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Performance of a Continuously Sampling Hand-Held Digitizer for Residual-Limb Shape Measurement

William M. Vannah, PhD
David M. Drvaric, MD
Joseph A. Stand III
Jeffrey A. Hastings
Janet E. Slocum, PhD
David M. Harning, CPO
George E. Gorton

ABSTRACT

Conventional pivoting-arm digitizers can be cumbersome when recording unusual limb shapes. The purpose of this study was to assess the accuracy and ease of use of a pencil-style, hand-held digitizer.

The digitizer sampled continuously, recording 120 points per second. An interactive visual display of the sampled data's quality was provided to guide the user in positioning the digitizer. The digitizer's accuracy was quantified with pilot tests (using residual limbs) and with formal tests (using three abstract shapes: a cylinder, a square column and a lobed column). Master molds with these shapes were digitized and carved, and various diameters of the resulting shapes were measured using a dial caliper. These measurements were compared to measurements of the master molds.

The average errors were less than 1.0 mm for all diameters. A worst single error of 3.22 mm was observed in measuring across the corners of the square column. The standard deviations of the errors also were less than 1.0 mm for all diameters except one. Across the corners of the square column, the diameter measurements varied by a standard deviation of 2.13 mm.

Key Words: Prosthesis; Artificial Limbs; Topography; Computer-Aided Design; Computer Graphics.

Introduction

A digitizer is a device that measures the positions of points in three-dimensional space. Digitizers can be used to check the dimensions of a machined part or to record the shape of an object so it may be stored in a computer. Geometrically irregular shapes, such as the shape of a hand-drawn curve, can be stored in a computer by sampling the positions of a large number of points along the curve. The curve can be re-created later by fitting a line to these points. The accuracy of the re-creation depends on the number of recorded points and the accuracy of the digitizer in recording each of the points. In orthotics and prosthetics (O&P), digitizers are used to record the irregular three-dimensional shapes of body parts. The same digitization process occurs; that is, the irregular physical shape is digitized by recording a large number of points on the surface of the shape. Computer-aided design (CAD) programs use these points to create graphic images, which then can be modified by a clinician working at a computer cathode ray tube (CRT). After modification, the shapes are sent to numerically controlled machinery for automated fabrication.

Most digitizers in O&P CAD are either of pivoting-arm (1-3) or optical (4,5) designs. Computed tomography (CT) and nuclear magnetic resonance (NMR) scans have been used in research but remain too expensive for routine clinical use (6). All of these systems use a geometrically regular pattern of sampling points to assure all areas of the surface are adequately sampled.

A hand-held, free-floating digitizer offers advantages such as an ability to be used on a wide variety of shapes without reprogramming or constructing a new mechanism, and it should be less expensive because a positioning mechanism is not required. However, the lack of a positioning mechanism makes it more difficult to guarantee the data points will be recorded in a geometrically regular manner; thus, some areas of the shape may be undersampled. The existing orthotic CAD application of a hand-held digitizer, which is the first of its type, records a data point only when the operator presses a mouse button (7). By pressing the button only when the digitizer is positioned correctly, it is possible to sample points in a geometrically regular pattern. The need to manually enter each point slows down data acquisition.

This article describes an alternative method in which a hand-held digitizer is operated in a continuously sampling mode. An interactive visual display of sampling quality is provided, allowing the operator to be certain all areas of the surface are adequately represented.

Ultimately, the accuracy of a digitizer also is a function of the method used to re-create the three-dimensional surface from the individual data points. The objective of this work is to quantify the accuracy of a hand-held digitizer and an accompanying surface reconstruction method.

Methods

Instruments

A free-floating, three-dimensional digitizing unit, the Fastrack device^a, was used. This device records the three-dimensional position and orientation of a lightweight transducer element at 120 samples/second. A transmitter creates a low-frequency magnetic field, allowing the position and orientation of the transducer element to be calculated from the signal generated in three miniature, mutually orthogonal antennae located within the transducer. The error due to the Fastrack device itself, in the present application, was 0.13 mm (root mean square error) or 5.33 mm (worst single error) (8).

A pivoting-arm ART Digitshape digitizerb was tested as a reference. The resolution for the ART digitizer was set at 72 points per slice (identical to the free-floating digitizer), and this digitizer's calibration was verified before each test session.

Software

When using a hand-held digitizer to continuously collect data, the sampling may not be totally accurate, and the data points may not be evenly spaced. In O&P, some areas of the limb may be very sparsely sampled while other areas are heavily or even redundantly sampled. For this reason, it was necessary to create a method to reconstruct the surface elevation map from data points collected with various sampling densities.

To reconstruct the surface in this study, data were continuously collected while the digitizer moved along a random path on the surface. A cylindrical coordinate system was used; thus, the data points represented surface elevations (that is, radii from a vertical axis, as is the convention in O&P CAD systems). As the samples were recorded, an image of the surface was generated in real time on an adjacent CRT.

The color of any small area of the image was defined so as to represent the quality index of the data sampled in that area. The quality index for a given small area was defined as a function of the number and standard deviation of the samples taken in that area. Thus, the color-coded image interactively guided the sampling process. Samples more than two standard deviations away from the current average for a given area were temporarily eliminated, and a new average and standard deviation were calculated. Areas in which no samples were taken were assigned the average value of nearest neighbors and were assigned a "dummy" color code indicating low quality.

The method is further detailed in Figure 1 . First, a cylindrical coordinate system was established by choosing three ordered noncoplanar points on the socket shape. The surface was divided into pie-slice-shaped compartments of 5 degrees in angle, 3.175 mm in vertical height and extending from zero to infinity in radius. Next, the digitizer was moved about the limb, continuously recording position data. As each position was recorded, the position was converted to cylindrical coordinates. The elevation (that is, radius) of the position then was placed into a compartment based on the angular and height coordinates. The mean and standard deviation of the elevation stored in that compartment then were recalculated.

Next, the compartment was drawn on the CRT, with position and size dictated by the angular, height and mean elevation values; color was dictated by the standard deviation. The process of sampling, calculating and drawing the color-coded shape on the screen continued until the operator pressed a keyboard key, causing an interrupt. At that point, the graphics could be altered (e.g., image rotated so the back could be seen), the coordinate system could be re-established, or the system could move to postprocessing.

When sampling was complete, the elevation map was then altered by a sampling density-weighted smoothing filter. This was done by adjusting the elevations in some compartments based on the average of the elevations of the surrounding compartments. The amount of smoothing applied at each compartment was an inverse function of the number of data points in that compartment. If there were 50 points in a compartment (the maximum), then no smoothing was applied. If there were no points in a compartment, then full smoothing was applied; that is, the elevation was set to the average of the surrounding compartments.

Between 0 and 50, the smoothing was linearly varied. The search for surrounding compartments first included those compartments immediately adjacent to the compartment being sampled; thus, the first search included nine compartments. If the total number of data points in the compartments searched was fewer than 200, the search was widened to include the next layer of compartments. Thus, the first attempt searched a nine-compartment square; if necessary, the second attempt searched a 25-compartment square, and so on. Thus, heavily sampled areas were smoothed less both because fewer (if any) of the original elevations were replaced and because these areas had smaller search zones. Finally, the smoothed elevation map then was input to the Shapemaker CAD softwarec so the shape could be carved.

Pilot Study

Previous testing of digitizers has consisted of measuring the total and/or segmental volumes of residual-limb molds (6,9,10). Limb-mold volume tests were used in the initial evaluation of the present system. However, it was discovered that hand-held digitizers have an advantage in these tests that may not be relevant clinically; this point may be illustrated by describing one of the limb-mold pilot tests.

A transtibial positive mold was digitized seven times with the hand-held digitizer and seven times with the pivoting-arm digitizer. Various measurements of the shape then were calculated using the Shapemaker CAD software. Specifically, the length and volumes from the midpatellar tendon level to the distal end were calculated. In addition, the circumference and the A-P and M-L diameters at the midpatellar tendon level were measured.

The results (see Table A and Figure 2 ) indicated the two digitizers had essentially equal repeatability in diameter and circumference measurement, but the pivoting-arm digitizer was markedly less repeatable in length and volume measurement. These results demonstrated the limb-mold volume tests give an advantage to the hand-held indentor that may not be clinically relevant in all cases. The fixed helical path followed by pivoting-arm digitizers means an error can occur in the vertical position of the points used to establish the coordinate system. This error has an average value equivalent to one-fourth the pitch of the helical path, or 6.35/4=1.59 mm for the commonly used pitch in pivoting-arm digitizers (2,11). Conversely, the hand-held digitizer can establish these points with repeatability equivalent to the accuracy of the digitizing unit (0.13-mm root mean square error observed in this application) (8). This difference in digitizer design was reflected in the length and volume results (see Table A) as these quantities were most affected by the vertical position of the points establishing the coordinate system. The midpatellar tendon level diameter and circumference measurements were much less affected, presumably because these quantities did not change significantly if the measurement plane was moved a few millimeters proximally or distally.

This design difference may have no actual clinical effect because the shape of a given mold will remain internally consistent. However, the design clearly affected the results in repeatability testing. Therefore, the experimental protocol was changed to one in which measurements could be made without reference to an external coordinate system. This meant that residual-limb shapes could not be used.

Data Collected

Three abstract shapes were used: a 89.1-mm-diameter cylinder, a 78.7-mm square column and a lobed column (see Figure 3 and Figure 4 ). The lobed column had a base diameter of 101.6 mm with a 3.33-mm amplitude sine wave superimposed to either side of the base diameter. The frequency of the sine wave was such that 10 cycles took place about the circumference (101.6 mm) of the column. The vertical spacing of the sine wave was identical; that is, 10 cycles in a distance equivalent to one circumference of the column. The 10-Hz frequency was an informal preliminary estimate of the high-frequency limit of socket shapes.

Figure 5 shows the frequency distribution of surface undulations occurring in a typical transtibial socket. This graph depicts the frequency distribution of the surface in all directions, calculated by two-dimensional discrete Fourier transform. The horizontal axis, frequency, expresses the number of sinusoidal oscillations occurring per 101.6p mm along the limb's surface. For example, if a graph of this type were generated for the lobed column, all of the power would occur at 10 cycles. As a second example, the smooth cylinder would show all power at a frequency of zero. For the typical transtibial limb, the frequencies of most surface contours are in the 1 to 4 cycles region (see Figure 5) , and nearly all of the contours have frequencies less than 10 cycles. (This estimate was based on an informal examination of the frequency spectra of residual-limb shapes. A more rigorous examination of this issue is underway; these results will be reported at a later date.) The 3.33-mm amplitude corresponded to the most curved path machinable with the 18-mm-diameter round-end mill.

These shapes were chosen to allow diameters to be measured without knowing the orientation of the coordinate system. The cylinder and square column masters did not vary in cross section. When maximum and minimum diameters were measured from the lobed column, it was obvious where these extremes were. Thus, the accuracy of reestablishing the coordinate system did not affect the results; only the reproduction of the shape was important.

Each master column was digitized twice with the hand-held digitizer and twice with the pivoting-arm digitizer. The resulting surfaces were exported to Shapemaker and carved using the standard clinical procedure. The carver used in all testing was the ART Carveshapeb.

The diameters of the shapes were measured as follows: The diameters of the cylinders were measured at seven locations on horizontal planes spaced every 2 cm down from the top surface. The square columns were measured at eight locations (across the flats and across the corners on four planes evenly spaced from the top to the bottom of the column). The lobed columns were measured across pairs of minimums and pairs of maximums. There were 10 sinusoids about the circumference: five pairs of minimums and five pairs of maximums on each plane. Seven planes were measured for a total of 35 minimum diameters and 35 maximum diameters on a given lobed blank. The master columns also were measured in an identical manner. A machinist's dial caliper (resolution 0.025 mm) was used for all measurements.

Results

For the hand-held digitizer, the average error was less than 1 mm for all tests (range 0.78 to 0.28 mm). The standard deviation (SD) was less than 1 mm for all tests with this digitizer (range 0.93 to 0.63 mm) except for measurement across the corners of the square column (2.13 mm). A worst single error of 3.22 mm was observed in measurement across the corners of the square column.

For the pivoting-arm digitizer, the average error was less than 1 mm in all tests (range 0.86 to 0.18 mm) except for measurement across the minimums of the lobed columns (2.16 mm). The SD was less than 1 mm in all tests with this digitizer (range 0.76 to 0.15 mm).

The results are indicated in Table B . Two blanks were used in each test case. In no case were the average measurements for the two blanks significantly different; thus, they were combined.

Figure 2 illustrates the performance of the digitizer on a transtibial residual-limb shape. In clinical use, data recording times of 90 seconds have been typical. It should be noted that for some test shapes-particularly those with an abundance of small contours (e.g., the lobed column)-data sampling with the hand-held method took substantially longer (typically 300 seconds).

Discussion

The objective of the study was to assess the accuracy and ease of use of a hand-held digitizer and an accompanying surface reconstruction algorithm. The average errors were under 1.0 mm in all tests with this digitizer. The SDs of the measurements also were under 1.0 mm except for measurements across the edges of the square column, where the variability in the measurements was greater. The behavior of the digitizer about the corners is shown in Figure 3 . In particular, note the surface about the corners has many small bumps while the surface of the flats is smooth. This behavior is a physical manifestation of the mathematics of the algorithm and is attributable to the portion of the algorithm that discards outlying data values within a compartment.

A square corner creates an infinitely rapid change in the slope of the elevation map. In compartments that span the degree corner, the data values tend to fall into more than one group. The algorithm assumes some of these groups are outliers and deletes them from the average elevation calculation. The bumpiness at the corners occurs because, given variable sampling density, the choice of which of the data groups is real and which is outlier becomes a function of which area was inadvertently sampled more densely; thus, the algorithm randomly shifts back and forth, producing the bumpy surface. This explains why, although the average error is small, the standard deviation is large.

Square corners are not common soft-tissue shapes (even the tibial crest has an ample radius); thus, this problem may be only of theoretical interest. However, at least two methods can be used to fix the problem. One method is to simply increase the resolution; that is, to decrease the compartment size. A second is to employ more sophisticated filtering methods, e.g., low-pass filter the surface map (12,13). Low-pass filters can remove the high-frequency random error ("noise") from a discretized signal but also may remove actual features of the signal. However, the topographic method described here has an advantage in that the measurement quality feedback code is available. Specifically, filtering may be applied so the characteristics in each area are adjusted to the local quality of the data. For instance, the pass frequency of a low-pass filter may be raised in areas of high quality. We assume areas where a precise surface contour is important will be more heavily sampled. The surface elevation map should, in general, converge to the "correct" profile with increased sampling; therefore, heavily sampled areas should not need much filtering.

For this reason, the pass frequency can be raised in these areas, and the low-pass filter will not affect these areas as much. Conversely, in areas that are smooth or where the precise surface contour is relatively unimportant, the surface may be very lightly sampled, relying on heavy filtering to produce a smooth socket shape.

While these improvements are possible, the data indicate neither is necessary for the digitizer to meet currently published criteria for digitizer accuracy. These criteria have specified accuracy in a radius measurement of 1 percent, or 1 mm, for clinical usefulness (9,10). Previous tests of various digitizers have reported repeatabilities in radius measurement ranging from 0.5 to 2.0 percent (6,9,10).

Note that measurements of the master lobed column produced SDs of 0.20 to 0.42 mm. These values reflect the variation actually present in the master combined with that occurring when measuring a soft plaster and cornstarch solid with calipers. Consequently, roughly one-third of the SD observed in the lobed column tests may be due to master variation and measurement technique rather than digitizer error.

This method of topography implies the following assumptions:

• The average of the samples for a given small area converge to the "correct" value with repeated sampling.
• The data are normally distributed about the actual population mean.

Another limitation is that although more than one field may be displayed on the surface, the number of fields that can be simultaneously displayed is limited by visibility.

Clinical Relevance

Hand-held digitizers are more flexible in use than current systems. These digitizers will record a wide variety of shapes without reprogramming or constructing a new digitizer-positioning mechanism. Aside from prosthetic sockets, the authors have begun using the system described here for TLSOs (see Figure 6) . In the authors' practice, pediatric TLSOs often are too narrow and distorted to digitize with a pivoting-arm digitizer. In such cases, the authors attach a second receiver to the TLSO mold and base the orientation of the coordinate system on this second receiver. This, in essence, attaches the coordinate system to the mold, and the coordinate system "floats" with the mold. The mold then does not need to be rigidly fixed and can be placed on a surface or held.

Relevant to a manually sampling hand-held digitizing system (7), the continuously sampling system discussed in this article may be faster and more user-friendly. Over-sampling provides a number of benefits. Error correction automatically occurs with further sampling. Variable sampling density means areas of the surface contour that are more critical (e.g., the fibular head) can be sampled more heavily, resulting in higher accuracy in the final topograph in that area. Also, if the orientation of the coordinate systems of the object and sensor becomes corrupted, the corruption is evident in that a string of areas of poor quality appears, trailing behind the current sensor position. This string of poor-quality color indicates the need to reorient the coordinate systems. Disorientation of the coordinate systems is a common problem in geometry measurement with living subjects (e.g., CT and NMR scans).

Conclusion

The accuracy of the hand-held digitizer and surface reconstruction algorithm was better than 1 mm in clinically relevant cases. Specifically, the accuracy was better than 1 mm even across high-amplitude, high-frequency surface oscillations but did break down over the discontinuity presented by the edges of the square column.

A hand-held digitizer may have a broader range of application than digitizers currently in use in prosthetics and orthotics and should also have a lower cost. Using the digitizer in a continuous sampling mode allows the shape to be recorded quickly and, through the use of a quality feedback code, may allow improved accuracy in critical surface contours. Increased resolution and more sophisticated filtering algorithms are being investigated as future improvements.

Acknowledgements

The authors gratefully acknowledge the support of Shriners Hospitals for Children Research Program Grant 9510.

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