2D-DFT
3. Import the sampled components B x and B y into the unit for localization of AMM point coordinates x and y
3.6.3 Gradient filter based algorithms
Classical gradient filter analysis methods from image processing [78, chap. 10], [59], applied in most cases for edge detection, were considered as possible candidates for the tracking algorithms as well. The gradient filter based approaches imply convolution operation:
Mf`(x; y) = Mf(x; y) h(x; y) (3.40) whereMf(x; y)is the magnitude image,h(x; y)is applied filter kernel mask andMf`(x; y)is the processed result where the information about gradients is contained in.
The gradient filter based approaches have here a general disadvantage in relation to the de-tectivity possibility of the AMM points. The range of area where the AMM points is localizable shrinks from all image sides by the half size of the filter kernel mask. Application of boundary conditions described in Section 3.4.1 do not lead to an information win. The second disad-vantage of filter based methods is the filter mask size itself. This disaddisad-vantage is resulting in case of a combination of derivatives and coupled filter functions. Depending on coupled filter function a particular kernel size is required in order to approximate the filter function shape sufficiently so that no filter kernel rotations variance13occurs [39], [59].
Laplacian of Gaussian filter based method
This algorithm is based on calculation of curvatures contained in the image, in similarity to the LoG convolution approach for angle information extraction described in Section 3.3.4. The applied Laplacian of Gaussian filter kernel function is given in Equation 3.25.
The idea in this approach is to evaluate the curvature of the magnitude imageMfincluding a low pass filtering by using of the Laplacian filter kernel. By convolution ofMfwith the LoG mask an image of local curvature values is obtained. The coordinates of the image where the highest value is present represents the localized AMM point. The signal flow diagram of the LoG filter based tracking algorithm is shown in Figure 3.24.
13Rotation variance describes the property of a filter kernel. There should be no change in the obtained charac-teristics after the convolution with the filter mask if an object is rotated in the image. If the filter kernel does not exhibit such effect then is said to be rotation invariant. In such case the filter kernel coefficients must become zero at their boundaries. See also a explanation in Section 5.3.1.
Bx components
By components
( ).
2( ).
2Filter
Filter
LoG filter kernel x-coordinate
y-coordinate
max( )
Mf
*
Figure 3.24: Signal flow diagram of Laplacian of Gaussian filter based tracking algorithm.
In addition, before the computation of Mf the values in Bx and By could be suitable prepro-cessed by a low pass filter application.
The pros and cons of this method are as follows:
• Advantages
- Filtering: The Gaussian filter mask is included in the kernel.
- Rotation invariance: It is possible to obtain the invariance by suitable approximating of the filter kernel.
• Disadvantages
- Field deformation: Deformation of the shape of the components is associated with filtering by included Gaussian filter mask.
- Minimum required mask size: In order to obtain rotation invariance a minimum kernel size of 7 x 7 elements is required [39].
- Localization at the boundaries: Due to the image filter based approach the localization area shrinks at the boundaries.
Derivative of Gaussian filter based method
At the next candidate for the tracking algorithm the Derivative of Gaussian filter (DoG). The idea behind is to smooth the image first in order to obtain more noise stability and apply afterwards a direct gradient filter. By partial derivatives of Laplacian filter kernel functionG(x; y)given in Equation 3.23 the filter masks for the gradients detection result in:
DoGx = @G(x; y)
@x = x
24e
x2+ y2
22 (3.41)
for gradient extraction in x-direction as well as in y-direction:
DoGy = @G(x; y)
@y = y
24e
x2+ y2
22 (3.42)
For this gradient filter based algorithm, the square operation needs to be aplied in two steps.
In the first step to determine the magnitude imageMf = Bx:2+By:2 and in the second step to calculate the gradients contained in the magnitude imageMf, after the partial gradient filter application. This approach represents the classical method for viewpoint independent edge detection in the image [59]. But especially in the present case, the interest is in the gradient contained in the image which is equal to zero, in accordance to the localization of maxima by 1st derivative. The signal flow diagram is depicted in Figure 3.25.
Filter
Filter
( ).
2( ).
2x-coordinate y-coordinate
min( )
X-derivate of Gaussian filter kernel
Y-derivate of Gaussian filter kernel
*
*
( ).
2( ).
2Bx components
By components
Mf
Figure 3.25: Signal flow diagram of derivative of Gaussians filter based tracking algorithm.
The advantages and disadvantages look as follows:
• Advantages
- Filtering: Same as previous.
- Rotation invariance: Same as previous.
- Higher accuracy in comparison to LoG: Indeed a better localization of AMM point can be obtained due to considering of all points contained in the mask for each gradient in one direction exclusively.
- Evaluation: More suitable evaluation is given due to threshold level = 0.
• Disadvantages
- Field deformation: Deformation of the shape is present here as well.
- Minimum required mask size: Same as previous.
- Localization at the boundaries: Same as previous.
- Computational effort: Two additional square operations are required in order to obtain
viewpoint independent localization of AMM point. This is especially important for AMM localizations at 45 encoder magnet position.
Prewitt filter based method
Derived from the previous method the Prewitt gradient filter was taken into the consideration.
This filter type comes without a low pass filter kernel and it is rotation variant. This was cho-sen consciously in order to investigate firstly, the effect if no filter mask is combined with the gradient filter and due to this no field deformations is caused and secondly, how is the impact of rotation variance in the present problem. The alternative filter including a slight low pass behavior would be the Sobel operator.
The processing procedure is the same as for the previous candidate. As a proposal it is sug-gested to apply a filter kernel mask of 5 x 5 elements what is a good compromise between the minimum required symmetrical mask of 3 x 3 and 7 x 7 elements. The filter kernels of a Prewitt operator consisting of 5 x 5 elements for gradient calculation in x- and y-direction is given by:
hP x =
2 1 0 1 2 2 1 0 1 2 2 1 0 1 2 2 1 0 1 2 2 1 0 1 2
; hP y =
2 2 2 2 2
1 1 1 1 1
0 0 0 0 0
1 1 1 1 1
2 2 2 2 2
(3.43)
The application of Prewitt gradient filter leads to the following assets and drawbacks:
• Advantages
- No explicit field deformation in comparison to LoG and DoG approaches: There is no low pass filter kernel included.
- Implicit Filtering: A filtering effect is still existent if the filter mask is increased. In this case the gradient extraction is more averaged over the spatial area.
- Higher accuracy in comparison to LoG and DoG: Due to smaller area of the extraction mask size and no inclusive low pass filtering the smoothing effect is smaller.
- Evaluation: As well as valid for DoG approach, the threshold level for the detection of the AMM point is equal to zero.
• Disadvantages
- Noise sensitivity: Due to direct gradient extraction a higher noise impact is existent.
However, this is compensated by an increase filter kernel size.
- Rotation variance: Rotation variance is unfortunately ot avoidable for this approach.
This may probably affect the localization of AMM points during rotation in a negative way.
- Localization at the boundaries: Same as previous candidates.
- Computational effort: As well as all gradient based approaches two additional square operations are required also for this method.