NAME:

SSAN:

Please Exclude (specify homework or exam):

  
   

Question 1A [5 points]

  

Question 1B[5 points]

  

Question 1C[5 points]

  

Question 1D[5 points]

  

Question 1E[5 points]

  

Question 1F[5 points]

  

Question 1 [30 points]

  

Question 2A [5 points]

  

Question 2B[5 points]

  

Question 2C[5 points]

  

Question 2D[5 points]

  

Question 2E[5 points]

  

Question 2 [25 points]

  

Question 3 [25 points]

  

Question 4[20 points]

  

FINAL EXAM TOTAL

  
   

Homework Average

  

Exam Average

  

Weighted Course Total

  

Rules:

  1. This examination is open book and open notes.
  2. It is a take-home examination
  3. It is due on or before noon, 12/18/97. It should be deposited in the designated wooden lock box outside NEB 224.
  4. You may spend at most 8 hours working on the exam. You do not need to do this in a single sitting, but I do expect you to keep track of the time spent and stop when the 8 hours are up. I think it can be completed in rather less time than that.
  5. Answers should be typed double space or otherwise neatly written. I cannot grade what I cannot read. Also leave generous margins for me to make annotations on.
  6. Please bind this cover sheet to your answers, since it includes a grade sheet.
  7. Number the pages and write your name on every page.
  8. Of course, you are expected to work alone on this exam. You are not to consult with any other person about the questions, answers to questions, or anything else about this exam.
  9. Question 1

 

Consider the 3D digitizer shown below

 

This tracker consists of two CCD video cameras located a distance D apart. The cameras track passive spheres located in space some distance in front of the cameras. The origin of the tracker coordinate system is located midway between the two lens centers. Camera 1’s lens center is located at (0,-D,0), and Camera 2’s lens center is located at (0,+D,0) in the coordinate center of the tracker. Both cameras point outward in the positive "Z" direction. Each camera has a nominal focal length . Image processing software is used to locate the centroid of the images of the passive markers on the two cameras. The position (in mm) of the centroid of a marker A on camera 1’s image detector is and the position on camera 2’s detector is . If we assume a simple pinhole camera model, the value of for a sphere located at will be given by

 

with a similar expression for .

 

1A (5 points): Given and , describe a method for computing , assuming the simple camera model above. Your answer should include appropriate formulas.

 

1B (5 points): Suppose that D = 500 mm, f = 10 mm, = (1.3637,6.3637), and =(1.3637,-2.7273). What is the value of (x,y,z)?

 

1C (5 points) Suppose that our measured values for D and f are inaccurate so that the true values are (D+D D) and (f+D f), respectively. Similarly, suppose that the centroid finders are subject to errors of (D u,D v). Estimate the error in your calculation of .

 

1D (5 points) Describe a suitable calibration method for determining and , including a sketch and enough formulation so that I can follow your approach. How will the calibration affect your estimate of ?

 

Suppose now that we have discovered that the lenses of the cameras are slightly distorted, so that the value for image sensor 1 is given by

 

 

with a similar expression for camera 2. Suppose also that we know that g and h are smooth functions with

 

 

but that they are otherwise unknown.

 

1E (5 points): Describe a suitable calibration method to correct for this distortion, again including a sketch and enough formulation so that I can follow your approach.

 

1F (5 points): Assuming that the basic image processing circle finder locates circles on the image sensors to an accuracy of , estimate the number of calibration points required to determine any position within a 1m 1m 1m cube centered at (0,0,1m) with respect to the tracker to an accuracy (0.1mm, 0.1mm, 0.1mm).

 

Question 2

 

Suppose that we have successfully calibrated the tracking device in Question 1 to an accuracy of mm in each dimension. Consider now the rigid body pointing device shown below

 

 

The pointer tip is at (200,0,-10) and it has four spheres at

 

(-20,20,0), (20,20,0),(-20,-20,0) and (20,-20,0)

 

with respect to the pointer coordinate system, as shown in the sketch.

 

2A (5 points) Describe (give formula) how to compute the position of the pointer tip with respect to the tracker.

 

2B (5 points) Estimate the accuracy of your answer to question 2A.

 

2C (5 points) Suppose that the pointer shaft has been bent. Describe a simple method for determining the new position of the pointer tip with respect to the pointer coordinate system.

 

2D (5 points): Suppose that the tracking sensor is mounted on a mobile stand placed approximately 1m from a patient’s head. The mounting stand is subject to various vibrations that cause it to twist by 0.01 degree at unknown frequency. What will be the effect on how accurately the pointer can be located with respect to the tracking device? Give a number and describe a simple and robust method for overcoming this type of disturbance when using such devices for surgical navigation.

 

2E (5 points): Briefly discuss the trade-offs between passive systems such as the one described here and active tracking systems such as the Optotrak™ or Pixsys™, which track sequentially flashed light emitting diodes.

 

Question 3 (25 points)

 

In class, we discussed the use of intraoperative 3D pointing systems for registration to 3D preoperative models. We also, discussed 2D-3D methods that rely on intraoperative x-rays. Discuss the relative advantages and disadvantages of each approach, using specific clinical applications discussed in class to justify your answer. Your answer should also appropriately cite specific systems, authors, and algorithms, but should not recite all the detailed steps of individual algorithms.

 

Question 4 (20 points)

 

Several authors have explored the use of intraoperative imaging ultrasound in conjunction with 3D tracking systems for intraoperative registration to preoperative patient models. Describe one such application and discuss briefly the advantages and disadvantages of this sensing modality for the application chosen. Your answer should include a concise description of how the ultrasound equipment is integrated with the other equipment and which registration algorithms are appropriate, as well as an evaluative comparison against competing approaches.