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.

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Pink Glove Video

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.

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].

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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

• Image creation: generation of digital images and their resulting properties.
• Image modification: operations that can be performed on digital images including denoising, enhancement, and compression.
• Image analysis: techniques for extraction of biomedical data from images, including feature recognition.

The issues and techniques discussed here are general: they apply to images acquired using various methodologies, including X-ray radiography, X-ray computed tomography, MRI, ultrasound, and PET. Note that while image processing can be performed on
analog images (e.g., by way of optical Fourier techniques), this overview concentrates on the processing of digital biomedical images.

Digital Image Creation

Image Generation

Analog images, having continuously varying color or gray-scale representations, can be stored as digital images for use in computer-based systems. The rules that apply to the digitization of one-dimensional (1D) signals (e.g., voltage versus time) also apply to the digitization of spatial images (e.g., gray level versus position): the images must be sampled and quantized with enough fidelity that they truly represent their analog image counterparts. This, of course, depends on the application for which the images will be utilized. Scanning, the initial step in the image digitization process, includes the division of the picture into a number of small regions called picture elements, or pixels (see Figure 1).

The scanning operation represents the image by a grid consisting of m rows × n columns. For example, in the image shown in Figure 1, m = 288 and n = 432, for a total of 124,416 pixels. Each pixel is addressed by its (m,n) coordinate within the matrix. The scanning operation is accompanied by a sampling operation, which transforms the representative brightness of each pixel into an analog voltage level or other measurable signal. In the case of an X-ray radiograph, this operation might be performed by a photomultiplier tube.
A good example of a common line scan sensor is a flatbed scanner, which uses an array of discrete silicon imaging elements, called photosites, to produce voltage output signals that are proportional to the intensity of the incoming light at each linear location. These solid state devices can be shuttered at high speeds (e.g., 1/10,000 of a second), and their precision typically ranges from 256 to 4096 elements [Gonzalez]. Solid state area sensors also exist, containing grid sizes from 32 × 32 pixels up to 1280 × 1024 pixels.
Finally, each pixel in the grid is assigned an integer that corresponds to its brightness level. This quantization operation is similar to the quantization performed on 1D signal data. For example, in an 8-bit gray-scale image, the integer 0 may represent black, the integer 255 may represent white, and the integers in between will represent various gray scales that correspond to image intensity. The general convention in gray-scale images is that larger integers represent brighter pixel values (or pixels exhibiting greater signal
strength).

Figure 1. Eight-bit, gray-scale digital image consisting of individual square pixels, or
picture elements.

As illustrated in Figure 2, two-dimensional (2D) spatial images must obey sampling constraints that are similar to those imposed on 1D, time-based signals. When 1D signals are digitized with an analog-to-digital (A/D) converter, they must be sampled at a frequency greater than or equal to twice the highest frequency component in the signal. Likewise, 2D images must be sampled at a spatial frequency greater than or equal to twice the highest spatial frequency component in the image. Otherwise, these undersampled images will display the type of aliasing depicted in Figure 2. Note that to redisplay a digitally stored image, a D/A conversion is often required (e.g., for a cathode ray tube in a television set).

Figure 2. Simple illustration of aliasing in a digital image line [after Seeram, p. 55].

Some images are not produced on native rectangular grids. B-mode ultrasound images, for example, are often reconstructed from fan-shaped beams or radially-distributed ultrasonic reflectance data (see Figure 3). Production of a 2D image requires that these radial line data be interpolated. These interpolation schemes can affect the quality of the overall images, especially if the individual line scans are separated by large angles.

Figure 3. Examples of rectangular images constructed from non-rectangular imaging
modalities [Fetal Image: http://www.biosound.com/image5.html; OCT/IVUS: CLEO
’96 Proceedings, p. 55]

Why Digitize an Image?

The reasons for digitizing 2D image data are very similar to the reasons for digitizing 1D data, revolving primarily around (1) the ability to transmit image data “noise free” and (2) the potential for processing these images with computational tools. These tools
provide assistance in the following areas:

• Image Enhancement/Restoration: image improvement, possibly through the
  reduction or removal of noise, artifact, or unnecessarily fine detail (e.g.,
  processing (1) degraded images of unrecoverable objects or (2) images from
  experiments that are too expensive to duplicate)
• Image Analysis: extraction of information with/without interpretation
• Pattern Recognition: structures and patterns are “seen” and recognized
• Image Detection: identification of certain shapes, contours, or textures while
  disregarding other image features
• Geometric Transformation: rotation and/or scaling of the image
• Data Compression: Reduction in image size for storage or transmission

Part II of Biomedical Image Processing will discuss image properties and DICOM Image Standards.  This information was compiled by Steve Warren of Kansas State University using the following references:

[1] Seeram, Euclid. Computed Tomography: Physical Principles, Clinical
Applications, and Quality Control, W.B. Saunders, Philadelphia, © 1994, ISBN 0-
7216-6710-4
[2] Shung, K. Kirk, Michael B. Smith, and Benjamin Tsui. Principles of Medical
Imaging, Academic Press, San Diego, © 1992, ISBN 0-12-640970-6
[3] Rosenfeld, Azriel and Avinish C. Kak. Digital Picture Processing, Second Edition,
Volume 1, Academic Press, San Diego, ©1982, ISBN0-12-597301-2.
[4] Rosenfeld, Azriel and Avinish C. Kak. Digital Picture Processing, Second Edition,
Volume 2, Academic Press, San Diego, ©1982, ISBN 0-12-597301-2.
[5] Gonzalez, Rafael C. and Richard E. Woods. Digital Image Processing, Addison-
Wesley, Reading, MA, ©1993, ISBN 0-201-50803-6.
[6] Teuber, Jan. Digital Image Processing, Prentice Hall, New York, ©1989, ISBN 0-
13-213364-4.

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The majority of physical processes are affected, to one degree or another, by temperature.  Whether these changes are exploited as a technique for temperature transduction or used to identify temperature induced artifacts, the measurement of temperature is of fundamental importance.  Some physical effects used as the basis for temperature transduction are listed below:

1. Thermal expansion
2. Thermocapacitive effect
3. Thermochemical effect
4. Thermoelectric effect
5. Thermoresistive effect
6. Pyroelectric effect
7. Radiation effect

Thermal expansion is the basis for function of mercury or alcohol thermometers while thermocouples employ the thermoelectric effect.  Like the resistance temperature detector (RTD), the thermistor is also a temperature sensistive resistor.  Thermistor is an acronym for thermally sensitive resistors.  While the thermocouple is the most versatile temperature transducer and the Platinum RTD is the most table, the word that best describes the thermistor is sensitive.

Of the three major categories of sensors, the thermistor exhibits by far the largest parameter change with temperature.   Thermistors are generally composed of semiconductor materials.  Although positive temperature coefficient units are available, most thermistors have a negative temperature coefficient (TC); that is, their resistance decreases with increasing temperature.  The negative T.C. can be as large as several percent per degree Celsius, allowing the thermistor circuit to detect minute changes in temperature, which could not be observed with an RTD or thermocouple circuit.  The price we pay for this increased sensitivity is loss of linearity.  The thermistor is an extremely non-linear device that is highly dependent upon process parameters.  Consequently, manufacturers have not standardized thermistor curves to the extent that RTD and thermocouple curves have been standardized.

An individual thermistor curve can be very closely approximated through the use of the Steinhart-Hart Equation:

Here,

= temperature in degrees kelvin for i = 1,…,n (number of data points),

= resistance of the thermistor (ohms) for i = 1,…,n, and

A, B, C = Curve-fitting coefficients.

The choice of temperature transduction technique depends on a variety of factors, including temperature range, cost, size, weight, power consumption, sensitivity, aging effects, etc.  Another prime consideration that affects a choice is the range over which measurements are to be made and the precision of the readings.  Other considerations that may not be so obvious include selection of a thermal mass to be compatible with the bandwidth requirements of the anticipated temperature variations.

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Next, you can choose to set up the MATLAB code in a few different ways. First make sure that both the numerator and denominator are in acceptable forms.

[1 2 3] is the same as saying

Using an example:

We can write the numerator and denominator MATLAB codes as:

>>numerator=[1];

>>denominator=[1 8];

For a more complex problem we can bypass the long and tedious expansion process and use the convolution function in MATLAB.

Here, the denominator is also represented by (s+1)*(s+3+3j)*(s+3-3j). Being three seperate parts, we can use convolution once for two of them, then use convolution again with the remaining part. Just look at the example:

conv( A , conv( B , C ) )  —> denominator = conv( [1 1], conv( [1 3+3*j], [1 3-3*j] ) );

After we define our numerator and denominator in MATLAB, we can use the root locus function then set our axis parameters as follows.

>>rlocus( numerator, denominator )

>>axis([-10 10 -10 10])

Your plot should look similar to the following for this example:

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For the remainder of the month, Engineersphere.com will donate $3 to the American Cancer Society for every guest post that is made.  Engineers can help in more ways than you thought!

You can donate too at the above link. Or to any of these foundations.

http://www.cancer.org/docroot/don/don_0.asp

http://www.relayforlife.org/

http://www.acscan.org/

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The integral of a function is represented like so: and it can be thought of as a sum of areas like so:

Here the integral is performed on the function from point a to point b, which would make it a definite integral because the bounds are defined.  Our indefinite integral is the same procedure, except missing the bounds, which makes indefinite integral operation require a little twist.

We all know that the derivative of a constant is zero.  For instance, the derivative of 5 is equal to zero.  Once performing this derivative, we should still be able to perform the anti-derivative on this new function (zero) to obtain the original equation (5).  But how will we know what number permeates from performing an anti-derivative of zero.  An indefinite integral is called indefinite because the bounds are not defined on the integration, like so:  .  When we perform our indefinite integral we represent this long-lost constant by the letter ‘C’.

Before we integrate our golf cart velocity equation, lets go ahead and look at the laws of integration:

The integral can also be split up into separate individual integrals if there is addition in the function you are integrating.

and whenever you add these integrals together, you only need to account for 1 of the constants (C).

So the integral of our velocity will go as follows:

or

We can solve for our constant, C, if we are given initial conditions, such as the golf cart was moving at y’(0) = 1 m/s when we began collecting our data.  Otherwise, we leave the integral in this form.  If you would like to learn how to perform a definite integral, refer to our article on definite integrals.

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