequential algorithm, that is, with a single core only. The CPU used in this work is an Intel® Core i7-970 processor, while the GPU is an Nvidia® GeForce GTX 580. All computations on the GPU are performed in MATLAB via the Jacket plugin developed by AccelerEyes®. The version of MATLAB is R2010b, and the Jacket version is 2.0 (build a15607c).

The timing method on the CPU is the usual MATLAB tic; m-code; toc; procedure. However, this method is not suited for timing m-code on the GPU. For this, we refer the reader to the method proposed by [8]. In Figures 2-5, the time taken for one simulation "dot" or result "dot" is the average of a hundred simulations.

To test performance, five different algorithms are considered for computing the gradient:

MATLAB singlethread naive (section 4.1)

MATLAB singlethread convolution (section 4.2) [REFERENCE CASE]

MATLAB Jacket naive

MATLAB Jacket convolution

MATLAB Jacket GFOR + convolution

Note that all results differ with respect to machine accuracy in double precision, and that the padding of the images is computed on the fly to save computer memory. This is because padding is very cheap in terms of computing cost, when compared to the cost of evaluating the gradient.

Case (4), MATLAB Jacket convolution, is the GPU equivalent of reference case (2) with a CPU. Case (3) is the GPU equivalent of case (1) with a CPU. The last case (5), MATLAB Jacket GFOR + convolution, is a special case that is not available on the CPU. To explain this, let us suppose that the user wishes to evaluate the gradient of

In all the figures showing performance behavior, the y-axis is in log scale. Figures 2(a) and 2(b) show the performance speedup with the 2D 2nd and 14th order isotropic gradients as a function of image size. For large images, speedups of

Figures 3(a) and 3(b) show the same situation, but with the 3D 2nd and 8th order isotropic gradients. For large images, a speedup of

Figures 4(a) and 4(b) show the speedup with the 2D and 3D isotropic gradient as function of the isotropy order at a fixed image size. In 2D, images are 992x992, and in 3D they are 110x110x110. For high isotropy, speedups of

Figure 5 shows the speedup with the 3D 8th order isotropic gradient as a function of

For both 2D and 3D cases, as the isotropy order or the number of images to evaluate simultaneously increases, the speedup that can be achieved using the GPU also increases. This is to be expected, since the computational complexity increases. Nevertheless, the chances of obtaining a speedup of

We must remember that the speedups were computed with the reference case, which uses convolution. It is important to note that a speedup of

The lattice Boltzmann method is a computational approach that is mainly used for fluid flow simulation with its roots in the field of cellular automata [9]. This method is particularly useful for solving complex flow systems, such as multiphase flows, in porous media where classical approaches based on the Navier-Stokes equations, like finite volumes or finite elements, encounter some difficulties. The method we use is based on the original Ph.D. thesis of [10]. Since then, there have been several improvements. However, these enhancements are outside the scope of this chapter, and will not be described.

The isotropic gradients we present here are useful for simulating immiscible multiphase flows, where the orientation of the various fluid interfaces has to be computed very frequently. The gradients of the density of each fluid color (phase) define the normal orientation of the interface, and special operators are used to keep the different interfaces between the fluids defined. Moreover, the norm of these gradients serves as a means to introduce a certain amount of surface tension between the fluids.

Suppose we wish to simulate the behavior of three fluids. At a certain point in the algorithm, three image gradients need to be computed, corresponding to the interface normal of each fluid density. Here is where the methods presented in this chapter become useful, a situation described in section (4.5) with

In multiphase flow simulation, calculation of a high order isotropic gradient is the most expensive part of the method, and the use of Jacket, has enabled us to reduce the computational cost by an astonishing amount. This type of calculation would not have been possible, in a reasonable time, by applying plain MATLAB.

We end this section with a simulation example that shows the spinodal decomposition of a three phase flow. This flow consists of an initial random mixture, where each phase self-agglomerates until a steady state is achieved. Figure 6 shows the spinodal decomposition at various times (in lattice Boltzmann units). Note that this simulation is given for illustration purposes only, and that spinodal decomposition has been quantitatively studied by [11].

In this section, we give several convolution kernels in the form of MATLAB m-code, which is very useful for calculating 2D/3D isotropic gradients on a square or cubic lattice. Most of the weights were taken from Ref. [1].

In this work, a detailed description of isotropic gradient discretizations and convolution products has been presented. These isotropic gradients are useful, and superior to anisotropic discretizations. This is especially true in the field of flow simulation, when the lattice Boltzmann method is used. However, high order isotropic gradients are computationally expensive. To address this issue, we combined the convolution product with the Jacket plugin in MATLAB and GPU hardware, which enabled us to achieve high computational speedups (up to

We extend our special thanks to Pavan Yalamanchili from AccelerEyes for his quick response to our queries and his generous support. We applied the sequence-determines-credit (SDC) approach to our listing of authors [12]. This work was supported by a grant from the NSERC (Natural Sciences and Engineering Research Council of Canada).

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