Introduction Formulation of the . . . Straightforward . . . Identification of Human Gait Why Fourier-Based . . . Why Fourier-Based . . . in Neuro-Rehabilitation: Shift Detection: . . . Towards Efficient Algorithms General Case Conclusions Acknowledgments Naga Suman Kanagala 1 , Martine Ceberio 1 , Title Page Thompson Sarkodie-Gyan 2 , Vladik Kreinovich 2 , and Roberto Araiza 3 ◭◭ ◮◮ ◭ ◮ 1 Department of Computer Science 2 Department of Electrical and Page 1 of 10 Computer Engineering 3 Bioinformatics Program Go Back University of Texas at El Paso Full Screen 500 W. University El Paso, TX 79968, USA Close contact vladik@utep.edu Quit
Introduction 1. Introduction Formulation of the . . . Straightforward . . . • Fact: many neurological diseases drastically decrease Why Fourier-Based . . . the patient’s ability to walk w/o physical assistance. Why Fourier-Based . . . • Examples: stroke, traumatic body injury, and spinal Shift Detection: . . . cord injury General Case • Extensive rehabilitation is needed to re-establish nor- Conclusions mal gait. Acknowledgments Title Page • At present: rehabilitation requires gait assessment by ◭◭ ◮◮ highly qualified experienced clinicians. ◭ ◮ • Problem: difficult to access, high costs. Page 2 of 10 • It is desirable: to automate gait assessment: Go Back – to make rehabilitations easier to access, and Full Screen – to decrease the rehabilitation cost. Close Quit
Introduction 2. Formulation of the Problem in Precise Terms Formulation of the . . . Straightforward . . . • A gait is measured by the dependence x ′ ( t ) of some Why Fourier-Based . . . characteristic on time. Why Fourier-Based . . . • Example: the acceleration or the angle between differ- Shift Detection: . . . ent parts of the foot. General Case • The gait assessment means comparing Conclusions – the recorded patient’s gait with Acknowledgments Title Page – a standard (average) gait x ( t ) of healthy people of the same age, body measurements, etc. ◭◭ ◮◮ • Problem: patients walk slower. ◭ ◮ Page 3 of 10 • Solution: appropriately shift and “scale” the standard gait. Go Back • Resulting formulation: find the values t 0 and λ for Full Screen which Close x ′ ( t ) ≈ x ( λ · t − t 0 ) . Quit
Introduction 3. Straightforward Algorithm and Its Limitations Formulation of the . . . Straightforward . . . • Given: the patient gait x ′ ( t ) and the standard gait x ( t ). Why Fourier-Based . . . • Find: the values t 0 and λ for which Why Fourier-Based . . . x ′ ( t ) ≈ x ( λ · t − t 0 ) . Shift Detection: . . . General Case • Straightforward idea: try all possible shifts and scal- Conclusions ings. Acknowledgments Title Page • Limitations: this is computationally very intensive. ◭◭ ◮◮ • Objective: to design an efficient algorithm for finding ◭ ◮ the optimal combination of a shift and a scaling. Page 4 of 10 • Our idea: adjust the known image referencing tech- niques that use Fast Fourier Transform. Go Back Full Screen Close Quit
Introduction 4. Why Fourier-Based Methods Formulation of the . . . Straightforward . . . • Simplest case: find the shift t 0 for which Why Fourier-Based . . . x ′ ( t ) ≈ x ( t − t 0 ). Why Fourier-Based . . . • Notation: let n be the number of moments of time for Shift Detection: . . . which we know x ( t ). General Case • Natural formalization: least squares method – find t 0 Conclusions ( x ′ ( t ) − x ( t − t 0 )) 2 d t . def � that minimizes I = Acknowledgments Title Page • Simplification: ◭◭ ◮◮ � � � ( x ′ ( t )) 2 d t + x ( t − t 0 ) 2 d t − 2 x ′ ( t ) · x ( t − t 0 ) d t. I = ◭ ◮ • Analysis: the first two terms do not depend on t 0 . Page 5 of 10 • Conclusion: find t 0 for which the convolution Go Back def x ′ ( t ) · x ( t − t 0 ) d t is the largest. � J ( t 0 ) = Full Screen • Computation time: we need n convolutions, with n Close steps each; overall time O ( n ) · O ( n ) = O ( n 2 ). Quit
Introduction 5. Why Fourier-Based Methods (cont-d) Formulation of the . . . Straightforward . . . • Fact: convolution is one of the main techniques in sig- Why Fourier-Based . . . nal processing. Why Fourier-Based . . . • Fact: we can compute convolution J ( t 0 ) faster: Shift Detection: . . . – first, we apply FFT to the original signals, resulting General Case in functions F ( ω ) and F ′ ( ω ); Conclusions – then, for each frequency ω , we compute the product Acknowledgments Title Page def = F ( ω ) · ( F ′ ) ∗ ( ω ); P ( ω ) ◭◭ ◮◮ – third, we apply FFT − 1 to the resulting function ◭ ◮ P ( ω ), and get the desired convolution J ( t 0 ). Page 6 of 10 • Finally, we find t 0 for which J ( t 0 ) → max. Go Back • FFT requires O ( n · log( n )) steps, multiplication and search for t 0 is O ( n ). Full Screen • So, we find t 0 in time Close O ( n · log( n )) + O ( n ) = O ( n · log( n )) ≪ O ( n 2 ) . Quit
Introduction 6. Shift Detection: Resulting Algorithm Formulation of the . . . Straightforward . . . • Ideal case: x ′ ( t ) = x ( t − t 0 ), hence: Why Fourier-Based . . . • F ′ ( ω ) = e 2 π · i · ( − ω · t 0 ) · F ( ω ); Why Fourier-Based . . . def • here, the ratio R ( ω ) = P ( ω ) / | P ( ω ) | is equal to Shift Detection: . . . R ( ω ) = e 2 π · i · ( − ω · t 0 ) ; General Case • thus, the FFT − 1 of R ( ω ) is equal to I ( t ) = δ ( t + t 0 ); Conclusions Acknowledgments • so, t 0 is the only value for which I ( − t ) � = 0. Title Page • In practice: x ′ ( t ) ≈ x ( t − t 0 ), so: ◭◭ ◮◮ • we apply FFT to the original signals x ( t ), x ′ ( t ) and ◭ ◮ compute their Fourier transforms F ( ω ) and F ′ ( ω ); Page 7 of 10 • we compute the product P ( ω ) = F ( ω ) · ( F ′ ) ∗ ( ω ) Go Back and the ratio R ( ω ) = P ( ω ) / | P ( ω ) | ; • we apply FFT − 1 to R ( ω ) and get I ( t ); Full Screen • we find t 0 for which | I ( − t 0 ) | → max. Close Quit
Introduction 7. General Case Formulation of the . . . Straightforward . . . • General case: x ′ ( t ) ≈ x ( λ · t − t 0 ) . Why Fourier-Based . . . • Analysis: the magnitudes M ( ω ) = | F ( ω ) | and M ′ ( ω ) = Why Fourier-Based . . . | F ′ ( ω ) | differ by scaling: M ′ ( ω ) ≈ (1 /λ ) · M ( ω/λ ) . Shift Detection: . . . • Idea: in log frequencies ρ = log( ω ), scaling becomes General Case shift-like: ρ → ρ − b , where b = log( λ ). Conclusions Acknowledgments • Resulting algorithm: Title Page – transform M ( ω ) and M ′ ( ω ) to log frequencies; ◭◭ ◮◮ – use the above FFT-based algorithm to determine ◭ ◮ the corresponding shift log( λ ); Page 8 of 10 – from the corresponding “shift” value, reconstruct the scaling coefficient λ ; Go Back – re-scale x ( t ) to x ( λ · t ) and use the same FFT-based Full Screen algorithm to compute the shift t 0 . Close Quit
Introduction 8. Conclusions Formulation of the . . . Straightforward . . . • Many neurological diseases drastically decrease the pa- Why Fourier-Based . . . tient’s ability to walk without physical assistance. Why Fourier-Based . . . • To re-establish normal gait, patients undergo extensive Shift Detection: . . . rehabilitation. General Case • At present, rehabilitation requires gait assessment by Conclusions highly qualified experienced clinicians. Acknowledgments Title Page • To make rehabilitations easier to access, it is desirable ◭◭ ◮◮ to automate gait assessment. ◭ ◮ • In this paper, we design a fast algorithm that uses Fast Fourier Transform for gait assessment. Page 9 of 10 Go Back Full Screen Close Quit
9. Acknowledgments Introduction Formulation of the . . . This work was supported in part: Straightforward . . . Why Fourier-Based . . . • by NSF grant HRD-0734825, Why Fourier-Based . . . Shift Detection: . . . • by Texas Department of Transportation Research Project General Case Conclusions No. 0-5453, Acknowledgments • by the Japan Advanced Institute of Science and Tech- nology (JAIST) International Joint Research Grant 2006- Title Page 08, and ◭◭ ◮◮ • by the Max Planck Institut f¨ ur Mathematik. ◭ ◮ Page 10 of 10 Go Back Full Screen Close Quit
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