

Interactive TutorialsDerivative FiltersDerivative filters provide a quantitative measurement for the rate of change in pixel brightness information present in a digital image. When a derivative filter is applied to a digital image, the resulting information about brightness change rates can be used to enhance contrast, detect edges and boundaries, and to measure feature orientation. The tutorial initializes with a randomly selected specimen image (captured in the microscope) appearing in the lefthand window entitled Specimen Image. Each specimen name includes, in parentheses, an abbreviation designating the contrast mechanism employed in obtaining the image. The following nomenclature is used: (FL), fluorescence; (BF), brightfield; (DF), darkfield; and (POL), polarized light. Visitors will note that specimens captured using the various techniques available in optical microscopy behave differently during image processing in the tutorial. Positioned to the right of the Specimen Image window is the Output Image window that displays the specimen image after a derivative filter has been applied. To operate the tutorial, select an image from the Choose A Specimen pulldown menu, and select a derivative filter from the Sobel Operation pulldown menu. Visitors should explore the effects that the various Sobel operations have on the appearance of the output image. The Sobel derivative filter is based on a convolution operation that can produce a derivative in any of eight directions depending upon the choice of a 3 x 3 kernel mask. These convolutions are very useful for edge enhancement of digital images captured in the microscope. Edges are often one of the most important features in a microscopic structure, and can often be utilized for measurements after appropriate enhancement algorithms have been applied. Two examples of Sobel derivative kernel masks are illustrated below:
Convolution of the specimen image with the former of the two kernel masks listed above amounts to a horizontal derivative filtering operation, while convolution with the latter amounts to a vertical derivative filtering operation. For notational purposes, if B(x, y) is the brightness value of a pixel located at the coordinates (x, y) in an image, then the horizontal and vertical finite partial derivatives of B represent the relative change in brightness in each direction and can be denoted respectively as: The derivative images produced through convolution with Sobel and related kernel masks are grayscale images that encode highfrequency spatial detail in the direction of interest as sharp transitions in brightness between light and dark. In the tutorial, examples of Sobel filters include the Horizontal Edges and Vertical Edges options in the Sobel Operation pulldown menu. Often, microscopists apply derivative filters from a 45degree angle to simulate differential interference contrast (DIC) images. Examples of these filters are listed below and are available in the Sobel Operation pulldown menu. Sobel Derivative Filters for DIC Effects
The Sobel derivatives in two orthogonal directions may be combined as the square root of the sums of their squares to obtain a measure of their magnitude that is independent of orientation. This measure is called the Sobel operator. The Sobel operator provides excellent boundary enhancement, and is one of the most commonly used methods for this purpose. This operation is illustrated in the All Edges option in the Sobel Operation pulldown menu of the tutorial. It is also possible to calculate an angular direction for each pixel occurring in a gradient (or along an edge) with the Sobel operator. This is accomplished by determining the arc tangent of the ratio of the brightness value partial derivatives, as illustrated in the equation below: When angle measure is scaled to the display grayscale range, the resulting image provides an indication of direction for every pixel. Because each pixel is displayed according to its direction alone, the visual impression of this image can be overwhelming, but not very informative. A more satisfactory representation can be obtained when the direction information is scaled according to the corresponding magnitude information to produce an image that shows both the edges and their orientation. In the tutorial, this corresponds to the Edge Direction (Intensity) option of the Sobel Operator pulldown menu. In addition, the HSI color space can be used to obtain a colorcoded representation of the magnitude and direction information. The angular direction information can be represented by the hue component of HSI, and the magnitude information can be represented by the intensity component. This representation corresponds to the Edge Direction (Hue) option of the Sobel Operator pulldown menu in the tutorial. When the hue image is converted to binary by thresholding according to a range of hues (not addressed in the tutorial), a pixel count for each color (direction) can be obtained for the purpose of edge analysis. Contributing Authors Kenneth R. Spring  Scientific Consultant, Lusby, Maryland, 20657. John C. Russ  Materials Science and Engineering Department, North Carolina State University, Raleigh, North Carolina, 27695. Matthew J. ParryHill, John C. Long, Thomas J. Fellers, Laurence D. Zuckerman, and Michael W. Davidson  National High Magnetic Field Laboratory, 1800 East Paul Dirac Dr., The Florida State University, Tallahassee, Florida, 32310. BACK TO DIGITAL IMAGE PROCESSING TUTORIALS Questions or comments? Send us an email.© 19982021 by Michael W. Davidson and The Florida State University. All Rights Reserved. No images, graphics, scripts, or applets may be reproduced or used in any manner without permission from the copyright holders. Use of this website means you agree to all of the Legal Terms and Conditions set forth by the owners.
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