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Tear Film

Page history last edited by PBworks 16 years, 8 months ago

Tear Film

 

Summary of Project

Using high speed films of unforced blinks, we are using various interpolation techniques along with edge detection to make a computer simulation of a blink.  This data will be further used as moving boundaries to a PDE that will model the tear film distribution across the surface of the eye.  Hopefully this will help in treating eye disorders such as dry eye and many others.

 

Group members

 

 

 

 

   

Above is an example of interpolation of the eyelid postion.  This is very similar to the procedure we are using for the motion of the eyelids, only we are interpolating a series of high speed frames and linking them into a video file.  Fortunately for us, we do not have to manually pick 15 points on each lid for every third frame when using the video file.  In the algorithm we are using, Matlab detects the edges with a canny edge detection program, picks a set amount of points on the edges and then represents a polynomial that best fits all of these points.

 

Capturing the Unforced Blinks

We have arranged a filming set in which the subject sits approximately 4ft away from the high speed camera.  Resting their head on the a guillotine appearing frame, we typically film for about 30 sec at 250 frames/sec.  Behind the camera two halogen lights, one set high aiming downward and the other set low facing upwared, are focused directly on the subjects face to minimize shadows.  Because of the extremely large variations in eye shape of humans we are very interested in what race, gender, age and other factors effect the accuracy of this algorithm. 

 

Fliqz has shut down their service. To access this video, email support with this video id: 23585
Fliqz has shut down their service. To access this video, email support with this video id: 23586

 

Applied Mathematics

 

  •  Least Squares
  • Used in finding the coefficients to the interpolated polynomials.
  • Often in Linear Algebra and other fields of mathematics we are faced with dealing with "over determined" systems of equations.  Although in many cases we cannot solve these equations for exact solutions, there is a technique that allows us to obtain very close approximations.  The theory of Least Squares is to not solve the typical system Ax=b , but instead solve the normal system (AT)Ax=(AT)b, for x, (note Ax is not equal to b in general), which is the best we can do for singular values of A.  Without a complete derivation of why this works, it is important to observe that the normal system is simply an error minimizing function.  If we say that the error vector E =min(||b-projW b||), where W is the column space of A.  We construct x such that projW b=Ax, which is orthogonal to AT .  Thus we now have E=min(||b-Ax||).   Because b-Ax is orthogonal to the column space of A, we can write (AT)[b-Ax ]=0.  This statement can be rearranged to the normal system defined before, and thus guarantees a minimum error E
    Polynomial Interpolation
  • Used in creating/recreating the eyelid edge position at every third frame. 
  • After using a Matlab function to capture a set amount of points on the edge of the eyelid, we try to find a polynomial that will fit all of these points with as little oscillation as possible.  Due to the smooth shape of the eyelid we are forced to use low degree polynomials whose coefficients are determined using Least Sqares.  Furthermore, after finding these polynomials at each frame, we use a "smoothing" effect which is the idea of splines.  Splines is a variation of interpolation where instead of finding one polynomial to fit all points, we construct a series of polynomials, in small intervals, allowing us to create continous second derivatives which guarantee smoothness to the overall approximating function.

 

Algorithm Accuracy Analysis

 

Subject # Age Gender(M/F) Race                Contacts(Y/N)

Rating in increasing accuracy 

(1-5)

           
           
           
           
           
           
           
           
           
           
           
           
           
           
           
           
           
           
           

 

Sources

 

  1. Elementary Linear Algebra, Bernard Kolman, David R. Hill
  2. Elementary Numerical Analysis, Kendall Atkinson
  3. Introduction to Scientific Computing, Charles F. Van Loan
  4. Numerical Analysis, Richard L Burden, J. Douglas Faires

 

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