- Optimage 3 3 1 – Image Optimization Using Advanced Algorithms For Beginners
- Optimage 3 3 1 – Image Optimization Using Advanced Algorithms Using
. A genetic algorithm (or GA) is a search technique used in computing to find true or approximate solutions to optimization and search problems. (GA)s are categorized as global search heuristics. (GA)s are a particular class of evolutionary algorithms that use techniques inspired by evolutionary biology such as inheritance. Optimization algorithms widely used today. (a) Deterministic Algorithms. They use specific rules for moving one solution to other. These algorithms are in use to suite some times and have been successfully applied for many engineering design problems. 3 (b) Stochastic Algorithms.
- 2 Cooley65 Cooley, J. 'An Algorithm for the Machine Calculation of Complex Fourier Series.' 19, 297-301, 1965. Mitchell, Marwan Y. Ansari and Evan Hart 'Advanced Image Processing with DirectX® 9 Pixel Shaders' - From ShaderX2 - Shader Programming Tips and Tricks with DirectX 9, 2003.
- Simulation optimization problems into those with (1) a finite number of solutions; (2) continuous decision variables; and (3) discrete variables that are integer-ordered. The paper describes procedures for each of these classes. Perhaps the most recent survey, 5, classifies simulation optimization algorithms.
Updated DirectX* 11 Sample Available!!!
This code sample has been updated for DirectX 11* and is available at Fast Fourier Transform for Image Processing in DirectX* 11. If you need a DirectX 10 version of this code sample, please download the sample on this page.
Download DirectX 10 Sample Code
FFTDX10.zip [ZIP | 6.36MB]
Introduction
The Discrete Fourier Transform (DFT) is a specific form of Fourier analysis to convert one function (often in the time or spatial domain) into another (frequency domain). DFT is widely employed in signal processing and related fields to analyze frequencies contained in a sample signal, to solve partial differential equations, and to preform other operations such as convolutions. Fast Fourier Transform (FFT) is an efficient implementation of DFT and is used, apart from other fields, in digital image processing. Fast Fourier Transform is applied to convert an image from the image (spatial) domain to the frequency domain. Applying filters to images in frequency domain is computationally faster than to do the same in the image domain.
This document will not go into the theory of FFT but will address the implementation of the algorithm in converting a 2D image to the frequency domain and back to the image domain (Inverse FFT). Once the image is transformed into the frequency domain, filters can be applied to the image by convolutions. FFT turns the complicated convolution operations into simple multiplications. An inverse transform is then applied in the frequency domain to get the result of the convolution.
The sample application is developed in DirectX 10 and demonstrates the forward and inverse transforms of the image to the frequency domain and back. Applying the filters is pretty straight forward once the transform takes place, and hence is not discussed here.
The application uses ping-pong textures, a common technique used in many GPGPU (General-purpose computing on graphics processing units) applications. Ping-pong textures involve a pair of texture surfaces that a shader uses both as input as well as output data. The shader program uses one texture as input to do some computation and writes the output to the second texture. Subsequent iterations will swap the input and output textures (thus the input from previous iteration becomes the output in the current iteration and so on).
The next sections give a brief description of FFT and its advantages and application in image processing, implementation of the algorithm in DirectX10 and conclusions.
Fast Fourier Transform
Fourier Transform decomposes an image into its real and imaginary components which is a representation of the image in the frequency domain. If the input signal is an image then the number of frequencies in the frequency domain is equal to the number of pixels in the image or spatial domain. The inverse transform re-transforms the frequencies to the image in the spatial domain. Alfred 3 powerpack 3 7 2 (951). The FFT and its inverse of a 2D image are given by the following equations:
Where f(m,n) is the pixel at coordinates (m, n), F(x,y) is the value of the image in the frequency domain corresponding to the coordinates x and y, M and N are the dimensions of the image.
As can be seen from the equation, a naïve implementation of this algorithm is very expensive. But the beauty of FFT is that it is separable, namely, the 2D transform can be done as 2 1D transforms as shown below (shown only in the horizontal direction) - one in the horizontal direction followed by the other in the vertical direction on the result of the horizontal transform. The end result is equivalent to performing the 2D transform in the frequency space.
The FFT that's implemented in the application here requires that the dimensions of the image are a power of two. Another interesting property of the FFT is that the transform of N points can be rewritten as the sum of two N/2 transforms (divide and conquer). This is important because some of the computations can be reused thus eliminating expensive operations.
The output of the Fourier Transform is a complex number and has a much greater range than the image in the spatial domain. Therefore to accurately represent these values, they are stored as floats. Furthermore, the dynamic range of the Fourier coefficients are too large to be displayed on the screen, and hence, these values are scaled (usually by dividing by Height*Width of the image) to bring them within the range of values that can be displayed [3].
The next section will go into the implementation details of the FFT algorithm and its inverse in a GPGPU application. Iflicks 2 v2 6 2.
Implementation details
Since the FFT is a divide and conquer algorithm, the various steps can be implemented in multiple passes in a shader by rendering the result of each pass to a texture. These steps are called butterflies, because of the shape of the>
The shader looks-up the butterfly texture for indices of the two pixels to be combined and the weight and computes the result by performing the complex math a+wb or a-wb. To make matters simple, the sign is encoded as part of the weight and hence only the addition is performed in the shader program all the times. The result is returned with the real value in the first three components and the imaginary value in the fourth component.
To transform the image from frequency domain to the spatial domain the exact same operations are performed but on the frequencies. Since the frequencies are not in the range to be displayed on the screen they are scaled by a factor of 1/(Width*Height) as shown below.
Where ScaleFactor is set by the application to 1/(Width*Height). Finally the result is displayed in the normal mode by concentrating the lowest frequencies towards the center. The below figures show a gray scale image and the result after applying FFT on it.
Figure 1: The original image displayed in gray scale
Figure 2: The normal mode of the image after applying the FFT
Future work and Conclusions
This DirectX 10 application not only implements the FFT and its inverse but is also intended to serve as a framework for implementing image processing algorithms. A CTexture class is implemented for handling 1D and 2D textures operations. The CTexture class supports various texture formats and can be easily extended to support 3D textures. Future work will focus on making the framework a plug-in architecture allowing developers to write image processing filters and 'plug-in' their algorithms into the framework.
This document detailed the implementation of FFT and its inverse for transforming a 2D image from the spatial domain to the frequency domain and back. The advantage of representing an image in the frequency space is that performing some operations on the frequencies is much more efficient than doing the same in the image space. Many of the convolutions are just multiplications in the frequency domain (the computational cost in the image space is O(N2) vs. O(Nlog(N)) in the Fourier space for N points). This enables efficient implementations of very large convolutions in image processing and other algorithms/operations in many fields.
References and Related Articles
[1] http://en.wikipedia.org/wiki/Fast_Fourier_transform
[2] [Cooley65] Cooley, J. W. and Tukey, O. W. 'An Algorithm for the Machine Calculation of Complex Fourier Series.' Math. Comput. 19, 297-301, 1965.
[3] Jason L. Mitchell, Marwan Y. Ansari and Evan Hart 'Advanced Image Processing with DirectX® 9 Pixel Shaders' - From ShaderX2 - Shader Programming Tips and Tricks with DirectX 9, 2003
[4] Thilaka Sumanaweera, Donald Liu 'Medical Image Reconstruction with the FFT' - GPU Gems 2: Programming Techniques for High-Performance Graphics and General-Purpose Computation
[5] http://en.wikipedia.org/wiki/Butterfly_diagram)
About the Author
Raghu Muthyalampalli is a Sr. Software Engineer working on client enabling in the Software Solutions Group that enables client platforms through software optimizations. Prior to working at Intel he had a short stint at eBay working as a Software Engineer. He earned his Master's degree in Computer Science from the University of Illinois at Chicago. His email is raghu.r.muthyalampalli@intel.com.
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Installation
Optimage 3 3 1 – Image Optimization Using Advanced Algorithms For Beginners
The following are the steps to install the OptiMole plugin
- In your WordPress Administration Panels, click on Add New option under Plugins from the menu.
Click on upload at the top. - Browse the location and select the OptiMole Plugin and click install now.
- Go to Media -> OptiMole and follow in the instructions on how to enable the service.
FAQ
How many images I can optimize with each plan?
The number of images that you can optimize or store is unlimited, we care only about the number of visits you have per month.
What happens if I exceed plan limits?
Once you exceed these, we will contact you and kindly ask to upgrade to the plan that fits you best.
What Content Delivery Network (CDN) do you use?
For both FREE and Paid plans we use AWS CloudFront CDN with more than 200 locations around the globe.
I'm already using a CDN, can I use that instead of yours ?
Short answer, YES. We can help you integrate your default CDN but it will require some additional work from our side and is available to Enterprise plans.
I'm already using an image optimization plugin, why should I switch to OptiMole?
You don't need to change your existing optimization plugin, image optimization is just a small part of what we do, if you are happy with ShortPixel for e.g, feel free to continue to use it, OptiMole would then take care only of serving your image at the RIGHT size, advanced cropping and smart lazy-loading.
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Yes, we do. By default, Optimole handle images delivered from your ajax content from admin-ajax.php ( not logged in users ) as well as WordPress REST API routes.
Can I remove the blurry placeholder from the lazyload effct ?
Yes, you can. We have bundled this tweak into a plugin you can install while you have Optimole active. The blurry placeholder will be removed and the images will be still lazy-loaded. You can find the plugin tweak here -> http://bit.ly/optml-rm-lqip
Will the original images be deleted?
Absolutely No. We use your original images as sources when deliver the optimized images.
What is the difference between the Auto, High, Medium, Low compression levels?
A higher compression might result in a small loss of image quality. Selecting the auto level will let Optimole choose the minimum size with no loss in the quality of your picture.
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For now we support jpg, png and svg format.
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Yes. You need to add define('OPTML_DISABLE_PNG_LAZYLOAD',true);
to your wp-config.php
file.
Can I disable optimization for a certain image ?
Yes, you can follow this code snippet and replace the sample image with the one you need: