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EOS740_Fall2006_Syllabus
Course: EOS 740, Fall 2008School: George Mason
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740 EOS 001, CRN 73827, Hyperspectral Imaging Systems FALL SEMESTER 2006 Credit Hours: 3 Description: This course will provide the requisite materials to understand hyperspectral imaging (HSI) technology and its many applications. The emphasis will be on the scientific principles involved and the application of the technology to real world applications. Topics that will be covered include hyperspectral concepts...
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740 EOS 001, CRN 73827, Hyperspectral Imaging Systems FALL SEMESTER 2006
Credit Hours: 3 Description: This course will provide the requisite materials to understand hyperspectral imaging (HSI) technology and its many applications. The emphasis will be on the scientific principles involved and the application of the technology to real world applications. Topics that will be covered include hyperspectral concepts and system tradeoffs, data collection systems, calibration techniques, HSI data processing software and data processing techniques, classification techniques, and case studies. The data processing techniques will include NDimensional Space, Scatterplots, Spectral Angle Mapping, Spectral Mixture Analysis, Spectral Matching, Mixture Tuned Matched Filtering, and other techniques. Applications and case studies will include environmental, medical, agricultural, and others. Ground, airborne, and spaceborne hyperspectral systems will be covered. Course Objective: To provide students with an introduction to modern hyperspectral remote sensing techniques and the basic fundamental physics involved in this technology. The course will (1) prepare the student to undertake graduate research in image hyperspectral processing and related areas, (2) prepare the student to participate in professional activities in this field of study, (3) broaden the student's background in the general field of spectral remote sensing and image processing, and (4) prepare the student to explore finding applications of this enabling technology to areas of interest to specific users. Prerequisites: An introductory course on Remote Sensing, or Earth Science, or Digital Image Processing, or Space Science, or Permission of Instructor. Text: Remote Sensing Digital Image Analysis: An Introduction, John A. Richards and Xiuping Jia, 3rd Revised and Enlarged Edition, Springer-Verlag, Berlin, 1999. Grading: Assigned Project and Oral P...
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Michigan State University - MATH - 1825
1.1 Real NumbersA. SetsA set is a list of numbers: We separate the entries with commas, and close off the left and right with and .The empty set is the set containing nothing:. It is given the symbol .B. Special Sets1. Natur
Michigan State University - MATH - 1825
1.2A Fractions ReviewA. SimplifyingTo simplify a fraction, we divide top and bottom by common factors. If we choose the largest (greatest common factor), then we can do it in one step. Examples: 1. Simplify 2. Simplify B. Multiplying a
Michigan State University - MATH - 1825
1.4A Properties of Exponents IA. Product RuleNotice the following: In general,Thus, when we multiply powers, we add exponents. Example: Find B. Quotient RuleNotice the following: Similarly, In general,
Michigan State University - MATH - 1825
1.4B Properties of Exponents IIA. Summary of Rules 4. Zero Power:, if6. Switch Rule:"up goes down, down goes up" "hit every entry"7. Multiple Power Rule:B. SimplifyingWe now want to be able to simplify expressions using several
Michigan State University - MATH - 1825
1.4C Scientific NotationA. Scientific NotationExtremely large or extremely small numbers are sometimes written in scientific notation. Scientific Notation looks like # , where # is a number between andB. Converting to Scientific NotationMove the
Michigan State University - MATH - 1825
1.5A Terms and ExpressionsA. Definitions1. Terms: things separated by plus/minus signs 2. Expressions: collection of terms; doesn't have an equals sign3. Coefficient: number in front of term: 3B. Important ViewpointSuppose we have an expressi
Michigan State University - MATH - 1825
1.5B Simplifying ExpressionsA. Clearing ParenthesesWe apply the distributive property. Example 1: Solution SimplifyAnsExample 2: SolutionSimplifyAns 1
Michigan State University - MATH - 1825
1.6 Evaluating Expressions/FormulasA. EvaluationTo evaluate an expression/formula, just plug the number in (surrounded by parentheses). Example 1: Solution EvaluatewhenAnsSolution Ans Example 2:Evalua
Michigan State University - MATH - 1825
2.1A Equations and CheckingA. Equations vs. Non-equationsWe have to first make sure we recognize the difference between an equation and a nonequation. An equation has an equals signAn equation has us solve for an unknown (say ). A non-equation as
Michigan State University - MATH - 1825
2.1B Solving First-Degree EquationsA. Equation RuleYou can do anything (add/subtract/multiply/divide) to one side of an equation, so long as you do it to the other side. For instance, suppose we have the equation . .We could, if we
Michigan State University - MATH - 1825
2.1C Special CasesA. DiscussionSometimes when solving an equation, " " disappears! If you get something false, like This is sometimes written as .,If you get something true, like , , etc., then the equation is always true. Every real numbe
Michigan State University - MATH - 1825
2.2 Literal EquationsA. DiscussionThese are equations with many variables in them. The goal is to solve for one of them. To do this, treat all other variables as if they were numbers. Strategy: Move everything with the desired variable to one side
Michigan State University - MATH - 1825
2.3 Absolute Value EquationsA. Absolute ValueRecall that means distance from the origin.B. Strategy1. Draw a number line and mark the location(s) that have the required distance. 2. Rewrite the problem without absolute value signs using t
Michigan State University - MATH - 1825
2.4 Problem SolvingA. Strategy1. Rewrite the problemmaybe with a pictureleaving out the useless information. 2. Let what you don't know. Then write down as many things as you know about . Note: If there are several things you don't know, cho
Michigan State University - MATH - 1825
2.5 More Problem SolvingA. Two Similar UnknownsWhen you have two unknowns that both add to a number, say and B. Mixture ProblemsWhenever we have two types of items that we put together to form a third, we have a mixture problem. The gene
Michigan State University - MATH - 1825
2.6 Linear InequalitiesA. Review of Number InequalitiesExamples: [ is bigger than ] [ is less than or equal to : it's less than!] is greater than or equal to[To compare fractions: Find a common denominator Example: Is Solution and , so since o
Michigan State University - MATH - 1825
2.7A AND or ORA. ANDThe word "AND" means both must be true. Goal: Given two shaded number lines, determine where they are both shaded inB. Examples of ANDExample 1:Find the AND graph. Solution We shaded on a new line, where it is shaded on bot
Michigan State University - MATH - 1825
2.7B Triple Inequalities; More AND/ORA. Triple InequalitiesA triple inequality is an "object" like Meaning: Graph:We want all numbers between "Sandwich"andExample: SolutionGraphNumbers betweenand : open circle atB. Comments on Triple
Michigan State University - MATH - 1825
2.7C Solving Compound InequalitiesA. MethodA compound inequality is an inequality pair that is joined in AND or OR. To solve these, we 1. Solve each inequality separately. 2. Graph each answer and form the AND/OR graph. 3. Write down the appropriat
Michigan State University - MATH - 1825
2.8 Absolute Value InequalitiesA. StrategyRecall: Absolute Value means distance from the origin 1. Draw a number line and mark the locations that have the required distance. Remember distance is always a "positive" idea. 2. Use the shaded number li
Michigan State University - MATH - 1825
3.1 Introduction to LinesA. Rectangular (Cartesian) Coordinate Systemorigindirection direction in inB. Lines/Point Plotting1. A line is represented by an equation containing 2. To draw a line: a. c. "Randomly" pick values for . Plot points an
Michigan State University - MATH - 1825
3.2 SlopeA. SlopeThe slope of a line measures "how much it is tilting".rise runBy definition, sloperise . runB. Finding Slope1. Pick two points on a line. Pick one as the first point and the other as the second point. 2. Going from the fi
Michigan State University - MATH - 1825
3.3 Equations of LinesA. Point-Slope Formula1. Formula: 2. Justification: Multiplying byyields the equation.1 B. Slope-Intercept Formula1. Formula:Here " " is the -intercept. 2. Justif
Michigan State University - MATH - 1825
3.5 Relations and Functions: BasicsA. Relations1. A relation is a set of ordered pairs. For example,2. Domain is the set of all first coordinates:3. Range is the set of all second coordinates:B. FunctionsA function is a relation that satisfi
Michigan State University - MATH - 1825
A. Graphing FunctionsB. ExamplesTo graph a function, we make a table of Then plot points and conne
Michigan State University - MATH - 1825
4.1 Systems of Linear EquationsA. IntroductionSuppose we have two lines. We have three possibilities:the lines intersectthe lines are parallelthe lines overlap (coincide)Thus, if we have a system of linear equations; that is, "two line equ
Michigan State University - MATH - 1825
4.3 ApplicationsA. CommentsSometimes when we solve word problems, we end up with a system of linear equations to solve.B. ExamplesExample 1: 372 people attended a concert. Floor seats cost $20/each. Balcony seats cost $12 each. If the ticket sal
Michigan State University - MATH - 1825
5.1A Polynomials: BasicsA. Definition of a PolynomialA polynomial is a combination of terms containing numbers and variables raised to positive (or zero) whole number powers. Examples of Polynomials NOT polynomials B. Terminol
Michigan State University - MATH - 1825
5.2 Dividing PolynomialsA. Dividing Polynomials By MonomialsTo divide a polynomial by a monomial, we form a fraction with the monomial in the denominator. Then divide the denominator into each term. Example 1: Solution Divideby
Michigan State University - MATH - 1825
5.4 Introduction to FactoringA. Factoring out GCFFactoring is "undoing" the distributive property/multiplicationWe can often factor by "stripping out" the largest factor common to all terms, and then writing what's left inside parentheses.B. Ex
Michigan State University - MATH - 1825
5.5A Factoring Trinomials: An Introductory DiscussionA. IntroductionBefore discussing AntiFOIL, the powerful technique for factoring trinomials, we will first explain where the method comes from. We will discuss the technique itself in the next sec
Michigan State University - MATH - 1825
5.5B AntiFOILA. AntiFOIL technique1. Given , multiply . This is your "magic number".2. Split the middle term into two, with signs given by the TSP. Start with one of the numbers as being the number 1. 3. Multiply the middle two numbers
Michigan State University - MATH - 1825
5.5C More on AntiFOILA. Discussion1. Trinomials that are not necessarily of the form using AntiFOIL. Just do the logical thing. can still usually be factored2. When factoring any trinomial, you should factor out the GCF first (if there i
Michigan State University - MATH - 1825
5.6 Special Factoring FormulasA. Perfect Square Factoring1. Perfect Square Factoring Formulas: and 2. To use: if the first and last terms of a trinomial are squares, try writing a perfect square; then use the s
Michigan State University - MATH - 1825
5.7A Generalized Factoring IA. General Factoring Strategy1. First try to factor out the GCF. 2. Decide how many terms you have, and do the following: a. Two terms: look for I. Difference of Squares: II. Difference of Cubes: III. Sum of Cubes: b.
Michigan State University - MATH - 1825
5.7B Generalized Factoring IIA. More on Differences of SquaresSome difference of squares problems are trickier. Be careful with minus signs! Example 1: Solution Two terms: Difference of Squares FactorAnsExample 2: SolutionFacto
Michigan State University - MATH - 1825
5.8 Polynomial Equations Solvable By FactoringA. Zero Product PrincipleThe zero product principle says that if a product of factors is zero, then one of the factors must be zero.Example 1: SolutionSolve the equationBy the zero product prin
Michigan State University - MATH - 1825
6.1A Rational Expressions and Rational FunctionsA. IntroductionAn algebraic fraction is called a rational expression. For example, and are rational expressions. A rational function is a function whose output formula Thus if , then
Michigan State University - MATH - 1825
6.1B Simplifying/Multiplying/Dividing Rational ExpressionsA. Simplifying Rational ExpressionsTo simplify a fraction, we divide top and bottom by common factors. To do this, we have to have the top and bottom factored! Example 1: Solutio
Michigan State University - MATH - 1825
6.2A Finding Common DenominatorsA. IntroductionWhen we add/subtract fractions, we need to find a common denominator. In an algebraic fraction, we work with factors, so we need to learn how to find common denominators from factors.B. LCM of Number
Michigan State University - MATH - 1825
6.2B Adding/Subtracting Rational ExpressionsA. Method1. Factor the denominators and find the LCD. 2. Rewrite each fraction with the common denominator by multiplying top and bottom of the original fraction by appropriate factors. 3. Add/subtract nu
Michigan State University - MATH - 1825
6.3 Complex FractionsA. IntroductionFractions that have fractions in the numerator and/or denominator are complex fractions. For example,is a complex fraction.To simplify these, we treat them as three problems in one. "Do the n
Michigan State University - MATH - 1825
6.5A Solving Literal Equations with Rational ExpressionsA. MethodWe combine the previous techniques of rational equations with previous literal equations. We treat all variables, other than the one we solve for, as constants.B. ExamplesSolutio
Michigan State University - MATH - 1825
6.5B Applications of Rational EquationsA. Proportion ProblemsA proportion is an equation between two ratios. Some problems explicitly mention ratios, others use the idea of similar triangles. Triangles are similar if they have the same shape, but m
Michigan State University - MATH - 1825
7.1A Rational Exponents IA. IntroductionIn this section and the next, we consider variables whose exponents are fractions, i.e. . While we will give meaning to these in the later sections, we will for now review some old techniques while having fra
Michigan State University - MATH - 1825
7.1B Rational Exponents IIA. Factoring with Rational ExponentsIf we see variables raised to fractional powers, the greatest common factor includes the variable(s) raised to the smallest fraction. Here we revisit factoring out the GCF from Section 5
Michigan State University - MATH - 1825
7.2A Introduction to RootsA. Square Rootsinside, because, because, becauseNote: Any number multiplied by itself is never negative! Thus, there is no real number answer to the square root of a negative number. Thus,is not a real number.
Michigan State University - MATH - 1825
7.2B Roots, Radicals, and Rational ExponentsA. DiscussionConsider the symbol .If we were to multiply it by itself, and the standard rules would still hold, then (by adding exponents) . ThusHence,Thus it makes sense to ident
Michigan State University - MATH - 1825
7.2C Absolute Value and RootsA. Absolute Value DiscussionRecall that absolute value means distance from the origin.We think of absolute value of numbers as "make it positive", but of course that doesn't work for variables. (See Sections 2.3
Michigan State University - MATH - 1825
A. List of Squares/Cubes/Etc. To Memorize5. Higher Powers 4. Fifth Powers3. Fourth Powers2. Cubes1. Squares7.3A Simplify
Michigan State University - MATH - 1825
7.3B Adding and Subtracting RadicalsA. Adding and Subtracting Like TermsThe object underneath the radical is called the radicand. Radicals with the same index and the same radicand are considered like terms. and and and are like terms
Michigan State University - MATH - 1825
7.4A Multiplying RadicalsA. Radical Product RuleWe have the radical product rule: Thus, for example, (true ifand are positive).This rule is really a property of exponents in disguise. Why? Note: To use the radical product rule, th
Michigan State University - MATH - 1825
7.4C Dividing RadicalsA. MethodWe use the "root of a fraction rule" in reverse to start the problem.B. ExamplesExample 1: Solution SimplifyExample 2: SolutionSimplifyThus we have Now convert back:
Michigan State University - MATH - 1825
7.4D RationalizationA. IntroductionWe do not consider fractions with roots in the denominator to be completely simplified. For instance, an example would be .To simplify these, we use a different "division-type" process called rationalizatio
Michigan State University - MATH - 1825
7.5 Radical EquationsA. Radical EquationsThese are equations with radicals in them. Here is the general strategy for solving them: 1. Isolate a radical (get one radical by itself on one side) 2. Eliminate the radical by raising each side to the app
Michigan State University - MATH - 1825
8.1A Plus/Minus1. NotationThese are a shorthand way of writing two solutions. 2. Minus Signs:3. In expressions involving or , we have two solutions. One by taking the "top" sign and one by taking the "bottom" sign. plus or
Michigan State University - MATH - 1825
8.1C Completing the SquareA. IntroductionBy the square formula, .Suppose you knew that were the first two terms of a perfect square, how could you figure out that the last term had to be ? Note: "half ofsquared is "
Michigan State University - MATH - 1825
8.1D Solving Quadratic Equations by Completing the SquareA. IntroductionSome quadratic equations can not be factored nicely, since the trinomial may be prime. If we use completing the square, we can solve all of them.B. Method1. Take the quadrat
Michigan State University - MATH - 1825
A. Derivation of the Quadratic FormulaWe can get a general formula for the solutions to by doing completing the square on the general equation.8.2A Quadratic Formula1
Michigan State University - MATH - 1825
8.2B Consequences of the Quadratic FormulaA. Discriminant Since,we see that ifB. Discriminant ExamplesExample 1: Solution " " ! Ansno real solutions1How many real solutions does
Michigan State University - MATH - 1825
8.4 More on Literal Equations; Pythagorean TheoremA. Literal EquationsWhen solving literal equations for a variable, sometimes roots and/or quadratic formula must be used.Solution Multiply by Divide by::AnsSolution Quadratic formula:Ans