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Journal of Neuroscience Methods 171 (2008) 264–270
Contents lists available at ScienceDirect
Journal of Neuroscience Methods
journal homepage: www.elsevier.com/locate/jneumeth
Spiral analysis—Improved clinical utility with center detection
Hongzhi Wang a,b , Qiping Yu a , Mónica M. Kurtis a , Alicia G. Floyd a ,
Whitney A. Smith a , Seth L. Pullman a,∗
a
Clinical Motor Physiology Laboratory, Department of Neurology, Columbia University Medical Center, The Neurological Institute,
710 West 168th Street, New York, NY 10032, United States
b
Computer Vision Laboratory, Department of Computer Science, Stevens Institute of Technology, Hoboken, NJ, United States
a r t i c l e
i n f o
Article history:
Received 26 September 2007
Received in revised form 21 March 2008
Accepted 21 March 2008
Keywords:
Spiral analysis
Movement disorders
Parkinson’s disease
Essential tremor
Dystonia
Motor performance
Curve fitting
a b s t r a c t
Spiral analysis is a computerized method that measures human motor performance from handwritten
Archimedean spirals. It quantifies normal motor activity, and detects early disease as well as dysfunction in
patients with movement disorders. The clinical utility of spiral analysis is based on kinematic and dynamic
indices derived from the original spiral trace, which must be detected and transformed into mathematical
expressions with great precision. Accurately determining the center of the spiral and reducing spurious
low frequency noise caused by center selection error is important to the analysis.
Handwritten spirals do not all start at the same point, even when marked on paper, and drawing artifacts
are not easily filtered without distortion of the spiral data and corruption of the performance indices. In this
report, we describe a method for detecting the optimal spiral center and reducing the unwanted drawing
artifacts. To demonstrate overall improvement to spiral analysis, we study the impact of the optimal spiral
center detection in different frequency domains separately and find that it notably improves the clinical
spiral measurement accuracy in low frequency domains.
© 2008 Elsevier B.V. All rights reserved.
1. Introduction
Handwriting or drawing can be used to study physiologic issues
in normal controls (NCs) and in patients with movement disorders
such as Parkinson’s disease (PD), essential tremor (ET), and dystonia
(DY) (Elble et al., 1990, 1996; Forsyth and Ponce, 2003; Hogg et
al., 2005; Liu et al., 2005; Louis et al., 2006; Siebner et al., 1999).
The most commonly measured parameters relate to the kinematics
and dynamics of writing and drawing. Compared to handwriting,
drawing requires subjects to execute standard figures, e.g. straight
lines, circles or spirals so more comparable information, such as
drawing shape and consistency, can be evaluated (Elble et al., 1990,
1996; Hogg et al., 2005; Miralles et al., 2006; Pullman, 1998; Wang
et al., 2005).
Spiral analysis is a clinical test that records Archimedean
spirals drawn on a digitizing tablet connected to a computer
(Pullman, 1998). It is used to quantify normal motor activity as well as measure dysfunction in patients with movement
disorders (Saunders-Pullman et al., 2008). With an inking pen
that also functions as a computer mouse, subjects draw spi-
∗ Corresponding author. Tel.: +1 212 305 1331; fax: +1 212 305 0743.
E-mail address: sp31@columbia.edu (S.L. Pullman).
0165-0270/$ – see front matter © 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.jneumeth.2008.03.009
rals in 10 cm × 10 cm with the starting point marked. Data are
collected in the x, y and z (pressure) axes and provide virtual
tri-axial recordings of spiral kinematics and dynamics. This analysis extends spiral drawing, a commonly performed neurologic
test, into an objective and accurate measure of motor performance.
Spiral analysis extracts specific quantifiable indices from the
handwritten spirals, such as linear and angular drawing speed,
acceleration, time-related changes in all these measurements, loop
width and consistency in the x–y plane, pressure in the z-axis,
among many others. Abnormal movements that are superimposed
on the spirals such as tremors are also evaluated. Multiple indices
can be combined to provide more complex analyses of spiral drawing, including an overall degree of severity. Compared to simple
drawing tasks, e.g. a straight line, spiral execution is a multi-jointed
task involving the distal and proximal upper limb and thus provides
a window on a wide range of motor problems (Hogg et al., 2005;
Lacquaniti et al., 1987; Wang et al., 2005). As explained below, an
important feature of spiral drawing is that it can be transformed
into an expression using polar coordinates with complete fidelity
to the original spiral shape.
The transformation of spiral data from image to polar coordinates results in a low order polynomial curve representation that
simplifies the extraction of the properties of the Archimedean spi-
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H. Wang et al. / Journal of Neuroscience Methods 171 (2008) 264–270
Fig. 1. (a) An ideal spiral with the center (0,0) the starting point of the spiral, indicated by the open circle, and another point (−10,−10) indicated by the open square below
and left of the true center; (b) the polar transformation of (a) based on the proper center (0,0); (c) the polar transformation of (a) based on the inaccurate center (−10,−10).
Note the markedly different transformation curve with spurious low frequency noise.
rals. Transformation of the two-dimensional spiral drawing into its
polar expression is as follows:
x = r sin ! + x0 ,
r =
!
y = r cos ! + y0 ;
2
(x − x0 ) + (y − y0 )2 ,
arctan ! =
y − y0
x − x0
where (x0 , y0 ) is the start of the spiral drawing (x, y) is the spiral
image coordinate and (r, !) is its counterpart in polar expression.
The Archimedean spiral polar expression has a simple polynomial
relationship, i.e. ! = k + arn , where a, n are positive constants and k
controls spirals orientation. The polar equations of spirals can be
“normalized” to an orientation where k = 0 as in Fig. 1, where a = 5,
n = 1. From the above equations, it is evident that the spiral center
is critical to their polar transformations. When the center is incorrectly identified, the polar expression becomes irregular (Fig. 1c)
with superimposed low frequency oscillations, which may result
in less accurately measured spiral indices.
Spiral onset can be taken many different ways, for example as
the geometric center of the 10 cm × 10 cm outline or as the first nonzero pressure value after the subject starts drawing (Pullman, 1998;
Wang et al., 2005). However, many subjects, particularly movement
disorder patients, do not execute spirals in a controlled manner at
initiation. Rarely do patients start from the center mark making the
first non-zero pressure point unreliable as the spiral center. Furthermore, poorly executed spirals and abnormal movements such
as tremors, superimposed on the spiral shape, cause artifacts that
distort the spiral data, alter the polar coordinates, and result in
improperly defined spiral centers. High order polynomials cannot
be utilized to define the spiral and its center because any irregular polar expression can be approximated to yield the original,
erratically initiated spiral. Without specific restrictions on the polynomial function, any (x, y) coordinate may falsely yield an apparent
spiral center.
To our knowledge, there is no automatic method to optimize
spiral center detection, and reduce low frequency drawing artifacts
to improve the clinical utility of computerized spiral analysis. In this
report, we describe a regression-based approach to achieve these
goals. We then test our technique in a disease classification task to
document the improvement in the clinical utility.
2. Experimental methods
2.1. Data collection
We obtain 10 freely drawn (no constraints, no attachments and
no templates to trace) handwritten spirals from NC subjects (n = 10)
and patients with PD, ET and DY (n = 10 for each condition), yielding a total of 400 spirals for the study. All subjects are instructed to
draw Archimedean spirals as well as possible inside a 10 cm × 10 cm
square on an 8.5 in. × 11 in. white paper mounted on a graphics
tablet. The drawing pen is wireless and there are no other attachments to the subjects. The tablet has a resolution of 10 pixels/mm, or
an accuracy of 0.1 mm/pixel, an output rate of 200 points/s, and 256
levels of measurable pressure. This provides approximately 70 Hz
sampling per channel. The captured spiral data are then normalized to have the same time interval between consecutive points via
linear interpolation.
2.2. Optimization through regression analysis
The polar coordinates of a spiral may be represented by a curve,
R(C) = {ri (C),! i (C)}i = 1:n , given a center C, where i is the ordered index
to spiral point and n is the total number of spiral points. With any
given center, there are infinite low order ideal Archimedean spirals passing any two consecutive spiral points, but an actual drawn
spiral cannot follow the ideal spiral definition exactly due to the
irregularities and inconsistencies of drawing different parts of the
spiral. We define that the local optimal center of an actual drawn
spiral is the one that is the most consistent with all spiral points.
Hence, the corresponding optimal spiral should have the least fitting error.
We perform the regression analysis in polar domain instead of
the image domain to take advantage of its simpler expression. We
assign R̂(C) = {ri (C), !ˆ i (C)}i=1:n to be the curve obtained by fitting
a low order polynomial to the curve R(C). We use the least square
fitting method, i.e. minimize the sum of squared residual error, to
obtain the fitting curve. We describe how to test the effects of the
order of polynomials in the next section. The optimal center, C̄, gives
the minimum fitting error:
"n
|! (C) − !ˆ i (C)|
i=1 i
C̄ = arg minC F(C) = arg minC "n−1
|! (C) − !i+1 (C)|
i=1 i
(1)
where F(C) is a normalized curve fitting error for center C. The
denominator is for normalization purpose so that F(C) is invariant to
the scale of the spiral. For simplicity, we use the least square fitting
method, i.e. minimize the sum of squared residual error, to obtain
the fitting curve. The curve obtained by least square fitting may
not minimize the sum of absolute residual errors, the numerator in
Eq. (1), but provides a reasonable approximation. Though one can
further refine the curve via gradient descent optimization, we find
that the least square curve fitting is sufficient to give good center
detection results. The normalized fitting error provides a measure
of spiral irregularity.
We then use the gradient descent approach to minimize the F(C)
function. Adjusting C in the direction of the derivative of function
F(C) with respect to the center, dF(C)/dC, allows C to converge at
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H. Wang et al. / Journal of Neuroscience Methods 171 (2008) 264–270
an local optimal center corresponding to a local minimum of F(C).
For simplicity, instead of the continuous searching space we only
search for optimal centers in the discrete image pixel coordinates.
The following summarizes our optimization procedure:
(1) Initialize the center estimate,
C0 = (x0 ,y0 ), to"
be the image center
"
0
xi /n and y0 =
yi /n.
of the spiral, i.e. x =
i=1:n
i=1:n
(2) Compute the F(C) for each pixel in the window within a small
neighborhood of the estimated center.
(3) If the pixel giving the best F(C) in the neighborhood window is
different from the current choice, move the center estimate to
this pixel, and repeat steps 2 and 3. Otherwise, set the current
choice as the local optimal spiral center.
The choice of the local searching window size is a compromise
between global optimal solution and computational costs. A large
local searching window has a better chance of finding the global
optimal solution, but at a higher computational cost. In this study,
we use a 21 × 21 pixel searching window, or 441 pixels. The optimization requires several seconds to converge using Matlab (The
MathWorks, Inc., Natick, MA) with a 2 GHz CPU computer.
2.3. Analysis of computer-generated spirals
To test our spiral center detection technique, we apply our algorithm using computer-generated spiral data measured in pixels.
The spiral data are synthesized with the polar expression using
! = r˛ , where ˛ = {0.8, 0.9, 1, 1.1, 1.2} and 50 ≤ r ≤ 400 with a constant step of 1 pixel. Thus, each spiral contains 351 sample points.
To test the robustness of this method, we deteriorate r and ! by
adding random noise evenly distributed over a radius [−5,5], in
pixels, and angle [−0.25,0.25], in radians, respectively. Image coordinates are then calculated from the polar coordinates and utilized
for test, with the image coordinate of the ground truth spiral center
as (0,0). For each ˛, we randomly generate 10 synthesized spirals,
resulting in a total of 50 spiral constructions. We defined the center detection error as the Euclidean distance between the detected
center and the ground truth.
We test the sensitivity of our optimization algorithm by initializing the center estimate to be the image spiral center plus random
noise evenly distributed in [−50,50]. We then search for optimal
centers using the algorithm outlined above for each spiral using
1st, 2nd, and 3rd order polynomial fitting separately, resulting in a
total of 150 trials.
2.4. Multi-scale analysis
To demonstrate the utility of our approach, we analyse the
impact of the local optimal center at different frequency domains
separately. To this end, we first decompose a spiral into different frequency domains using the Gaussian pyramid technique.
Then we compare the accuracy of the measured spiral indices,
evaluating drawing irregularities, with and without our optimal
center at each frequency domain separately. Widely applied in computer vision studies, the Gaussian pyramid technique is a form
of image processing that utilizes decreasing pixel resolution representations of the spiral as a form of low-pass filtering. As in
center detection, the Gaussian pyramid technique is applied on the
polar coordinates. We use the following repeated smoothing and
down-sampling procedures (Forsyth and Ponce, 2003; Lindeberg,
1994):
(1) Applying the Gaussian kernel method to remove high frequency
signals by smoothing the spiral curves. This smoothing was performed to avoid the aliasing effect of the next down-sampling
2
2
procedure. The Gaussian kernel is G(x) = e−x /2" , where " is
the deviation. We used " = 2, which as shown in (Yu, 2004) is
sufficient to avoid aliasing.
(2) Down-sampling the spiral by 2, where the jth element of the
down-sampled spiral was the 2jth element of the original spiral. Since the unit in the coarser scale had twice the magnitude
unit from the previous iteration, the spiral scale was coarser
at half the width and height of the previous drawing, with
one-fourth the area. The coarser spiral was visually equivalent to being observed at a distance twice that of the previous
scale.
Each of these iterations is applied to spirals already processed
for optimal center detection, gradually eliminating spuriously
low frequency oscillations in the polar domain. As the digitizing tablet samples at approximately 70 Hz per channel, according
to the Nyquist theorem (Forsyth and Ponce, 2003), the spirals
were recorded without aliasing up to ∼35 Hz. When the sampling
frequency is halved in 2nd scale, only signals up to ∼17 Hz are
preserved and the differences between 1st and 2nd scale spirals,
the radial oscillation speeds described below, contained signals
between 17 and 35 Hz. The amplitude of the radial oscillation
speeds corresponds to the signal power in this frequency domain.
Similarly, the signals preserved in 3rd, 4th and 5th scales are ∼8,
∼4 and ∼2 Hz, respectively. Since tremors and most other clinically
abnormal movements are composed of signals with frequencies
greater than 2 Hz, the 5th scale spiral is effectively without tremor
or abnormal movements. In our analysis, we decompose spirals into
five scales. Hence, we need to perform down sampling four times
for each spiral. Compared with other global frequency analysis
approaches, e.g. Fourier transforms, the Gaussian pyramid technique accurately locates the original signals at different frequency
domains in the original spiral (Fig. 3).
To measure the difference between the local optimal center and
pre-defined center with measured spiral quantities, we compare
the separation ability between clinically normal and abnormal spirals drawn by patients with PD, ET and DY. If the local optimal center
obtained by our method is better than pre-defined centers, the corresponding measured spiral quantities should be more accurate and
give better separation performance.
We compute the oscillation speed of tremors and other abnormal movements that superimposed on the processed spirals as
spiral quantities to capture the degree of abnormality. Given the
radius curve of a spiral {(C)}si=1:n at a scale s in the polar domain, we
compute the oscillation speed for each spiral point as follows:
(1) Remove the high-frequency abnormality from the spiral curve
by a smoothing algorithm using the same Gaussian kernel
above. Let the smoothed radius curve be {r̄i (C)}si=1:n .
(2) Compute the oscillation amplitude, {ai (C)}si=1:n , as the radial difference between the original and the abnormality free radius
curve, i.e. {ai (C) = ri (C) − r̄i (C)}si=1:n .
(3) Compute the radial oscillation speeds, {vi (C)}si=1:n , for spiral points as the absolute difference between the radial
oscillation amplitudes of consecutive spiral points, i.e.
{vi (C) = ai (C) − ai−1 (C)}si=1:n−1 . The points with higher radial
oscillation speeds are considered as more likely containing
abnormality than those with lower radial oscillation speeds.
According to the central limit theorem, the distribution of the
average radial oscillation speed at each of the five scales is computed and assumed to be Gaussian for both normal and abnormal
spirals (Bishop, 1995; Lacquaniti et al., 1987). This is performed
by modeling a class mean and a class variation (#s , " s ) for each
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H. Wang et al. / Journal of Neuroscience Methods 171 (2008) 264–270
267
Fig. 2. Row 1, a sample computer-generated spiral used in our test. Row 2, spiral drawn by a normal control. Rows 3–5 are representative spirals drawn by patients with
increasingly abnormal spiral execution. Column 1: spirals with the detected local optimal centers, where the x-axis and y-axis are labeled in pixels. Column 2: polar expressions
with first non-zero pressure points taken as the spiral centers (polar expression with initial center for the synthesized spiral). Column 3: polar expressions with locally optimal
centers. The fitted curves are also illustrated in columns 2 and 3. For columns 2 and 3, the x-axis is labeled in radians, the y-axis is labeled in pixels.
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H. Wang et al. / Journal of Neuroscience Methods 171 (2008) 264–270
Fig. 3. Left: the multi-scale representation for a sample spiral from finer to coarser scales. The spiral is displayed in pixels at 0.1 mm/pixel, the resolution of the digitizing
tablet. The corresponding radial oscillation speed traces (right) contain high frequency signals sequentially removed from one scale to the next (see text for details). x-Axis:
spiral points. y-axis: unit in 1 mm/s. A level radial oscillation speed (horizontal line) is shown in the right traces for comparison cross-different scales, and corresponds to a
constant power at different frequency domains. Note that tremors in this example are present only in the first three scales.
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H. Wang et al. / Journal of Neuroscience Methods 171 (2008) 264–270
scale which are estimated from training data. Given two estimated
classes (#s1 , "1s ), (#s2 , "2s ) and a testing spiral with measured average radial oscillation speed xs , classification was performed at each
scale by evaluating the following probability ratio:
rs =
(1/"1s ) exp[−(xs − #s1 )2 /2"1s2 ]
(2)
(1/"2s ) exp[−(xs − #s2 )2 /2"2s2 ]
If r > 1, the probability that the new spiral belonged to the first
class was greater than the probability that it belonged to the second
class.
We perform pair-wise classification tests between NC against
patients with movement disorders, classifying NC vs. PD, NC vs.
ET and NC vs. DY at each of the five scales separately. To avoid
the bias introduced by partitioning the data into training and testing sets, we apply the “leave one out” cross-validation strategy
(Bishop, 1995) for each classification test. At each iteration, one
spiral from each of the paired testing classes is chosen for testing
and the remaining spirals are used as training data. This training
and classification is repeated until every combination of the spiral
pair is selected for testing. We need to perform 100 × 100 or 10,000
training and classification iterations for each classification experiment. We measure the classification performance at each of the five
scales to obtain profiles in each frequency domain. We also measure the classification performance using the aggregate of the five
scales to determine the overall classification performance using all
information (Table 1).
3. Results
From the computer-generated spirals, the average center detection errors using 1st, 2nd, and 3rd order polynomial curve fitting
are 10.91, 10.78, and 10.58 pixels, respectively. The number of found
local optimal centers with error larger than 20 pixels, ∼5% of the
largest radius, is 2, 2, and 2, respectively. It is evident that this
approach shows robustness against noise and initialization. Furthermore, we did not observe large difference using different order
polynomials for curve fitting. For the remainder of the testing, we
use only 3rd order polynomial fitting. Fig. 2 gives an example of the
synthesized spiral used in this test.
Fig. 2 illustrates sampled data using our algorithm for improved
spiral center detection in a series of actual drawn spirals of increasing abnormality. Note that the optimal center location behaves as
a filter that removed the low frequency sinusoidal curve from the
polar coordinates caused by an inaccurate center. High frequency
tremors and other abnormal movements, however, are not affected.
Employing the Gaussian pyramid technique, we separate and
quantify superimposed tremors and other complex movements.
Fig. 3 shows an example Gaussian pyramid series for a sample
patient spiral. Since smoothing with Gaussian kernel is equivalent
to low-pass filtering, the spiral at the next coarser scale only contains the low frequency signal of the previous scale. The difference
between two consecutive scales reflects the remaining high frequency signals. Hence, multi-scale analysis decomposes spirals into
different frequency domains.
Fig. 4. The mean radial oscillation speeds of the spiral data measured with predefined spiral centers for normal controls and patients (NCpre, ETpre, DYpre, PDpre),
and with optimally determined spiral centers (NCop, ETop, DYop, PDop). x-Axis: scale
index for multi-scale analysis. y-Axis: the average radial oscillation speed (mm/s).
Our optimal center detection did not significantly affect the classification performance at the1st and 2nd scales, which indicates
that using the local optimal centers does not affect high frequency
spiral information. However, at the 3rd, 4th and 5th scales, our
method significantly improved the classification performance as
shown by average correct classification rates in Table 1. For DY
and ET, the improvements are notably the greatest at 7% and 12%,
respectively. For further demonstration, the average radial oscillation speed at each of the five scales is represented for each spiral.
A direct comparison of the mean average radial oscillation speeds
for each clinical group – based on spirals with centers detected by
previous methods and by spirals with optimal center detection – is
illustrated in Fig. 4.
4. Discussion
Computerized spiral analysis is used as a clinical test to record
Archimedean spirals drawn on a digitizing tablet (Pullman, 1998).
While highly successful in quantifying aspects of upper limb motor
control (Elble et al., 1990; Louis et al., 2006) and in diagnosing and
detecting early changes in movement disorders such as Parkinson’s
disease (Saunders-Pullman et al., 2008), defining the center of a
drawn spiral can be problematic. In this report, we improve on the
detection of the center of the Archimedean spiral based on regression analysis. With the new algorithms, the locally optimal spiral
center was shown to result in a more regular polar transformation.
We found that accurately determining the center of the spiral subsequently reduces spurious low frequency noise and improves the
overall utility of spiral analysis.
Applying the new center selection process to spirals drawn by
normal controls and patients with movement disorders resulted in
Table 1
Classification results list by percentage correct
1st scale
NC: optimal
NC: predefined
2nd scale
3rd scale
4th scale
5th scale
All five scales
PD
ET
DY
PD
ET
DY
PD
ET
DY
PD
ET
DY
PD
ET
DY
PD
ET
DY
97
98
98
98
84
79
97
99
100
100
88
88
85
82
99
99
88
83
74
68
88
83
81
76
57
58
76
64
73
66
98
98
100
99
90
89
Multi-scale results comparing control subjects to each patient group using optimal and pre-defined centers. NC = normal control; PD = Parkinson’s disease; ET = essential
tremor; DY = dystonia.
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modest but improved classification rates in low frequency domains.
We found that the improvement is as high as 12%, which is
notable considering the clinical subtleties and overlapping kinematics of movement disorders (Deuschl et al., 2001). This result
demonstrates that our local optimal spiral centers can significantly
improve the accuracy of measured spiral indices at low frequency
domains by removing artifacts caused by inaccurate spiral centers.
Spiral centers calculated with the new algorithm revealed
greater analytical accuracy, improved clinical classification rates
and decreased false positive tremor assessments compared to spiral analysis without optimal center detection. The center detection
algorithm, combined with the Gaussian pyramid and multi-scale
spiral representations, enhanced the clinical utility of spiral analysis. These new methods thus offer an improvement in spiral analysis
measurements and diagnostic utility in assessing clinical motor
performance.
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