Sub-pixel edge detection - emsys - Hogeschool voor Wetenschap ...

DE NAYER Instituut 

Onderzoeksgroep Digitale Technieken 

J. De Nayerlaan 5 

B-2860 Sint-Katelijne-Waver 

Tel. (015) 31 69 44 

Fax. (015) 31 74 53 

Email: ppe@denayer.wenk.be 

gvd@denayer.wenk.be 

bvd@denayer.wenk.be 

Website: http://emsys.denayer.wenk.be 

Sub-pixel edge detection 

Version: 1.2 

HOBU-fund research project 030106 

Title : Multifunctional embedded digital camera system 

Project manager : ing. Patrick Pelgrims 

Project engineers : 

ing. Gerrit Van de Velde 

ing. Björn Van de Vondel 

Copyright (c)2004 by Patrick Pelgrims, Björn Van de Vondel and Gerrit Van de Velde. 

This material may be distributed only subject to the terms and conditions set forth in 

the Open Publication License, v1.0 or later (the latest version is presently available at 

http://www.opencontent.org/openpub/). 

Sub-pixel edge detection October 2004 



© OPL v 1.0

Abstract. 

The recent evolution in digital electronics enables the execution of different, highperformance 

tasks on compact embedded systems. 

For some applications it is necessary to perform very precise measurements, 

based on a photographic image. The application mentioned here is part of a 

laser-scanning device. An object is scanned by means of a laser and the reflected 

beam is captured by an image sensor. It is very important to locate the edges of 

the object as exact as possible. The algorithm described below will produce a list 

of accurate coordinates of the position of this edge. 

1. Introduction 

From September 1, 2003 until august 31, 2005 researchers from the Hogeschool 

voor Wetenschap & Kunst, campus De Nayer are conducting an IWT/HOBU-fund 

research project about embedded camera systems and applications for such 

systems. The main reason for this research is that a large amount of the image 

processing tasks are done on a PC, which implies a substantial amount of data to 

be transferred from the camera to the image processing unit and vice versa. The 

researchers are convinced that, thanks to the recent evolution in embedded 

electronics, it has become possible to perform a certain amount of image 

processing in the camera itself. The application as described below serves as an 

example. 

2. Problem definition 

When performing a laser 

scan of a certain object, the 

reflected laser beam is 

captured by a CMOS or 

CCD imaging sensor. This 

produces a very dark image 

(nearly black) with a very 

bright white line, drawing the 

contours of the scanned 

object (Fig. 1). Due to 

refraction and the limited 

number of samples, the line 

is not represented with 

sharp edges, but becomes 

rather blurred. 

The challenge for this 

algorithm is to determine the 

Fig. 1. image after object scanning 

exact position of the edges of the object, by performing a sub-pixel edge detection 

procedure on the given image. 




© OPL v 1.0

3. Sub-pixel edge detection 

When a bright line on a black background is viewed on sample-level, column by 

column, theoretically the pattern obtained is an ideal step-shaped waveform. In 

real-life, this waveform is not that ideal, but the line will be a bit more blurred. On 

sample-level, the blurring will result into a Gaussian curve. (Fig. 2). This will make 

the detection of the positive and negative edges even more complex. 

The first problem to look at is how to find an 

edge Let’s get back to the ideal 

representation. When calculating a point-topoint 

derivative, a ‘peaked’ signal is 

obtained. The maximum and minimum of 

this differential signal are the positions of 

the both edges (positive and negative). So, 

the solution to the first problem is quite 

simple: calculate the first order derivative 

and locate maximum and minimum. (Fig. 3 ) 

What changes when conditions are not 

ideal The effect of non-ideal edges is the 

fading of the sharp edges in the 

Fig. 3.: the first derivate function to locate 

the edges 

Fig. 2.: ideal and realistic sample-view of 

a line 

derivative signal. It might be more 

difficult to locate maximum and 

minimum, but results can be found. 

The challenge starts to occur when a 

certain degree of precision is needed. 

Theoretically the boundary for the 

precision lies with the Nyquistboundary. 

Changes at higher 

frequencies than the sample rate 

divided by two cannot be detected. 

Theoretically that is, because in 

practice the sub-pixel edge detection can be used to achieve higher precision. 

The algorithm consists of three main parts: 

• Part 1: As in the theoretical case, the point-to-point derivate function is 

calculated. This function has only valid values on the sample moments of 

the original values. 

• Part 2: Calculate the second order derivative. 

• Part 3: The cubic interpolation technique is used to create a continuous 

function out of the sample values obtained in step 2. 

• Part 4: To know the exact position of the positive and negative edge, the 

zero-crossing position for the function from part 3 has to be determined. 

The complete sub-pixel edge detection procedure is illustrated in Fig. 4. 

In the following paragraphs the algorithm will be explained more in detail. For 

further information on the algorithm or its performance, contact the authors. 




© OPL v 1.0

The adjacent figure (fig. 4) shows 

the different phases of the subpixel 

edge detection algorithm. The 

first figure (blue sample points), 

show the sample-view of the cross 

section of an arbitrary line. The 

goal of this procedure is to 

determine the exact position of the 

edges of this line. 

The first step to take is to calculate 

the first order derivative of this 

function. This is shown in the 

second part of the figure (red 

sample points). The location of the 

edges is on the maximum and 

minimum of this derivate function. 

To determine these positions, the 

second order derivative is 

calculated (green sample points). 

The edges are now located on the 

zero-crossings of this function. To 

determine these positions very 

accurately, this function is 

interpolated, as shown in the fourth 

part of the drawing. Finally the blue 

arrows, point out the exact 

positions of the positive (=rising) 

and negative (=falling) edges. 

Let us go on to the more detailed 

explanation of this procedure. 

Fig. 4.: Different steps in the sub-pixel edge 

detection 

4. First and second order derivatives 

The problem of finding the exact position of the edge of a sampled line is treated 

like a classical extremum problem. This means that, as explained above, the 

derivative functions have to be calculated. 

Any handbook on mathematics will state that the definition of a derivative function 

is as follows: 

dy 

d( 

f ( x)) 

= 

dx 

Figure 5 provides some more information. 

y2 

− y1 

= 

x2 

− x1 




© OPL v 1.0

In this implementation we respect this definition. For every sample S n , we define 

the derivative for that samples as 

S n+1 – S n-1 . To completely respect 

Y 

the mathematical definition we 

ought to divide this result by two, 

y2 

but since we are only interested in 

the maximum, we can omit this 

constant coefficient. For the second 

dy 

order derivative, the way to go is 

the same as for the first order 

y1 

derivative. Again, the ½ is omitted 

in the result. 

To limit the interval in which we 

x1 

x2 

X 

have to search for the zero- 

dx 

crossing of the second order 

derivative, the procedure is to find 

Fig. 5.: The derivative 

and log the maximum and 

minimum values of the first order derivative. These values are then used in the 

last part of the algorithm. 

5. Cubic interpolation 

To get a high precision determination of the edges of a line, an interpolation is 

implemented. For this purpose the cubic interpolation was chosen. The first thing 

to do is to define the number of subsamples to calculate. The algorithm will take 

up to four real samples into account (two samples on the left, y 0 and y 1 , and two 

samples on the right, y 2 and y 3 ) and calculates a number of subsamples in 

between those real sample values. 

The heart of this interpolation technique is trying to fit a third order polynomial 

onto the discrete function. The formula used is: 

3 

2 

y ( n) 

= a * t + a * t + a t + a 

0 1 

2 

* 

with the following definitions: 

n = the index of the current subsample (n ∈[1..numb_of_subsamp]) 

y = the interpolated value 

n 

t = 

N 

a = y 

a 

a 

a 

0 

1 

2 

3 

= y 

3 

0 

= y 

= y 

2 

1 

− y 

− y 

2 

1 

− y 

0 

− y 

− a 

0 

0 

+ y 

1 

After this interpolation, the results are used in the last step of the algorithm, where 

the zero-crossing is determined as accurately as possible. 

3 




© OPL v 1.0

6. Determination of the zero-crossing 

Remember that we determined the maximum and minimum values of the first 

order derivative. Here, we use those results to narrow down our scope of search 

for the zero-crossing. The interval where the search will be conducted, starts at 

minimum (or maximum) – numb_of_samples and ends at minimum (or maximum) 

+ numb_of_samples. 

The procedure will be 

Y 

Zero-crossing to the left 

Index-- 

Determine value (V) of f” 

at MAX_f’ 

Y 

V>0 

N 

Zero-crossing to the right 

Index++ 

V>0 V 0 

 

Zero-crossing to the left 

Y 

explained for the positive 

edge, the procedure for 

the negative edge is very 

similar. The algorithm is 

shown in figure 6. 

The first thing to do is to 

determine the value of 

the second order 

derivative at the point 

where the maximum of 

the first order derivative 

was found. By checking if 

this value is less than 

zero, the location of the 

zero-crossing can be 

predicted to be either on 

the left or on the right of 

the current value (Fig. 7). 

After this check, the 

algorithm graduately 

MAX 

V < 0 

 

Zero-crossing to the right 

Fig. 7.: prediction of zero-crossing point 




© OPL v 1.0

7. Results 

When this algorithm is applied to the picture shown in figure 1, figure 8 is 

produced. Note that this is just to give some feedback on the algorithm, because it 

is impossible to show the results with the same accuracy as they were calculated. 

For the more accurate results, a list is also provided with the exact coordinates of 

the positive and negative edges. 

Fig. 8.: resulting image 

column 14: positive edge 0.110 past row 62 and negative edge 0.980 past row 64. 












… 

Fig. 9.: List of exact coordinates 

8. Bibliography 

• R. A. D. Zanardini, Piecewise Cubic Hermite Interpolation, 2003 

• C. Steger, Evaluation of subpixel line and edge detection precision and 

accuracy 




© OPL v 1.0

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