Baharav_Capacitive Touch Sensing Signal and Image Processing Algorithms

Capacitive touch sensing : Signal and image processing algorithmsZachi Baharava and Ramakrishna Kakaralab a Corning b School West Technology Center, Corning Incorporated, 1891 Page Mill Rd., Palo Alto, CA of Computer Engineering, Nayang Technological University, Singapore ABSTRACT Capacitive touch sensors have been in use for many years, and recently gained center stage with the ubiquitous use in smart-phones. In this work we will analyze the most common method of projected capacitive sensing, that of absolute capacitive sensing, together with the most common sensing pattern, that of diamond-shaped sensors. After a brief introduction to the problem, and the reasons behind its popularity, we will formulate the problem as a reconstruction from projections. We derive analytic solutions for two simple cases: circular finger on a wire grid, and square finger on a square grid. The solutions give insight into the ambiguities of finding finger location from sensor readings. The main contribution of our paper is the discussion of interpolation algorithms including simple linear interpolation , curve fitting (parabolic and Gaussian), filtering, general look-up-table, and combinations thereof. We conclude with observations on the limits of the present algorithmic methods, and point to possible future research. Keywords: Capacitive Touch Sensor, Multi Touch. 1. INTRODUCTION Touch is one of the 6-senses humans use to feel and interact with their environment, and has long been a key component in human-computer interactions. It has come to the forefront in recent years with the explosion in electronic devices using it, most notably mobile phones. Prior to the development of touchscreen phones, touch sensing was used in many places including computer track-pads. With such a wide range of applications, there are many different requirements and constraints on the sensing system.1 For example, some systems need to sense both human-finger, stylus, pencil, or pen, while others are tailored only for sensing sub-sets of those. Some systems, like smart-phones, have to be “clear”, meaning one should be able to see the content (usually an LCD screen) underneath the systems, while other systems can be opaque (like trackpads). Yet some systems need to be able to determine the locations of multiple fingers touching the screen, while other systems need merely to determine whether a button was pressed or not. And of course, there are differences in the size-constraints, power consumption, accuracy requirements, cost, and so on. Moreover, the wide range of requirements gives rise to different methods to sense touch. The most common technique today is still resistive touch sensing, which relies on two conductive layers separated by a small space, and when the user touches, a conductive path is created between the two layers. Note that this method relies on the pressure created when the user touches, and not on the presence of the finger/object per-se. Other methods to sense touch include optical methods, acoustic-waves, resistive-nodes, and various types of capacitive sensing: mutual, absolute, and surface. For a general review of touch technologies, which is updated frequently, a very good resource is Jeff Walker’s presentation.2 For a treatment of the history of capacitive sensors see Baxter’s classic book.3 In this paper we are focusing on absolute capacitance touch sensing, used to determine the location of one finger over a sensing-area. This method used to be the most common method for capacitive sensing, though in the last three years the focus has moved in many segments to mutual capacitive sensing, which enables a more detailed sensing of the space by creating a capacitance-image of the area sensed. Mutual capacitance is Further author information: Zachi Baharav - [email protected] , Ramakrishna Kakarala [email protected] Absolute capacitance sensing systems can be divided in general into three main ingredients: physical sensor. This is the regime where those two bodies constitute a parallel plate capacitor.2.4 we have ǫA . and keep all of them as close to the finger as we can. 4. (3) . we will be wasting signal that could have been captured. In addition. we can deduce that the signal (or change of capacitance on the sensor) due to a finger will be proportional to the overlapping area between the finger and the sensor. . we focus in this paper on various methods to calculate the coordinate given the sensor data. used in the iPhone c and in other devices capable of multi-touch sensing. Since the capacitance between the finger and the trace can be thought of (in zero-order approximation) as a parallel-plate capacitor. We close with Sec. following up with the parallel-plate capacitor model for the interaction between the finger and the sensor. (6) describes the simulations and the results. and in Sec. the larger the signal is. (5) discusses different methods for calculating the coordinates of the finger. Typical structure of absolute capacitance sensing system. and determine its coordinate (location) over a rectangular area. (4) we analyze a special case. and d is the distance between the plates. and signal processing. which is the leftmost part in Fig. Never the less. and a finger location is calculated. PROBLEM FORMULATION In this section we describe an absolute capacitance sensing system. and point to possible future work on the subject. where the sensor is composed of thin-wires. We also derive an analytic result for square fingers in Sec. which is used to sense one finger. If we leave some area unused. the smaller the distance (d) between the finger and the sensor. we need to have as much trace-area (A) exposed to the finger. We touch briefly on projection-approach in Sec. This will lend itself to analytic solution. 1 is the element that serves as the capacitor to be sensed. this change is measured and analyzed. A is the area of the plates. Sec. (1) CP arallel−Capacitor = d where epsilon is the dielectric constant of the material between the plates. As the final goal of the sensing in our case is to determine a finger-touch coordinate. for example. 2. (2) we describe the general system we will be using. Also. (7) . The physical sensor. or where there is no real need for capacitive imaging capability. electronic measurement. absolute capacitance is still heavily used in many applications where a less-expensive solution is required. From (1) it follows that in order to have a large signal from an approaching finger. When a finger (or other conductive element) comes close enough to change this capacitance value. The structure of the paper is thus as follows: in Sec. We will delve into this further when we talk about the measurement shortly. and Sec. We will use this fact extensively later on in the analysis and simulation parts. We measure its capacitance to the external world. Touch Sensor Physical Sensor Electronic Measurement Signal Processing Figure 1.the method. but it is worth remembering that we want to cover the sensing area with electrodes as much as we can. and lay down the terms to be used in the sequel. 1. however we want to emphasize that this is far from a trivial problem. 2. Most of the methods today use techniques involving modulation of the signal. noise in the system. The most common pattern for sensing absolute capacitance is described in Fig. where the rows and columns have the special “concatenated diamond” shape. gesture recognition. In general. follow essentially the same idea. It is composed of two layers. and at the same time give good results in terms of responsiveness. tap or draw. accuracy (local and global). one can consider the measurements as “projections” of the capacitance “image” onto rows and columns. and gestures interpretation. as this is beyond the scope of this paper. etc). for example : noise (from LCD. and more. and is known as the diamond-pattern (for obvious reasons). averaging. like signal conditioning. and other noise-avoiding tricks to facilitate the measurement. not to produce disturbing instabilities that will frustrate the user. 1. and below it the rows-layer. whereas the background capacitance of the sensor can be around 50pF). and usually convert it to digital-representation to be processed by the signal-processing block. We will not delve into the measurement process. tracking. imposing known charge on the sensor and measuring the resulting voltage. This configuration of the electrodes complies with the goal of having maximum exposure to the finger. tiling the whole surface. or alternatively. etc). This involves both traditional signal-processing. there are two different methods to measure it: imposing known voltage on the sensor and measuring the resulting charge on it. though they apply to other configurations as well. one called the columns-layer. the last part in Fig. One needs to deal with various issue. In this pattern. is the part that measures the change in the physical property (capacitance). Most of our simulations will follow the use of this pattern. translates it to electronic signal. It does present the issue of the need for two layers. Illustration of the diamond pattern of electrodes. which is the most common pattern for absolute capacitance sensing. Signal processing. converts the signals from the sensor into finger coordinates. as otherwise the rows and columns will be short circuited. We will delve more into the requirements as we get to evaluate the different algorithms. need for high resolution in the reading in order to achieve good location accuracy (in the order of a few femto-Farads). since capacitance describes the proportion ratio between voltage and charge. filtering (smoothing).Electronic measurement. and in addition algorithmic parts to analyze the data and deduce finger(s) location. small signal-changes due to finger compared to background signal (the change due finger can be a few pico-Farads. This leads us to the . Other methods. which measure the complex impedance of the sensor. and most importantly. other electrical components. and symmetrical for rows and columns. The final result of this chain (sensor+measurement+analysis) should be robust to different usage modes (small/big finger. R1 R2 Rows R3 R4 C1 C2 C3 C4 Columns Figure 2. manufacturing variability in the sensor. and so on. the middle element in Fig. . we will look for a stable solution. which is “best” according to some criteria. Suppose that finger capacitance is represented by a two-dimensional (2-D) function f (x. Space does not permit a derivation of the transform. The Moto-labs study5 of finger tracing accuracy on various phones shows the devices on the market have significant errors when tracing straight lines. which lend themselves to analytic analysis. and analyzing various solution options. where the finger imprint on the capacitive image can be of any shape. however. where F (u. In the general case. that the Fourier transform at frequency ω of the projection at any angle φ is G(φ. ω) = F (ω sin φ. In other cases. 4. However. ANALYTIC SOLUTIONS In this section we address two particular cases. temperature. we leave those to future work because the purpose of this paper lies in investigating practical interpolation methods. The details are interesting. there are cases where one can look for exact solution. as we will show. We have too few measurements to enable a unique correct solution. One may wonder if this is a difficult problem: hasn’t it been solved already many times? After all. and we may have multiple fingers altogether. the study found that the resulting lines reconstructed on the screen are often wiggly. with Λ∗ denoting the dual lattice. and variations in the finger model (of how finger looks like). If Λ is a hexagonal 2-D sampling lattice having fundamental unit area A(Λ). PROJECTION APPROACH Though the purpose of this paper is to investigate interpolation algorithms. manufacturing). it is useful to consider the inherent projection of the finger capacitance onto the row and column traces as a type of Radon transform. v) is the 2-D Fourier transform of the function f (x. find the location of a finger on the sensor.following problem formulation: Problem definition: Given the projections of capacitance-image over the sensor. The rest of this work deals with addressing the above problem. Expanding a little more on this. but it is worth noting that the projection process may be modelled as follows. and the observed signal is therefore Fo (u) = F (u)H(u). However. The Fourier transform of the diamond-shaped aperture is denoted H(u).6 We know. u denoting 2-D vectors. note that this will be a 2-D sinc function rotated 45-degrees. and summing the resulting samples. though the robotic movement is straight. this is of course an ill-posed problem. Then the aperture serves as a lowpass filter. −ω cos φ). there are many devices on the market using this technology already. We may conclude that the current generation of touchscreen phones has not solved the problem. similar to what is studied in medical imaging. and used a robotic-finger to trace a grid pattern of straight lines on the screen of each one. by the Fourier slice theorem. then the Fourier transform of the samples is Fs (u) = 1 A(Λ) Fo (u + k) k∈Λ∗ (2) (3) We can now derive the Fourier transform of the projection applying (2) to (3). with x. The nature of the projection as shown in Figure 2 involves sampling the averaged value over a square aperture. y). where robust means accuracy and stability. we can elaborate: Expansion on problem definition: The resulting coordinates should be robust to variations in the measurements (noise. The study examined six different phones. 3. y) = f (x) with Fourier transform F (u). and further investigation in the areas discussed in this paper are necessary. depicting only the columns sensors. as depicted in Fig. and we will denote those values as c1 and c2 . Since we assume the response is proportional to the area covered by the finger.1 Wire Grid Sensor In this section we will derive a solution for the case where the sensing electrodes are simple wires. From substituting the first two in the third one. 4. This translates to the requirement that d > Rf > d/2. we immediately note that c1 and c2 are nothing but the respective chords of a finger-circle. respectively. d R1 d R2 R3 R4 R1 R2 R3 R4 2 ∗ Rf Rows C1 C2 C3 C4 C1 C2 C3 C4 Columns (a) (b) Figure 3. (b) A round finger of size 2 ∗ Rf superimposed on the sensor from (a). In this case. These comprise 3 independent equations. Fig. only columns 1 and 2 will have readings. and since we assume thin wires with negligible overlap between themselves. unless at the edges of the sensor.4. Since the response of the sensor to the finger is proportional to the area overlap between the two. the overlap between the wires reduces to such a small area. that we will also neglect that factor. (a) Basic configuration. In (a) the wire-grid is drawn. we get one (non-linear) equation with only one unknown Rf : d= Rf 2 − c1 2 2 + Rf 2 − c2 2 2 . with 3 unknowns: x1 . where d is the spacing between traces. 4 describes such a scenario. and the notations in Fig. c1 2 c2 Rf 2 − 2 Rf 2 − 2 (4) . 3. Wire-grid sensing electrodes pattern. With the help of some basic geometry. (8) . the response to a round finger is reduced to being proportional to the length of the appropriate chord in the finger-circle. Now. or the finger size. Moreover. it will always impact the reading on exactly two columns and two rows. and Rf . x2 . where we can assume each wire is much narrower than the pitch size. let us assume that the finger radius is such that. we can write the following 3 equations: x1 x2 x1 + x2 = = = d. 2 (5) (6) (7) . the constraint means that we obtain a non-zero reading on at least one trace Ri and one trace Cj . but for the case of a square finger. y) coordinates of the center of the finger. the fact that Rf is the same for both axes. we can use a two-dimensional look-up-table to solve the equation in real-time.The above is an exact solution for determining the x-coordinate and Rf . cj values. • Separation of variables . it is a closed solution. we see that. we can exploit this to solve the whole system of equations together in order to gain robustness to noise. There are a few things to note about this result: • Analytic solution . 4. aligned to the grid. the radius of the finger. Stating it yet differently. (9) Here again d is the spacing between the electrodes.2 Analytic solution for square finger We follow the method used above. Notations for analytic computation. . but not more than two of either the row or the column readings are nonzero. Considering the fact we have three unknowns ((x. creates the coupling that can be exploited to overcome noise. within this block. and thus is more robust. However. W .d x1 x2 Rf C1 C2 C3 C4 Figure 4. Assume that the width of the square. c2 ). then the finger fits within the basic block of Figure 5. we can use the same table for the X and Y axes. keeping in mind we do have 4-measurements for only 3 unknowns. and Rf the radius). it is not surprising we are able to find a unique analytic solution. satisfies a constraint similar to (4): d > W > d/2. The size of the LUT depends on the resolution we want in the X/Y directions. We now show that there are in general four symmetries. • Solving in real-time: LUT . but is considering the possibility the noise. and 4 measurements (2 on rows and 2 on columns). the finger position is not uniquely determined by the ri .Since we can write the equation in terms of Rf = Rf (c1 . Note that the diamond grid in Figure 2 is the same as a square grid rotated 45 degrees. Still.It is worth noting that the ‘X’ and ’Y’ coordinates can be computed independently. and that the contours of equal readings are hyperbolic in shape. and the resolution of our measurements. Suppose that we now have exactly one square finger. With respect to Figure 2. When it produces a nonzero reading on one and only one Ri and Cj . Denoting as before the readings on each trace R or C by lowercase r or c. In this very symmetric sensor case. is formally similar to the problem of finding the peak location of a continuous function to subpixel level. then so is (uy . SUBPIXEL INTERPOLATION Suppose that a finger gives readings on more than one electrode ri and more than one electrode cj .7 Below. ri+1 ). To gain insight into the estimators used in the table. with W = 75. (10) Neglecting the constant term W 2 on the right. A basic block of 2 × 2 diamond sensors is shown. cj √ ux . We can solve (10) for uy as follows: ri − (W − ux )W uy = . and the trace spacing is d = 100 2. ci+1 ). We describe briefly several popular algorithms for subpixel peak estimation. ri−1 is the row below the maximum. it can be seen that the contour lines of the 2uxuy term are hyperbolic in shape. ri−1 + ri + ri+1 (13) . The problem of finding the position to a resolution smaller than d. with the horizontally connected row traces and vertical column traces indicated. which is larger than a single diamond. but small enough to fit within the basic block. ux ). uy space. as well as (W − ux . (11) 2ux − W We obtain the same equation for the column reading on the plain electrodes in Figure 5. the spacing of the diamond-shaped blocks in Figure 1. ri . has width W . Similar equations hold for the column readings (ci−1 . on The contour lines were obtained by simulation. From the table. for example. W − uy ) and (W − uy . and similarly for the column readings. uy that fit within a basic block is therefore 25. hence there is a four-fold symmetry to the solution space. The range of coordinates ux . uy ) is a solution. The square finger. Table 1 summarizes four different algorithms for estimating subpixel location of the finger from the three samples (ri−1 . 5. let us consider. namely that cj = W (ux + uy ) − 2ux uy (12) Clearly. Figure 6 illustrates the symmetries by showing the contour lines of ri .W uy u x Ri Cj Figure 5. and W . ri + cj = W 2 . an equation which when combined with (10) and (12) gives us three equations to solve for ux . W − ux ). assumed to be aligned to the grid. uy ) is ri = ux uy + (W − ux )(W − uy ) = 2uxuy + W 2 − (ux + uy )W. From Figure 5 the reading on the hatched electrode for a square finger having position (ux . ri+1 is above. the equation is δ= ri+1 − ri−1 . uy . Note that by the algebraically-symmetric form in (10) and (12). if (ux . ci . and that the contour lines of the −(ux + uy )W term provide a linear correction. assume that ri is the maximum value of the row trace readings. the estimator δ using the center of mass method. Illustration of four-fold symmetry in contour lines for row (column) readings for a square finger that fits in a basic block. so moving diagonally upward in this figure is equivalent to moving the finger horizontally in Fig. note. up to the log function. respectively. uy both increase. we may set the 2 denominator ri−1 + ri + ri+1 to 1. This figure.ri+1 } ri−1 −ri+1 1 2 ri−1 −2ri +ri+1 Table 1. is best viewed in color. because the exponent of a Gaussian distribution is parabolic. as we now show. Estimator Gaussian Center of mass Linear Parabolic Equation ln ri−1 −ln ri+1 1 2 ln ri−1 −2 ln ri +ln ri+1 ri+1 −ri−1 ri−1 +ri +ri+1 ri+1 −ri−1 1 2 ri −min{ri−1 . Similarly. by dividing both top and bottom by the sum. and solving for the mean of the distribution. ri . see Fisher and Naidu7 for details. this implies |δ| ≤ 1 . ri+1 ) form a discrete probability distribution located on the indices (−1. for example. The arrows indicate the direction in which r or c increase. +1). and δ serves as the subpixel offset. For any value of δ. some insight may be gained by noting that the parabolic estimator is the solution of a second-order Taylor series expansion of the finger capacitance. 0.ri contours 25 20 15 r 10 5 5 10 15 ux 20 25 25 20 15 10 5 5 cj contours uy uy c 10 15 ux 20 25 Figure 6. The center of mass estimator may be derived by assuming that (ri−1 . the Gaussian estimator is similar. the linear estimator assumes that the shape of function is a straight line on both sides of the peak. The numerator has upper and lower bounds ri+1 ≥ ri+1 − ri−1 ≥ −ri−1 (14) . that r increases if ux . Note that we assume that ri is the peak value of the distribution. the estimated location of the peak is i+δ. Hence δ ≤ 2 and we can show similarly that δ ≥ − 2 . Subpixel finger position estimators The upper bound ri+1 is reached if ri−1 = 0. and others to follow. which means ri+1 ≤ 1 since ri+1 + ri = 1 and ri ≥ ri+1 by the 2 1 1 assumption that it’s the peak location. where i is the integer index of the maximum row reading. For the other estimators. 5. In (13). and so on). namely the distance between the center of any two rows (or columns) was discretized into 100 cells. the basic pitch. that of circular finger. and considered two simple case. and for each location compute the profiles. 9(b-c). and indeed one can see the different impacts of noise on the various algorithms. as described in Fig. 10. In (b) a single column is described. presents a fertile field for investigation. which is where we simulated it in the first place. The diamond pattern used is described in Fig. Taking it one step further. We then introduce multiplicative noise component to the raw profile data. We describe interpolation algorithms. .6. We used a Gaussian profile truncated to fit into a diameter of 3 columns and rows. We do this by placing the finger with a center on a known location of the sensor. white. and the results are given in Fig. 9(a). will also effect the different algorithms differently. Keep in mind we also know the exact location of the center of the finger. In (a) we describe a single row of diamond pattern. the projections approach. and calculate “simulated” location. The amplitude of the noise is varied. Non-separable algorithms. and a simulated finger. and compare results. which is the result of the interaction between the finger trajectory and the underlying periodic diamond pattern. different finger sizes. that of absolute capacitive sensing. we can then simulate the finger performing a trajectory over the sensor. SIMULATIONS AND DISCUSSION A Matlab simulation was used to compare the various methods discussed under different conditions. In addition. This will give us the reading of the rows and columns. One can compare the algorithms under different kind of noises (additive. Each row and column are colored differently. We used a round finger for the simulation. are also of paramount importance. Therefore. plausibly using an LUT implementation. it means that the diameter of the finger influence is 15mm. In our simulation. and evaluate their performance. which are shown in Fig. we can then compute a location according to various algorithms. Some algorithms are much more sensitive to noise. We can then compute the overlap between the finger (as weighted by the profile) and each of the sensors (rows or columns). and thus we have 12 + 16 = 28 different colors in the picture. Typical dimensions for the row and column spacing (and size) is about 5mm. The results of such a process are described in Fig. We simply put it each time in a new location. Specific algorithms which are optimal with regard to insensitivity to finger size or shape (within limits). Results for those cases will be shared in the presentation. 7. we can also calculate error measures. SUMMARY AND CONCLUSION In this work we discussed the most common method of projected capacitive sensing. that of a square finger. This will give us indication of periodic errors. We repeat the analytical approach for a second case of interest. Thus. we can evaluate the interaction between those. and added a profile to it’s capacitive characteristics. 11. This takes into account the thickness of the cover-glass and fringe-field effects. In addition. we derive analytically the solution for finger location. a pixel in the simulation is of size 50µm. and errors which depend on the location of the finger relative to the grid. and thus takes into account fringe-fields etc. We analyzed the problems involved in determining finger location from the most-common type of sensors. In addition. might give us a hint at the optimal solution. correlated. Fig. assuming a 5mm pitch. 7. and there are many more issues to explore. In (c) we combine rows and columns to create a full sensing panel. There are 16 rows in our simulated panel. Keep in mind the size is the one captured by the capacitive sensing electrodes. using inverse transform of the projections. Once we have a simulated sensor. and shapes. in terms of maximum-error and in terms of mean-error. This work just touched the tip of the iceberg. which is around 7mm to 15mm in diameter. 8 combines different views of the simulated finger. it is evident that some algorithms perform better than others. We formulated the problem as a reconstruction from projections. Since the pitch is 5mm. One can clearly see the periodic nature of the error. Using these profiles as a starting point. In the first case. diamond-shaped sensors. This has to agree with the expected finger size. There are 12 columns in our simulated panel. . Simulated finger profile used in the simulations: (a) Color depicting magnitude.4 3 1 2 3 0 1 2 3 (a) (b) Figure 8.7 0. (b) Cross section of profile.2 4 6 8 10 12 14 2 4 6 8 10 12 1 16 1 (a) Full panel: 28 traces 2 4 20 6 8 10 10 12 14 16 5 15 25 (b) 2 4 6 8 10 12 (c) Figure 7. Panel diamond pattern: (a) Single row of diamond pattern. (c) Combining rows and columns to create a full sensing panel.5 0.8 1 0. Finger cross section 1 Finger 0.(b) Single column of diamond pattern.9 0.6 2 0. .385795 // 0. (c) Response on the columns. traveling along a straight line.385795 // 0.700224 // 0. The exact travel location is depicted as a Blue line. Blue − exact route Red − COM (max/mean error = 0. (b) Response on the rows. (a) Simulating the finger at a different location. and for each location calculating the response on the traces (rows and columns).317378 [mm]) Black − Gaussian = 0.417365 [mm]) Green − Linear = 0.519937 // 0.252979 // 0.148013 [mm]) 500 550 600 650 (a) (b) Figure 10.148013 [mm]) 1600 1400 1200 1040 1000 800 600 400 940 200 920 100 200 300 400 500 600 700 800 900 1000 1100 1200 900 450 1020 1000 980 960 1100 1080 1060 Blue − exact route Red − COM (max/mean error = 0.519937 // 0.209413 [mm]) Cyan − Parabolic = 0.417365 [mm]) Green − Linear = 0. (b) Zooming in on part of the graph in (a). and from that the estimated computed location. where the others depict different interpolation algorithms (see title of picture for more details).209413 [mm]) Cyan − Parabolic = 0.252979 // 0. (a) Finger superimposed on the Diamond pattern.Array with finger area super−imposed 2 4 6 8 10 12 14 16 2 4 6 8 10 12 (a) Traces response: rows 10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 0 0 2 4 6 8 10 12 14 16 0 1 2 3 4 5 6 7 8 9 10 11 12 4000 6000 10000 12000 Traces response: cols 8000 2000 (b) (c) Figure 9.700224 // 0.317378 [mm]) Black − Gaussian = 0. G.532050 [mm]) Black − Gaussian = 1. [3] Baxter. Siciliano. P. Resnick R.236180 [mm]) Cyan − Parabolic = 0. [Online: accessed 8 Dec 2010]. J. R. [Online: accessed 10 Jan 2010]. 2nd ed.556204 // 0. (2009). and Naidu. B. J..” http://www. D. “A comparison of algorithms for subpixel peak detection.551196 // 0. [2] Walker.com/motodevelopment (2010). 385–404. Wiley (2004). L.Blue − exact route Red − COM (max/mean error = 0... Springer-Verlag (1996). O. and Khatib. eds. [Fundamentals of computerized tomography: Image reconstruction from projection ].” in [Springer handbook of robotics]. IEEE-Wiley (1996). R. W..446962 [mm]) Cyan − Parabolic = 1.. “Touch technologies tutorial.264948 // 0. Springer.. “Diy touch screen analysis. Same trajectory as in the previous image. Springer-Verlag. Advances in Image Processing. R.1.” in [in Image Technology. D..490748 // 0. Howe. (b) Noise is uniformly distributed with 0. REFERENCES [1] M. (a) Noise is uniformly distributed with amplitude of 0.332286 [mm]) Black − Gaussian = 0. “Force and tactile sensors.htm (2010)..” vimeo. T.432628 [mm]) Green − Linear = 0. [7] Fisher. [6] Herman. .559910 [mm]) Green − Linear = 1. B.237692 // 0. Multimedia and Machine Vision ]. R. only this time a multiplicative noise is added to the measurement.149167 // 0. W.235776 [mm]) 1100 1080 1060 1040 1020 1000 980 960 940 920 900 450 500 550 600 650 1100 1080 1060 1040 1020 1000 980 960 940 920 900 450 Blue − exact route Red − COM (max/mean error = 1. [4] Halliday D. [Fundamentals of Physics]. K.walkermobile.504834 [mm]) 500 550 600 650 (a) (b) Figure 11. Berlin (2007).906636 // 0.com/PublishedMaterial. [5] MotoLabs.3 amplitude. Cutkosky. [Capacitive Sensors: Design and Applications].676228 // 0. 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