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Posted by
Jeff
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Dec 30, 2009
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Differential Equations,
Linear Systems
Often during a course you will need to be able to solve a system equation for its roots. These roots can be complex, distinct, or repeated. These problems usually arise when working with linear systems or differential equations. A system equation is formatted as follows:
System Equation: 
For example purposes, I will solve a system equation with complex roots. A system equation with complex roots as a function of 
will appear in the following format (if it does not, you need to manipulate your equation to be in the form):
Roots: 
So we have 
which also equals 
so your first step is to look at your equation and determine your roots, then write out your 
equation with constants.
Example 
with initial conditions 
and 
so now we can write our equation as follows:
In order to solve for 
and 
we need to use our initial conditions. To evaluate the first derivative initial condition, we must first take the derivative of our 
that we just found.
evaluating this equation with t = 0 and the response equal to -4v, we get this: 
evaluating our equation with t = 0 and the response equal to 3v, we calculate 
Using these two equations, we calculate our constants:
and 
Fill these into our equation to determine the final result.
Now you know how to solve this common differential equations and linear systems problem, determine characteristic roots and modes, and write system equations. 
Related Blogs
Tags: complex roots, Differential Equations, linear system equation, Linear Systems, solve for complex roots, system equation, system equations
Posted by
Jeff
on
Dec 7, 2009
in
Biomedical Engineering
This post is continuing from Biomedical Image Processing – III. Enjoy
Basic Image Modification
While image processing operations can be performed in hardware (e.g., in cases where speed is paramount), many image processing operations are performed in software. This section discusses some of the basic processing operations performed on different types of images, including image arithmetic, point operations, and geometric operations. Many of these operations are driven by the need to isolate regions of an image or to improve the appearance of an image.
Like more advanced image modification techniques, these basic operations are rarely used alone. Rather, the individual operations are cascaded together to produce an overall desired effect.
Image Arithmetic
Image arithmetic represents a broad category of algorithms that use pixel-based processes for combining or extracting image information. For example, if images P1 and P2 contain pixels that are referenced by a row integer i and column integer j, Table 1
illustrates some arithmetic operations that can be performed on these images. In the table, the symbol C represents a constant value by which an image can be adjusted or scaled. Note that both images must be the same size for these operations to apply. These operations are primarily used as sub-steps in more complex image processing operations, rather than being useful on their own.
Table 1. Expressions for some common pixel-based image arithmetic algorithms.
Figure 13. Example of an image that has been brightened by simple pixel-based addition
Boolean operations are pixel-based, logical operations performed on sets of images (see Figure 14). Note that these operations are usually applied to two-color images (the pixel values are either “0” or “1”). A common example of a boolean operation is the use of NAND to identify objects that have moved between images (AND will yield the intersection of two images, denoting the stationary objects).
Figure 14. Boolean operations performed on images
Point Operations
Point operations, like image arithmetic, are pixel-based. However, these operations perform a mapping between image pixel intensity values and their representations on the new image. The most common point operations are thresholding, contrast stretching, and histogram equalization.
Thresholding
When an algorithm thresholds an image, the pixels whose intensities are above and below the threshold are assigned binary values. This technique is often used as an image preprocessing step before the image is passed to other processing algorithms. An example of thresholding is shown in Figure 15. In this figure, a slice of brain tissue (containing nervous cells and glia cells) is thresholded and then connected-component labeled so that the number of cells in the image can be counted.
Figure 15. Illustration of how thresholding and connected-component labeling can be used for cell counting applications [http://www.dai.ed.ac.uk/HIPR2/threshld.htm].
Histogram Equalization
Histogram equalization reassigns the intensity values of pixels in the input image so that the output image contains a uniform distribution of intensities (i.e., a flat histogram). This is illustrated in Figure 16 (compare this figure to the image/histogram in Figure 10).
Figure 16. An example of histogram equalization for magnifying the level of detail inportions of a gray-scale image.
The next post, Biomedical Image Processing – V, will discuss contrast stretching, Image Denoising and Enhancement, and digital filters for noise reduction. References for this post are listed in Biomedical Image Processing – I.
Tags: boolean operations, component labeling, digital filters, grayscale imaging, image modification, image processing, pixels, point operations
Posted by
Jeff
on
Dec 4, 2009
in
Biomedical Engineering
This post is continuing from Biomedical Image Processing – II.
DICOM Image Standard
Medical specialists have been slow to adopt widely accepted standards for image/film storage, display, and transmission. However, one standard has been adopted with reasonable success in the radiology community: DICOM (Digital Imaging and Communication in Medicine) has been progressively developed since 1983 by ACR/NEMA (America College of Radiologists and the National Electrical Manufacturers’ Association). DICOM defines a standard for the exchange and storage of medical images from various imaging modalities, including MRI, CT, and ultrasound. It focuses on the communication interface between a host computer and the scanner, but it also defines file format standards to which a system must adhere in order to be considered DICOM-compliant. Some examples of DICOM images are depicted in Figure 7.
Good starter links:
http://www.scispy.com/Standards/DICOM.html
ftp://ftp.philips.com/pub/ms/dicom/DICOM_Information/CookBook.pdf
Figure 7. Radiological images stored in the DICOM standard image format [Vepro
Computersysteme GmbH, Cardio Viewing Station, Version 4.41].
Image Analysis
Image analysis encapsulates a set of basic image processing operations whose purpose is to query, but not alter, an image. These operations include
- intensity histogram generation,
- information classification, and
- connected components labeling.
Intensity Histogram
An intensity histogram describes the distribution of pixel intensity information within an image. This information is displayed as the number of counts associated with each intensity level (see Figure 9). Histograms can also be ascertained for color images, where distributions are displayed for the individual red/green/blue values.
Figure 9. Depiction of a histogram for an 8-bit gray scale image (256 intensity levels).
Figure 10. Histogram: 256-level gray scale image (X-ray of a human abdomen)
[http://www.medphys.ucl.ac.uk/research/borg/research/NIR_topics/imaging_exp.htm].
Classification
Image classification includes a broad set of algorithms for identifying portions of an image that are related to one another. This is very closely related to segmentation, which physically separates these regions from other regions. For example, in a fluorescence image of cells, a researcher might be interested in an image processing algorithm that localizes cell nuclei (see Figure 11) for counting purposes.
Figure 11. Results of an algorithm that attempts to classify and segment portions of a
fluorescence image that correspond to cell nuclei.
Connected Components Labeling
Connected components labeling is a process whereby an algorithm scans an image and groups sets of pixels based on common features (such as pixel intensity). Once these pixel elements are grouped, they are all assigned the same value and labeled as a region. This process is illustrated in Figure 12. Connected component labeling is different from classification in that the algorithms make no judgments as to which components exhibit similar properties.
Figure 12. Example of component labeling based on nearest neighbor analysis
[http://www.dai.ed.ac.uk/HIPR2/label.htm].
Related Blogs
- Related Blogs on biomedical image processing
- Related Blogs on component labeling
Tags: biomedical image processing, component labeling, dicom, dicom image standard, image analysis, image classification, intensity histogram, m-mode
Posted by
Jeff
on
Dec 2, 2009
in
Biomedical Engineering
Continuing from Biomedical Image Processing – I
Image Properties
Once an image is stored in digital format, it can be described by a number of different parameters. Some of the relevant parameters are briefly discussed here. The traditional convention for an image coordinate system is depicted in Figure 4.
Figure 4. General convention for image coordinate systems.
While biomedical images are generally viewed in 2D, it is sometimes helpful to view gray-scale (monochrome) images in perspective, with the third axis being brightness. This is illustrated with “pseudoimage” data in Figure 5.
Figure 5. Pseudoimage data viewed in 2D and perspective modes. Data values (either a “0″ or a “2″) are placed within a 31-row by 24-column matrix.
Images can be described by a large number of different parameters. Some of these are listed here.
- Pixels (a.k.a. picture elements, pels, image elements) – Individual rectangular
elements that comprise an image. The term voxel describes a pixel’s 3D analog.
- Gray levels – An 8-bit, gray-scale image with 1024 × 1024 pixels requires a
megabyte of storage.
- Color depth – Usually reported as powers of 2, this can range from 2 colors up to 32-
bit color (a.k.a. True Color). Colors are often defined using RGB (red-green-blue) or
HSB (hue-saturation-brightness) combinations (see Figure 6).
- Aspect ratio – The scaling ratio between the x and y axes.
- Contrast – The relationship between the brightest and dimmest pixel in the image.
- Histogram – A binned representation of the gray levels, colors, or brightness levels
in the image.
Other notes:
- Because of the optimization properties of 2D Fourier transforms, it is often
advantageous to select image sizes whose number of rows and columns are both
powers of 2 (e.g., 512 x 512 array with 128 gray levels – comparable to a
monochrome TV image)
- In motion videos, images are displayed at a rate of 30 frames per second
RGB
Primary colors: red, green, blue
Secondary colors: yellow = red + green, cyan = green +
blue, magenta = blue + red.
White = red + green + blue
Black = no light.
[http://www.cecs.csulb.edu/~jewett/colors/rgb.html]
HUE: actual color
Measured in angular degrees around the cone starting and ending at red = 0 or 360 (so yellow = 60, green = 120, etc.).
SATURATION: purity of the color
Measured in percent from the center of the cone (0) to the surface (100). At 0% saturation, hue is meaningless
BRIGHTNESS: measured in percent from black (0) to white (100). At 0% brightness, both hue and saturation are meaningless.
[http://www.cecs.csulb.edu/~jewett/colors/hsb.html]
Figure 6. RGB and HSB color descriptions.
The next Biomedical Image Processing lesson will discuss image analysis, classification, and component labeling. Happy Holidays!
-Jeff
Tags: aspect ratio, image coordinate system, image processing, image properties, monochrome, pixel, voxel
