How to Reconstruct the Main Idea: Use . . . Reconstructing x . . . - PowerPoint PPT Presentation

Main Use of Radars: . . . Radars Provide . . . It Is Desirable to . . . Filtering Filtering: A Problem How to Reconstruct the Main Idea: Use . . . Reconstructing x . . . From Characteristic . . . Original Shape of a Radar Computations: . . . From Logarithmic . . . Signal? From L n to M n : cont-d Maximum Entropy . . . Alternatives to MaxEnt Matthew G. Averill, Gang Xiang, Vladik Kreinovich, Other Alternatives to . . . G. Randy Keller, and Scott A. Starks The Use of Expert . . . Pan-American Center for Earth and Environmental Studies Acknowledgments University of Texas at El Paso, El Paso, TX 79968, USA Title Page averill@geo.utep.edu, gxiang@utep.edu, ◭◭ ◮◮ vladik@utep.edu, keller@utep.edu, sstarks@utep.edu ◭ ◮ Patrick S. Debroux and James Boehm Army Research Laboratory, SLAD Page 1 of 18 Go Back Full Screen Close Quit

Main Use of Radars: . . . Radars Provide . . . 1. Radars are Important It Is Desirable to . . . Filtering • Radar measurements are used in many areas of science and engineering. Filtering: A Problem • Historically the first use of radars was in tracing airplanes and missiles . Main Idea: Use . . . Reconstructing x . . . • This is still one of the main uses of radars. From Characteristic . . . Computations: . . . • However, radars are used more and more in geosciences as well. From Logarithmic . . . • The information provided by airborne radars nicely supplements other remote From L n to M n : cont-d sensing information Maximum Entropy . . . Alternatives to MaxEnt – radar beams can go below the leaves, to the actual Earth surface Other Alternatives to . . . – and they can even go even deeper than the surface. The Use of Expert . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 2 of 18 Go Back Full Screen Close Quit

Main Use of Radars: . . . Radars Provide . . . 2. Main Use of Radars: Localization It Is Desirable to . . . Filtering • The main idea behind a radar is simple: Filtering: A Problem – we send a pulse-like radio signal, Main Idea: Use . . . Reconstructing x . . . – this signal gets reflected by the target, and From Characteristic . . . – we measure the reflected signal. Computations: . . . • The main information that we can get from the radar is the travel time . From Logarithmic . . . From L n to M n : cont-d • Based on the travel time, we can find the distance between the radar and the Maximum Entropy . . . target. Alternatives to MaxEnt • If we use several radars, we can thus get an exact location of the target. Other Alternatives to . . . The Use of Expert . . . • This is how radars determine the exact position of the planes in the vicinity Acknowledgments of an airport. Title Page • This is how radars produce high-accuracy digital elevation maps that are so important in geophysics. ◭◭ ◮◮ ◭ ◮ Page 3 of 18 Go Back Full Screen Close Quit

Main Use of Radars: . . . Radars Provide . . . 3. Radars Provide Additional Information It Is Desirable to . . . Filtering • If the targets were points , then after sending a pulse signal, we would get a Filtering: A Problem pulse back. Main Idea: Use . . . • In this case, the only information we are able to get is the distance from the Reconstructing x . . . radar to the point target. From Characteristic . . . Computations: . . . • In reality, the target is not a point. From Logarithmic . . . • As a result: From L n to M n : cont-d Maximum Entropy . . . – even if we send a pulse signal, Alternatives to MaxEnt – this pulse is reflected from different points on a target and Other Alternatives to . . . – therefore, we get a continuous signal back. The Use of Expert . . . Acknowledgments • The shape of this signal can provide us with the additional information about Title Page the reflecting surface. ◭◭ ◮◮ ◭ ◮ Page 4 of 18 Go Back Full Screen Close Quit

Main Use of Radars: . . . Radars Provide . . . 4. It Is Desirable to Determine the Probability Distri- It Is Desirable to . . . bution of the Reflected Signal Filtering Filtering: A Problem • In an airborne geophysical radar, pulses are sent one after another. Main Idea: Use . . . Reconstructing x . . . • As a result, individual reflections get entangled. From Characteristic . . . • We can still measure the probability distribution of the values of the reflected Computations: . . . signal. From Logarithmic . . . From L n to M n : cont-d • Our objective : to extract the information about the reflecting surface from Maximum Entropy . . . this distribution. Alternatives to MaxEnt Other Alternatives to . . . The Use of Expert . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 5 of 18 Go Back Full Screen Close Quit

Main Use of Radars: . . . Radars Provide . . . 5. Filtering It Is Desirable to . . . Filtering • Problem: the reflected signals are weak and covered with noise. Filtering: A Problem • Solution: to decrease the noise, we apply filtering – usually, linear filtering. Main Idea: Use . . . Reconstructing x . . . • What is linear filtering: From Characteristic . . . Computations: . . . – instead of the original signal x ( t ), From Logarithmic . . . – we consider a linear combination of this signal and the signals ate the From L n to M n : cont-d previous moments of time: Maximum Entropy . . . � y ( t ) = a ( s ) · x ( t − s ) . Alternatives to MaxEnt Other Alternatives to . . . s The Use of Expert . . . + This filtering decreases the noise and makes the distance measurement very Acknowledgments accurate. Title Page − On the other hand, it replaces the original possibly non-Gaussian signal x ( t ) with a linear combination of such signals. ◭◭ ◮◮ ◭ ◮ Page 6 of 18 Go Back Full Screen Close Quit

Main Use of Radars: . . . Radars Provide . . . 6. Filtering: A Problem It Is Desirable to . . . Filtering • Central limit theorem: as we increase the number of terms in a linear combi- Filtering: A Problem nation of several small random variables, the resulting distribution of a sum Main Idea: Use . . . tends to Gaussian. Reconstructing x . . . • Comment: this theorem is the main reason why Gaussian distributions are From Characteristic . . . so frequent in practice. Computations: . . . From Logarithmic . . . • Conclusion: after filtering, we get a distribution that is close to Gaussian. From L n to M n : cont-d • Problem: Maximum Entropy . . . Alternatives to MaxEnt – we have a probability distribution for Other Alternatives to . . . � The Use of Expert . . . y ( t ) = a ( s ) · x ( t − s ); Acknowledgments s Title Page – we want to reconstruct the original distribution for x . ◭◭ ◮◮ ◭ ◮ Page 7 of 18 Go Back Full Screen Close Quit

Main Use of Radars: . . . Radars Provide . . . 7. Main Idea: Use Logarithmic Moments It Is Desirable to . . . Filtering • Main idea: describe both distribution in terms of logarithmic moments. Filtering: A Problem • Characteristic function of a random variable ξ is Main Idea: Use . . . Reconstructing x . . . def χ ξ ( ω ) = E [exp(i · ω · ξ )] . From Characteristic . . . Computations: . . . • For the sum ξ = ξ 1 + ξ 2 of independent ξ i , From Logarithmic . . . From L n to M n : cont-d E [exp(i · ω · ξ )] = E [exp(i · ω · ξ 1 ) · exp(i · ω · ξ 2 )] . Maximum Entropy . . . Alternatives to MaxEnt E [exp(i · ω · ξ )] = E [exp(i · ω · ξ 1 )] · E [exp(i · ω · ξ 2 )] , Other Alternatives to . . . i.e., χ ξ ( ω ) = χ ξ 1 ( ω ) · χ ξ 2 ( ω ) . The Use of Expert . . . Acknowledgments • Hence, ln( χ ξ ( ω )) = ln( χ ξ 1 ( ω )) + ln( χ ξ 2 ( ω )) . Title Page i n · d n χ ξ ( ω ) = 1 def • So, if we define n -the logarithmic moment as L n ( ξ ) | ω =0 , we dω n ◭◭ ◮◮ conclude that L n ( ξ ) = L n ( ξ 1 ) + L n ( ξ 2 ) . ◭ ◮ • Comment. The factor 1 i n is added to make the moments real numbers. Page 8 of 18 Go Back Full Screen Close Quit

Main Use of Radars: . . . Radars Provide . . . 8. Reconstructing x From y : Main Idea It Is Desirable to . . . Filtering • Since y ( t ) = � a ( s ) · x ( t − s ) , we get Filtering: A Problem s Main Idea: Use . . . �� � Reconstructing x . . . ( a ( s )) n L n ( y ) = · L n ( x ) . From Characteristic . . . s Computations: . . . From Logarithmic . . . � �� � ( a ( s )) n • Idea: so, we can reconstruct L n ( x ) := L n ( y ) . From L n to M n : cont-d s Maximum Entropy . . . • In the ideal non-noise case , once we know the exact distribution for y , we Alternatives to MaxEnt can reconstruct the desired distribution for x as follows: Other Alternatives to . . . The Use of Expert . . . – first, we compute the logarithmic moments L n ( y ) of the signal y ; Acknowledgments – then, we compute the value L n ( x ); Title Page – finally, we use the Taylor series to reconstruct the logarithm of the char- acteristics function as ◭◭ ◮◮ ln( χ x ( ω )) = L 1 · i · ω + L 2 · i 2 · ω 2 + L 3 · i 3 · ω 3 + . . . ◭ ◮ So, we can determine the characteristic function χ x ( ω ) of the original distri- Page 9 of 18 bution x . Go Back Full Screen Close Quit