Informe final: Elaboración de tendencias, patrones y conclusiones

This section holds several numerical experiments. The expectations are higher robust- ness to noise due to the additional smoothing component which was already shown in Section 2.6.

Visualization of Orientation Data

Typically, orientation images are displayed in values of the HSV-colour space. Like the RGB-space it consists of three different values, which itself define colours of the same broad spectrum ( the mapping between HSV and RGB is bijective ). Figure 3.18 gives an impression of the three influences Hue, Saturation and Value:

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Hue defines the colour itself. In case of orientations the angle is mapped onto this scale. Especially because of its periodicity, the HSV-colour space is particularly suitable for any values which repeat itself alongside an interval. This is the case in orientation data.

Saturation scales the brightness of a colour. The higher this value is, the stronger the colour is present. On the other hand, the intensity fades into white the lower this term is.

Value scales the darkness of the colour. If it is very low the shade fades into black and thus visualize dark and greyish shades depending on the other values.

Fig. 3.18.: The HSV-cylinder (this graphic is originated in [72]) which describes the influence of the three values. Reason behind the use of the hue scale for orientation angles is the periodicity. The length of the orientation vector is usually coded in the saturation or value while the other is set to a constant 1 to give an impression of influence respective strength of the depict orientation vector.

The use of the hue scale is the classic approach to visualize two dimensional orientation data in image processing and is used in recent research papers. The addition to use either the saturation or the value as an indicator of the length of the orientation vector is a new approach which mainly arises from the use of multi scale algorithms. There, several orientations are summed up from different levels of detail. This results in areas with uncertain orientation values where many different orientations are summed up and the resulting vector has less influence than vectors where the orientation was directed in similar directions on all scales.

One special example on this phenomena is noise, which is usually present on higher more detailed scales which represent finer structures. Orientations on this scale are usually point into chaotic directions and therefore have an effect on the values of other scales. This phenomena of distorted orientations and their influence on significant scales will be discussed more detailed on the example consisting of lines and additional noise in the next segment.

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Example with Binary Lines

Line Function Orientation

1 y = 0 0 2 y = x π₄ 3 y = tan(π₈)x + 0.25 π₈ 4 x = 0 → π 2 5 y = −x 3π₄ 6 y = tan(π₆) − 1 π₆ 7 y = tan(6π₁₄) 6π₁₄ 8 y = tan(10π₂₂) 10π₂₂

Fig. 3.19.: The test pattern “Lines” which consists of 8 lines. The original size is 1024x1024 pixels and the lines have different angles as shown in the table. Lines 7 and 8 are drawn thinner and more close to each other which increases the difficulty to obtain orientation values by unsupervised algorithms.

Fig. 3.20.: The “Lines” test pattern with additional normally distributed noise. The left image shows a low noise level of 0.3 which results in a PSNR of 13.48 while on the right side the noise level is increased to 0.5 which leads to a PSNR of 9.39. Higher noise-levels with PSNR below 9 lead to high error levels. Even though the basic functionality of the algorithms still work on such high noise corruptions, the quality of empirical results on binary images is poor.

To accomplish an empirical study of the different presented orientation estimation approaches, a test pattern consisting of 8 single lines was created which is shown in Figure 3.19. The goal is to measure the exact angles of these lines under several noise conditions. To calculate the error of these measurements an angular measure is

106 3. Applications of Monogenic Wavelet Frames

Fig. 3.21.: These diagrams show the results of the presented algorithms. The upper one with the results of a noise-level of 0.3 and the bottom diagram on noise-level 0.5. On the x-axis, the index of a specific line as given in Figure 3.19 is used and on the y-axis the error- measure (3.42) is used to determine the quality of the algorithm. completeTKEO respective completeStructure denote the Riesz and MRA versions of their original counterparts. Here, MRA was used, but the results on all scales where used to reconstruct the data again. The main result of these diagrams is the fact, that TKEO algorithms generally perform more accurate and give better results than techniques which are based on the structure tensor.

suggested. Therefore the following approach is used to gain periodicity along the error observed by this measure d.

d(α, β) : = 1 − cos(2(α − β)) (3.42)

It is a commonly known periodic function for angular applications. The advantages of this exact procedure is the periodicity of π to neglect the influence of different directions, while the orientation does not differ (see Section 3.4).

The influence of noise is the central component in this test. Figures 3.20 shows two images which are noise-corrupted versions of the original version. In this case Gaussian distributed values with variance of the noise-levels 0.3 and 0.5 around mean 0 are used

3.4. Applications in Orientation Estimation 107

to alter the original values. Afterwards the images are clipped again at values 0 and 1. This leads to a PSNR of 13.48 respective 9.39.

The results of the different algorithms are shown in Figure 3.21. There, the lines are measured individually with masking techniques which also disables the estimation in sections where crossing occur. The concept of orientation of the phase cannot be explained properly so far in such regions. The algorithms result in an orientation-vector in each pixel of the image (in the case of MRA-algorithms these orientation-vectors appear on each scale and are summed up without normalization, which is the origin of the names “completeTKEO” and “completeStructure”). The single lines are masked by the original lines and the values are compared with the true values via the function d. On each line the median then is calculated along all values and plotted on the y-axis while on the x-axis the specific line is depicted.

There are several results, which can be found in these diagrams:

1. The multi resolution algorithms give smoother results along the lines. There are no sharp peeks as in line 3 or lines 6, 7 and 8 of the non-MRA algorithms. Higher robustness to noise mainly applies to the thinner lines which are bearably detected by the non-Riesz versions.

2. The error values of the TKEO are generally lower than the values of the structure tensor.

3. Depending on the line, the classic TKEO can outperform the MRA-TKEO. This especially happens in the case of lines 2 and 5. These are the diagonal lines.

Fig. 3.22.: This diagram shows the accumulated error level of all different lines from the example along the scales. While on the first few ones the noise is present which results in high error values, a frequency band with lower error values is between scales 3 and 7. On the coarser scales the features of the lines cannot be resolved any more which leads again to higher distances.

While the first two points are expected from mathematical concepts, which were dis- cussed in earlier sections, the third needs to be investigated in a deeper fashion. Also,

108 3. Applications of Monogenic Wavelet Frames

in further considerations, just the TKEO is in the focus due to its better performance along all example lines.

The key aspect to achieve better results is the concept of different levels of details. While in the first test, the orientations on all scales were summed up, a better approach is to individually sum up orientations on significant scales, where good results are present. Figure 3.22 shows the accumulated error values from all lines along the scales of decomposition on both introduced noise-levels. The main difference between the error values is located on scales 1 till 3, where the noise has a strong impact. On coarser scales beyond 7 the measurement errors are rising again. This can be explained through the absence of any features of the different lines in these low frequency areas and also because of a bad localization which is mathematically described by equation (1.71) of Remark 1.4.2 .

Therefore it is advised to accumulate the orientation vectors from those scales, which are neither influenced by noise nor estimate orientation on too low frequency areas. In this way results with well behaving orientation features can be extracted. The choice, which scales need to be used, depends on the kind of underlying data. In case of binary lines in an image of the size of 1024x1024, Figure 3.22 shows a typical behaviour. In other classes of images different scales might contain significant features. Nonetheless, in most cases which include noise, deleting the upper two scales and using the following four to five is a good value, which several experiments have shown.