# /smash/get/diva2:19612/FULLTEXT01.pdf | Experimental Results

**5.1.1 Description and Motivation of Method **

The implementation of a linear scale-space method involves smoothing with an isotropic

Gaussian kernel, which will suppress features smaller than a defined scale. The main idea is that distinct stable features of a fingerprint image are larger than the smallest representation

(i.e. a pixel), and that small enough features may be unstable. Hence smoothing detailed features would result in a more stable representation of the fingerprint pattern. Scale-space smoothing an image flattens it, and using to large a scale will suppress important structure information. It is therefore likely that this proposed method works best at smaller scales.

**5.1.2 Definition of Method Parameters and Boundaries **

The linear scale-space method is an uncommitted procedure where no analysis of the image precedes the smoothing. The only parameter to consider is the scale of the Gaussian kernel, referred to as the scale parameter, *t*. The Gaussian kernel suppresses most signal structure of size smaller than the standard deviation,

σ

[30]. The scale parameter is equal to the square of the standard deviation,

*t*

=

σ

2

. Apart from the size of a feature, its relative amplitude (i.e. contrast) is an additional important characteristic which decides at what scale the attribute is suppressed. Less distinct features (i.e. low-contrast structure) are smoothed at smaller scales than high-contrast structure.

The boundaries for the scale parameter, *t*, should include values which may render interesting results. The lower boundary is selected very small to be able to investigate the effect of smoothing barely noticeable to the naked eye. Scale-space filtering at a very detailed scale flattens small features without affecting the larger structures. The lower boundary has been

chosen to 0.1 (see Figure 5-1), which results in a 3x3 sized Gaussian kernel.

The selection of the upper boundary is more delicate since larger scales suppress larger features and may flatten the actual ridge structure. By visible assessment it has been decided that fingerprint images smoothed at scales up to *t* = 2 are possible to give improvements in the

evaluation measures (see Figure 5-1). An additional scale (of size 4) has been included in the

test to validate the response of *MNCC*

*feat*

and *UM*

*rel*

when distinct fingerprint structures are smoothed.

**5.1.3 Specification of Parameter Sample Values **

The linear scale-space method only depends on one parameter, thus the number of sampling values can be chosen fairly large without generating an incomprehensible amount of data.

Smaller scales are more interesting since it is presumed that detailed information may be more unstable over time than larger structures. By visual estimation it is decided that 6 sample values should be sufficient to accurately evaluate the effect of linear scale-space smoothing.

The sampling points are distributed between the specified boundaries as shown in Table 5-1.

*t *

0.1, 0.2, 0.5, 1, 2, 4

**Table 5-1: Parameter values for linear scale-space. **

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Three different sub-area fingerprint images, smoothed at the scales defined in Table 5-1, are

shown in Figure 5-1. The images have been selected to represent structures at different scale.

**Figure 5-1: Three different fingerprint sub-area images, original and scale-space filtered at scales t = {0.1, 0.2, 0.5, 1, 2, 4}. **

**5.1.4 Implementation of Linear Scale-Space **

The linear scale-space method has been implemented by convolution with a discrete Gaussian kernel. The kernel is normalised to give a maximum response of one at arbitrary scale.

The image borders are mirrored when convolving with the scale-space kernel, for the sake of preserving the mean grey-value intensity of the image.

**5.1.5 Results **

The overall results for the linear scale-space technique shows that the best performance is achieved for scale parameter *t* = 0.1, and the number of correct matches are decreasing at

coarser scales (see Table 5-2). There is however no single scale that performs better than

correlation of the initial images. The percentage of correlation pairs with improving *UM*

*rel*

and

*MNCC feat*

presented in Table 5-2 are compared to the initial evaluation measures.

*t *

**Higher UM**

*rel*

** (%) Higher MNCC**

*feat*

** (%) Higher UM**

*rel*

** and **

*MNCC feat*

** (%) **

0.1 40.9 97.3 40.0

**Number of correct matches**

666

0.2 36.6 95.5 34.9 665

0.5 32.8

1 29.2

2 20.4

4 10.8

93.5

88.0

75.7

61.8

30.9

27.3

18.9

10.0

654

634

580

493

**Table 5-2: Overall results for the linear scale-space method. Each correlation pair have been compared to the initial evaluation measures. **

The histogram of *UM*

*rel*

and *MNCC*

*feat*

measures for scale *t* = 0.1 are shown in Figure 5-2 and

Figure 5-3. It is difficult to visually estimate the differences between these histograms and

50

those calculated for the initial images (see Figure 4-10 and Figure 4-11). Closer numerical

analysis shows that there are more *UM*

*rel*

values decreasing than increasing at scale 0.1, which

is also shown in Table 5-2. The *MNCC*

*feat*

values are almost exclusively increasing.

50

40

30

20

10

0

-0.6

-0.4

-0.2

0 0.2

0.4

0.6

**Figure 5-2: Histogram of UM**

*rel*

** values for scale-space smoothed images ( t = 0.1). **

70

60

50

40

30

20

10

0

0.7

0.75

0.8

0.85

0.9

0.95

1

**Figure 5-3: Histogram of MNCC**

*feat*

** values for scale-space smoothed images ( t = 0.1). **

The linear scale-space method depends on one parameter, making it easy to evaluate the overall results for all parameter settings. The histograms for *UM*

*rel*

and *MNCC*

*feat*

at all scales, although not included in the thesis, have been evaluated. The *UM*

*rel*

values are normally decreasing at coarser scales. For the scales tested the *MNCC*

*feat*

are generally increasing, but the percentage of correlation pairs with improving *MNCC*

*feat*

are decreasing at coarser scales

(see Table 5-2). In other words many feature tiles get a lower match score when smoothed at

coarser scales. This was not anticipated and it will be further investigated for separate correlation pairs in the succeeding detailed analysis.

Following is an investigation of the effect that the linear scale-space smoothing process has on separate correlation pairs. Three different types of results will be examined. Firstly the best and worst cases (i.e. where the *UM*

*rel*

value diverges most compared to the initial evaluation measures) are considered. A total of 16 different correlation pairs were falsely matched initially, but correctly matched when scale-space smoothed. These correlation pairs are the third case which will be studied in detail.

The most obvious feature of the results is that the worst cases (i.e. when the calculated *UM*

*rel*

is considerably lower than the initial *UM*

*rel*

) always appear at coarser scales. Most commonly

51

at scale 4, but to some extent also at scale 2. The reasons for a decreasing *UM*

*rel*

at larger scales can be divided into two groups. The first case is in line with the predicted behaviour of the method, and it involves correlation pairs where the *MNCC*

*feat*

increases for larger scales.

Hence the single reason for a lower *UM*

*rel*

value is that there are other features within the image that also obtain a higher correlation value at coarser scales. This may occur at two separate occasions; firstly when the feature tile is not distinct enough (i.e. the initial *UM* is low), hence there are at least one further part of the fingerprint which is similar to the feature tile. This is often found in fingerprints of low quality, where the ridge structure is vague. In the second case the parts of the feature tile that makes it unique are so small that they are suppressed at larger scales, thus making the smoothed feature tile less distinct and more

similar to other parts of the fingerprint. The latter situation is depicted in Figure 5-4 where the

left-most valley is thinner than the adjacent ridges, hence the valley is smoothed faster than the ridges. The ridge ending blend with the neighbouring ridge and the feature tile gets less distinct. Although the *UM*

*rel*

value decreases considerably at larger scales, the example in

Figure 5-4 is still distinct enough at all scales to be singled out as the correct match.

**Figure 5-4: Feature tile from three weeks at scales t = {0, 0.1, 0.2, 0.5, 1, 2, 4}. **

The second group of fingerprints that produces significantly worse results when scale-space smoothed are those where the *MNCC*

*feat*

value decreases for larger scales. This effect was not anticipated, but results of this type are in fact more common than the group previously described. Feature tiles that most often give rise to this kind of result are those that are initially fairly similar but include structure that locally deviates in size or strength (i.e. contrast). When scale-space smoothing such features, the local mean grey value will differ within each of the feature tiles, and this will result in low *MNCC*

*feat*

values. In other words the structure is more similar at pixel scale for unsmoothed images of a feature than for the

smoothed versions of the same images. Figure 5-5 illustrates different readings of a feature

tile with this characteristic. The feature segment in the top row includes a ridge which is both wide and strong within the whole tile. In the centre row the feature is as strong as the top row in the upper part of the image, but weaker in the lower part. In the bottom row the circumstance is reversed compared to the centre row (i.e. strong in the lower part of the image, and weaker in the upper part compared to the top row). This will at coarser scales result in a lower *MNCC*

*feat*

value, thus decreasing the *UM*

*rel*

value.

52

**Figure 5-5: Feature tile from three weeks at scales t = {0, 0.1, 0.2, 0.5, 1, 2, 4}. **

Occurrences where the linear scale-space method performs notably better than the correlation of the unsmoothed images can also be divided into two groups; the case when similar features in the fingerprint are smoothed quicker than the feature tile, and secondly when deviating details are smaller than the distinct structure of the feature segment. The former occasion does not involve enhancement of the feature tile, but rather destruction of similar information in the fingerprint. For this type of feature segment the *MNCC*

*feat*

value is commonly equal or lower at coarser scales compared to the initial *MNCC*

*feat*

, but the *UM*

*rel*

is still increased since correlation values within the rest of the image decrease more that the *MNCC*

*feat*

.

The second type of feature tiles that perform better when scale-space smoothed include distinct structure that is larger and/or stronger than the deviating details. The main reason for an increasing *UM*

*rel*

value, when scale-space filtering this type of feature segment, is that the tile areas become more alike and thus the *MNCC*

*feat*

value is enhanced. Figure 5-6 shows an

example where the distinct structure is both larger and stronger than the deviating details, and

Figure 5-7 depicts a feature tile where spurious information is weaker than the distinct feature.

For the images in Figure 5-6 the highest *UM*

*rel*

value was calculated at scale 4, and for the

images in Figure 5-7 it was found at scales 0.5 and 1.

**Figure 5-6: Feature tile from three weeks at scales t = {0, 0.1, 0.2, 0.5, 1, 2, 4}. **

53

**Figure 5-7: Feature tile from three weeks at scales t = {0, 0.1, 0.2, 0.5, 1, 2, 4}. **

The final type of result that will be analysed in detail are the correlation pairs where the feature tile match initially failed, but succeed when the images are scale-space filtered. There are a total of 16 different correlation pairs that for one or more scales belong to this category.

Some of the initial false matches were due to misregistration. In these cases the maximum

*MNCC*

value ended up just outside the ground truth region, thus they could not be considered false matches. For the remaining occasions the most common reason for a negative initial

*UM rel*

is that the feature tile is not distinct enough. This is either because a low *UM*

*init*

or considerable differences between the feature tiles of the two correlation pair images. These correlation pairs result in such a small *UM*

*rel*

value that they cannot be considered reliable.

The remaining type of result is where the spurious information is apparent at detailed scales,

and the distinct features are of greater similarity at coarser scales. Figure 5-7 shows an

example of this type of result. The expectation of this type of effect was one of the main motivations to test the linear scale-space method. For the data set considered in this thesis there are however very few correlation pairs where this characteristic is found.

Conclusively the linear scale-space method, although in many cases improving evaluation measures at small scales, cannot be considered a method apt for fingerprints image enhancement. Firstly there were no singular scale found that gave an overall improvement of the evaluation measures. Secondly the cases when significant improvement could be proven it more often depended on destruction of similar features in the fingerprint than actual enhancement of the feature tile. There were occasions when the linear scale-space method enhanced feature tiles by suppressing unstable information, however this almost exclusively occurred for feature tiles with a large and strong structure. These are the features that are initially very distinct and not noticeably affected by the template ageing problem, hence they are not in actual need of image enhancement. It has been shown that distinct structure may appear at very detailed scales, and thus uncommitted smoothing is as likely to suppress distinct features as spurious information.

*5.2 Nonlinear Isotropic Diffusion *

*5.2 Nonlinear Isotropic Diffusion*

**5.2.1 Description and Motivation of Method **

The results from the linear scale-space method showed that relevant structure may appear even down to pixel scale. The proposal for testing nonlinear isotropic diffusion is that distinct

54

features are related to edges, and that smoothing structure that is not considered to be edges

(i.e. inside ridges and furrows) may enhance the evaluation measures.

Using the normalised Gauge first derivative as edge detector (see Chapter 4.4), the smoothing

process is controlled to blur parts of the images where the edge response is weak. This will not only blur intra-regional areas of ridges and furrows, but it will also have the effect of enhancing the edges (i.e. increasing the contrast).

**5.2.2 Definition of Method Parameters and Boundaries **

The first parameter to consider is the conductivity coefficient (i.e. diffusivity), since this measurement decides what type of features that should control the smoothing process.

Following the previous motivation an edge detection operator is a logical selection, and the method of choice is the first derivative of the Gauge axis perpendicular to the isophote. The

background on the selection of this method has been presented in Chapter 4.4 Edge Detection.

The first parameter that has to be decided is the scale of the Gauge derivative. The images in

the top row of Figure 5-8 represent ridge structure of different sizes, and the plots show the

response of the Gauge derivative for the centre row (marked with a red line) of these images.

The Gauge derivative have been calculated at scales, *t* = 0.0625 (top row of plots), 0.25

(middle row) and 0.56 (bottom row).

1

0.5

0

1

0.5

0

1

0.5

1

0.5

0

1

0.5

0

1

0.5

1

0.5

0

1

0.5

0

1

0.5

0 0 0

**Figure 5-8: Plots of Gauge first derivative for the centre row of the images in the top row. Plots calculated at scales t = {0.0625, 0.25, 0.56} (top to bottom). **

The edge response for the large (right column) and medium sized (centre column) ridge

structure in Figure 5-8, does not deviate much with the selection of the scale. For the small

structure (in the left column) the edge response is somewhat weaker at coarser scales. Since

the features in all of the images in Figure 5-8 are representing edges, the calculated edge

response should be as high as possible. Hence, a fixed value of 0.0625 for the scale of the

Gauge derivative, *t*, is appropriate to use when testing the nonlinear isotropic diffusion.

55

The next decision to take relates to what conductivity function that should be used to control

the diffusivity. By observing the edge responses in Figure 5-8 it can be determined that the

function should diffuse much for a conductivity coefficient lower than approximately 0.4, and halt the diffusion process for values over 0.5. In other words the response curve of the conductivity function should rapidly decrease around 0.4. The Weickert conductivity

function, presented in Chapter 2.2.3, possesses this property. Changing the conductivity

coefficient from the gradient magnitude to the absolute value of the Gauge first derivative, the function is written as:

*c*

*W*

=

⎧

⎪

⎪

1

− exp

⎛

⎜

(

1

−

*abs*

3 .

315

( )

/

λ

)

4

(

(

*s s*

=

>

0

0

)

)

Function curves for the Weickert conductivity function, with varying

λ

1

0.8

0.6

0.05

0.1

0.15

0.2

0.25

0.4

0.2

0

0 0.1

0.2

0.3

0.4

0.5

L

ω

0.6

0.7

0.8

0.9

**Figure 5-9: Weickert conductivity function, c**

*W*

**, of different **

λ

**. **

1

From Figure 5-8 and the plots in Figure 5-9 the boundaries for

λ

was initially decided to 0.15 and 0.25. When practically testing the method it was evident that the response from edges occasionally could be even lower than 0.4, thus the lower boundary of

λ

was changed to 0.05.

The final parameters to specify boundaries for, involve the scale-step and the number of iterations for the diffusion process. These factors decide the maximum amount of diffusion that will be performed (i.e. when the conductivity function is 1). The scale-step should be selected smaller than 0.25 to ensure a stable solution [26], and it was therefore set to a constant value of 0.2. Since the structure that is to be smoothed is fairly detailed (i.e. the width of ridges and furrows), the number of iterations could be chosen rather small.

56

The upper and lower boundaries of the number of iterations where selected to 5 and 10

respectively. This was determined through visual assessment of the images in Figure 5-10.

These images were calculated with different numbers of iterations for the diffusion process, and with the conductivity function set to a constant value of 1 (i.e. maximum homogeneous diffusion).

**Figure 5-10: Original image and maximum homogenous diffusion calculated for 5, 10, **

**15 and 20 iterations, with scale-step ds = 0.2. **

**5.2.3 Specification of Parameter Sample Values **

With the diffusivity and conductivity function decided, and the Gauge derivative scale and the diffusion scale-step set to fixed values, there only remain two parameters to specify sample values for. Namely the constant of the conductivity function,

λ

, and the number of iterations for the diffusion process.

The boundaries for the conductivity function constant have been set to 0.05 and 0.25 respectively. This constant decides which features are to be blurred, and is therefore essential for the result of the nonlinear isotropic diffusion process. Thus the number of samples chosen

should be relatively high. From the plots in Figure 5-9 it was decided to include five values in

the test;

λ

= {0.05, 0.1, 0.15, 0.2, 0.25}.

For the number of iterations it was decided that the boundary values (5 and 10) are enough and no further sampling values were used in the test.

The parameter values defined for the nonlinear isotropic diffusion method are summarised in

*t *

λ

0.0625

0.05, 0.1, 0.15, 0.2, 0.25

*ds *

0.2

No. of iterations 5, 10

**Table 5-3: Parameter values for nonlinear isotropic diffusion. **

**5.2.4 Implementation of Nonlinear Isotropic Diffusion **

The implementation of the scalar driven diffusion was adapted from the description given by

(

Rein van den Boomgaard in [26]. Boomgaard define the discrete approximation as follows:

*L s x*

+

,

*y ds c s x*

+

1 ,

*y*

=

+

*L s x c s*

,

*x*

,

*y y*

+

*ds*

)(

*L s x*

2

+

1 ,

[

*y*

(

*c s x*

−

,

*y*

+

1

*L s x*

,

*y*

+

*c s x*

,

) (

*y c*

)(

*s x*

,

*L s x y*

,

+

*y*

+

1

*c*

−

*s x*

−

1 ,

*y*

*L s x*

,

)(

*y*

*L s x*

,

) (

*y*

−

*c s x*

,

*y*

+

*L s x*

−

1 ,

*y c*

)

]

*s x*

,

*y*

−

1

)(

*L s x*

,

*y*

−

*L s x*

,

*y*

−

1

)

+ where *s* is the scale, *x* and *y* are the spatial position, *L*

*s*

is the image at scale *s*, *ds* is the scalestep and *c* is the conductivity function.

57

Readers interested in details regarding the discrete approximation are referred to [26] for further reading.

**5.2.5 Results **

The three parameter settings that results in the most number of correct matches are presented

in Table 5-4, and the three cases with the least number of correct matches are shown in Table

5-5. The number of accurate matches for the initial images was 666, hence there is no

combination of parameter values that renders a better overall result. Considering all parameter settings there were a total of 12 different correlation pairs that were initially false matches, but became successfully matched when scale-space filtered.

**iterations, **

λ

**Higher **

*UM rel*

** (%) **

5, 0.05 38.6

**Higher **

*MNCC feat*

** (%) **

46.8

**Higher UM**

*rel*

** and **

*MNCC feat*

** (%) **

24.1

**Number of correct matches **

665

10, 0.05

5, 0.1

34.9

31.1

37.3

33.9

20.0

17.8

664

658

**Table 5-4: The three best cases for scalar dependent diffusion. Percentile values are in comparison to the initial evaluation measures. iterations, **

λ

**Higher **

*UM rel*

** (%) **

10, 0.25 15.3

10, 0.2 19.3

10, 0.15 22.2

**Higher **

*MNCC feat*

** (%) **

47.0

29.9

20.8

**Higher UM**

*rel*

** and **

*MNCC feat*

** (%) **

11.4

11.4

11.5

**Number of correct matches **

536

579

617

**Table 5-5: The three worst cases for scalar dependent diffusion. Percentile values are in comparison to the initial evaluation measures. **

The percentile improvement of the *UM*

*rel*

can be compared to the results from the linear scale space method. The *MNCC*

*feat*

result conversely deteriorated. Less than half of the correlation pairs get a higher *MNCC*

*feat*

value when enhanced with the current method. This means that the most common case for the scalar driven diffusion makes an accurate match more difficult

(i.e. decreases the *MNCC*

*feat*

), while making a false match more likely (i.e. decreases the

*UM rel*

). The overall results presented in Table 5-4 and Table 5-5 suggests that the proposed

method is unsuitable as a method for fingerprint image enhancement. The following detailed analysis will investigate whether this is generally true or not.

The overall results for nonlinear isotropic scale-space shows that the best performance is achieved for

λ

= 0.05 and calculation over 5 iterations (see Table 5-4). The *UM*

*rel*

and

*MNCC feat*

histograms for the best case, are very similar to the histograms for the initial results

(see Figure 4-10 and Figure 4-11) and no additional information could be gained from them.

Therefore they are not included in this report.

The correlation pairs that performed best (i.e. significantly higher *UM*

*rel*

) were generally calculated with a

λ

-value of 0.2 or 0.25, and to some extent also for

λ

= 0.15. In other words the edge tolerance of the conductivity function was set fairly high, thus these cases also smoothed across edges. A high edge response value however, guarantee slower smoothing, even though the process is not stopped completely for high

λ

-values. The images included in

58

this group of correlation pairs are in other words fairly similar to the images filtered by the linear scale-space method. The difference is that strong edge structure is preserved, or rather

suppressed at a slower rate than non-edge structure. Figure 5-11 shows three examples of

correlation pairs where the *UM*

*rel*

was improved when the images were blurred by the scalar dependent diffusion method. The two columns to the left contain the initial images, and the two to the right depict the scale-space smoothed versions of these images.

**Figure 5-11: Three examples where the scalar dependent diffusion improved the UM**

*rel*

**value; (left) initial correlation pair, (right) scale-space filtered images. **

More common for the scalar driven diffusion were correlation pairs where both the *UM*

*rel*

and

*MNCC feat*

decreased. A closer study proves the proposed method to include some characteristics which makes it futile for enhancement of fingerprints. Consider for instance a fingerprint with a broken ridge structure. In such case the nonlinear isotropic scale-space method would smooth the valley created by the broken structure and thus enhance the contrast of the ridge gap. Another example is when part of the fingerprint is of low quality (i.e. smudged), which results in weak edges. In this case the scalar driven diffusion smoothes across the significant structure in the low quality area, while enhancing (or preserving) edges in areas of higher quality. When diffusing across ridge/valley-structure new edges will appear at the boundary between high and low quality areas, resulting in creation of new structure that is not apparent in the original fingerprint. For smudgy fingerprints there is little chance that the edge response in two images of the same fingerprint is the same, hence not only will the diffusion process distort the distinct structure in one image, but furthermore increase the dissimilarity between the two images.

59

Figure 5-12 depicts three examples where the *UM*

*rel*

value was decreased considerably due to this problem. The two image columns to the left are the initial correlation pairs, and the two columns to the right are the scale-space smoothed versions.

**Figure 5-12: Three examples where the scalar dependent diffusion noticeable decreased the UM**

*rel*

** value; (left) initial correlation pair, (right) scale-space filtered images. **

The problem illustrated in Figure 5-12 occurred frequently for the diffusion filtered images.

No single edge response value can be decided that accurately defines the edges of the actual fingerprint ridge and valley structure. Since the edge response for smudgy fingerprints appears very low, and for broken ridge structure is high.

The conclusion is that edges, as interpreted by the Gauge first derivative at detailed scale, are not reliable as definition of boundaries of distinct structures in fingerprints. Due to indistinctness of edges in low quality fingerprint images, the nonlinear isotropic scales-space method may perform uncontrolled piecewise smoothing across significant structure. This property solely classifies it as an ill fit method for fingerprint image enhancement.

*5.3 Nonlinear Anisotropic Diffusion *

*5.3 Nonlinear Anisotropic Diffusion*

**5.3.1 Description and Motivation of Method **

The two previous methods have suggested that unstable fingerprint features are not homogeneous and isotropic, nor can they exclusively be related to structure that is considered as edges. In other words, distinct characteristics may be found at any scale and are not only limited to edges. The approach of nonlinear anisotropic diffusion will investigate if important fingerprint attributes are connected to the orientation of structure at a certain scale. The suggestion is that deviations in orientation at detailed scales may be unstable; hence increasing the coherence locally would make two images of the same fingerprint more alike.

At first, this method may sound very similar to nonlinear isotropic diffusion (see Chapter 5.2),

however there are two essential differences between the previous method, and nonlinear anisotropic diffusion. Firstly, the former method used edges (which are details of level 3 scale) to control the diffusion process, while the current method considers features of coarser scales, namely ridge orientation (level 2 detail). In practice this difference is primarily due to the use of the so called integration scale,

ρ

, which decides at what scale the structure tensor

60

should be smoothed. The value of the integration scale decides at the size of the local area that the isotropy is enhanced for, the higher the integration scale, the larger the area. The second difference, compared to nonlinear isotropic diffusion, is that the diffusion process in the current method is never entirely stopped. This means that features cannot be enhanced in the same way edges were in the previous method. Though a certain type of structure is preserved, specifically data perpendicular to the preferred smoothing orientation, there is always some amount of blurring that will occur for the entire image. Relatively, structure will appear to be enhanced, but the nonlinear anisotropic diffusion will never increase the local contrast.

The method proposed in this section, and similar methods utilising nonlinear anisotropic diffusion, have previously been tested as a means of fingerprint image enhancement [19, 32,

33, 34] and have been shown to improve results of fingerprint verification. The reason for including the method in this thesis is to be able to compare it to the results rendered from the other methods implemented within this work. The similarities and differences between the method used in this thesis and the formerly proposed methods have been explained in Chapter

**5.3.2 Definition of Method Parameters and Boundaries **

The method implemented and tested in this section is the coherence-enhancing diffusion

filtering proposed by Weickert [28] and described in detail in Chapter 2.2.4. The structure

tensor will be used to analyse the fingerprint structure, hence the first parameter to specify is the scale that the structure tensor is to be calculated at, referred to as the observation scale.

Following the reasoning in Chapter 5.2.2, a small scale should be selected to guarantee an

accurate estimation of the structure of detailed features. The observation scale, *t*, is set to

0.0625, which renders kernels of size 3x3 that is used to calculate the structure tensor.

The next parameter to specify is the integration scale,

ρ

. It should be chosen large enough to suppress detailed orientation deviations, like ridge edge irregularities, but small enough not to destroy distinct features with deviating orientation, such as minutiae and singularity points.

Considering the images in Figure 2-22 the integration scale boundaries is set to 0.5 and 8.

This will allow for investigation of the effect of enhancing correlation at both a very small local scale, as well as for a rather large scale. A integration scale of 8 may seem like a too large value to render reasonable results when considering a diffusivity constant of 5

⋅

10

−

5

.

However, for larger diffusivity constants, an integration scale of 8 is a more reasonable choice, and that is why it is included in the evaluation of the method.

The parameters that are left to define are the number of iterations, the scale-step, *ds*, and the two constants involved when calculating the eigenvalues of the diffusion tensor,

α

and the diffusivity constant, *C*. To get a more comprehendible data set and simplify evaluation of the method the scale-step and furthermore the number of iterations is set to fixed values. In this way the

α

and *C* parameters single-handedly decide the amount of diffusion.

61

The calculation of the diffusion tensor eigenvalues follow the approach described by Weickert

[28], and have been explained in Chapter 2.2.4. It is repeated here for reader convenience.

λ

*1*

determines the diffusion perpendicular to the isophotes, and

λ

*2*

defines the amount of smoothing along the isophotes.

λ

1

=

α

λ

2

=

⎧

⎪⎩

α

*C* > 0,

α

+

∈

(

1

−

α

α

) exp

⎝

κ

*C*

(

κ

=

0

)

, and

κ

is the coherence measure as described in Chapter 2.2.4.

The

α

parameter decides the minimum amount of smoothing, and the maximum difference between diffusion along isophotes and perpendicular to them. Setting it to 0.01 gives a minimum ratio of 1/100. The scale-step is set to 0.2 and the number of iterations is decided to

50. These are the same parameter values that were used to calculate the images in Figure

α

-value, scale-step value and number of iterations, the diffusion perpendicular to isophotes will be constant, and the amount of diffusion along isophotes will be defined by the diffusivity constant, *C*, alone.

Once again it is convenient to turn to the images in Figure 2-22. The boundary values for the

diffusivity constant, *C*, is determined to 0.05 and 5

⋅

10

−

5

, through visual assessment. It should be noted that it is plausible that there exist combinations of

α

and *C*, which will allow a lower number of iteration steps without noticeably changing the result of the diffusion process. For example, if the number of iterations is decreased to half of the initial value, then the

α

-value should be doubled and *C* must be trimmed (decreased) to give results comparable to the previous parameter setting. This would improve the speed of the diffusion calculation.

However, this aspect is outside the scope of this thesis, and will not be examined further.

**5.3.3 Specification of Parameter Sample Values **

There are two parameters that sample values should be defined for, namely the diffusivity constant, *C*, and the integration scale,

ρ

. For both these cases the values have been chosen by

visual assessment (see Figure 2-22). The integration scale is sampled with five values, and the

diffusivity constant is sampled with four. The parameter values for the nonlinear anisotropic

diffusion method are summarised in Table 5-6.

*t ds *

0.0625

0.2

No. of iterations 50

α

0.01

*C *

0.05, 0.005, 0.0005, 5

⋅

10

−

5

ρ

0.5, 1, 2, 4, 8

**Table 5-6: Parameter values for nonlinear anisotropic diffusion. **

**5.3.4 Implementation of Nonlinear Anisotropic Diffusion **

The calculation of the diffusion tensor is following the coherence-enhancing diffusion

filtering method as proposed by Weickert [28] and described in detail in Chapter 2.2.4.

62

The discrete approximation of the tensor driven diffusion was adapted from the description given by Rein van den Boomgaard in [26], and is defined as follows:

*L s x*

,

+

*ds y*

2

(

(

*a s x*

,

*y*

=

−

1

*L s x*

,

+

*a y s x*

,

+

*y*

)

)

*ds*

4

*L x*

,

[

*y*

−

−

1

(

*s b x*

,

−

2

*y*

(

−

1

*a*

(

+

*x*

,

*y b s x*

+

1 ,

−

1

+

*y*

2

*a*

)

*L x x*

+

1 ,

,

*y y*

−

1

+

*a*

+

*x*

,

2

*y*

+

1

(

) (

*c*

+

*s x*

+

1 ,

*c y x*

,

*y*

+

−

1

*c*

+

*s x*

,

*y*

2

*c*

)

*L x*

,

*y*

)

*x*

+

1 ,

+

*y c*

+

*x*

,

(

*y b x s*

,

]

+

1

*y*

+

1

)

*L x*

,

*y*

+

*b*

+

*s x*

+

1 ,

2

(

*a y*

)

*x*

,

*L y x*

+

1 ,

+

1

+

*y*

+

1

*a x*

,

+

*y*

)

*L x*

,

*y*

+

1

+

*b x*

,

*y*

−

1 where

+

*b x*

−

1 ,

*y*

*L x*

−

1 ,

*y*

−

1

+

2

*c x*

−

1 ,

*y*

+

*c x*

,

*y*

*L x*

−

1 ,

*y*

−

*b x*

,

*y*

+

1

+

*b x*

−

1 ,

*s* is the scale, *x* and *y* are the spatial position, *L*

*s y*

*L x*

−

1 ,

*y*

+

1

is the image at scale *s* and *a*, *b* and *c*

are the components of the diffusion tensor as described in 2.2.4.

**5.3.5 Results **

The result of the three best and the three worst cases for nonlinear anisotropic diffusion are

presented in Table 5-7 and Table 5-8 respectively. Once again there is not a single parameter

combination that generates better overall results than the initially calculated evaluation

measures (see Chapter 4.5). Another conclusion that can be made is that the method performs

best for high values of the diffusivity constant, *C*, and worst for low values. This means that a high threshold for the anisotropy of the diffusion is preferred, resulting in relatively low difference between the smoothing along, and perpendicular to, the ridge structure.

*C*,

σ

0.05, 2

0.05, 4

**Higher **

*UM rel*

** (%) **

40.1

39.5

**Higher **

*MNCC feat*

** (%) **

97.0

97.3

**Higher UM**

*rel*

** and **

*MNCC feat*

** (%) **

40.1

39.5

**Number of correct matches **

664

663

0.05, 1 41.1 97.2 40.9 663

**Table 5-7: Three best cases for tensor driven diffusion. Percentile values compared to initial evaluation measures. **

*C*,

σ

5

⋅

10

−

5

, 2

5

⋅

10

−

5

, 4

5

⋅

10

−

5

, 8

**Higher **

*UM rel*

** (%) **

**Higher **

*MNCC feat*

** (%) **

**Higher UM**

*rel*

** and **

*MNCC feat*

** (%) **

25.0 88.2 24.9

**Number of correct matches **

562

24.5 88.5 24.1

22.4 91.2 22.3

562

563

**Table 5-8: Three worst cases for tensor driven diffusion. Percentile values compared to initial evaluation measures. **

The histograms of

*UM rel*

and

*MNCC feat*

for the best performing parameter setting (i.e. *C* = 0.05 and

σ

= 2) are very similar to the ones that were calculated for the linear scale-space method

(Figure 5-2 and Figure 5-3) and therefore they are not included. The

*MNCC feat*

value is almost

exclusively increasing for the three best cases that are presented in Table 5-7. This suggests

that the reason why more than half of the correlation pairs get a lower *UM*

*rel*

value is due to an increase of the *MNCC* value for structure that is similar to the feature tile.

Considering all parameter settings there were a total of 24 different correlation pairs that were initially false matches, but were successfully matched when scale-space filtered. This is far better than for the two previously tested methods, which had 12 respectively 16 correlation pairs that rendered this type of result.

63

The detailed analysis will start by examining the correlation pairs that performed worse (i.e. had a substantially lower

*UM rel*

value compared to the initially calculated

*UM rel*

). There were generally two different reasons for a decreasing

*UM rel*

value for scale-space filtered images.

Firstly, two images of the same feature tile that locally differs in local mean-grey value may enhance this deviation when they are scale-space smoothed. This problem is very similar to the problem described for the linear scale-space method, when the *MNCC*

*feat*

value decreased

at larger scales (see Chapter 5.1.5 and Figure 5-5). This problem is depicted in the two top-

most rows in Figure 5-13. The two images to the left are the initial correlation pair, and the

two to the right are the scale-space filtered images.

The second problem, which is the more common, involves images with very detailed structure that are of low quality. In these cases the orientation estimation that is calculated by the structure tensor may deviate between two images of the same feature tile. Hence, the diffusion process will enhance the coherence in different direction for the two images. This will render very poor results correlation-wise, since new structure that does not exist in the original

fingerprint may be created. This problem is illustrated in the two bottom rows in Figure 5-13.

The worst cases almost explicitly appear for the two lowest value of the diffusivity constant,

*C*, which are the situations involving maximum diffusion. It should however be noted, that even though the problem is not as serious for higher *C*-values (i.e. less diffusion), they still exist for the type of correlation pairs that include this type of problem.

**Figure 5-13: Four examples where the tensor driven diffusion noticeable decreased the **

*UM rel*

** value; (left) initial correlation pairs, (right) scale-space filtered images. **

The nonlinear anisotropic diffusion method was able to correctly match 24 correlation pairs that failed initial correlation. The images for these correlation pairs were investigated and they could be divided into three groups, depending on the reason for the improvement. Firstly there were 6 correlation pairs from the same sub-area group, where the initial mismatch was due to a very similar feature that got a higher correlation value. These false matches are due to a poor selection of a distinct feature, rather than the template ageing problem, therefore they are not investigated further. There were also a few correlation pairs that rendered false matches

64

due to misregistration between the images. Again, this problem is not within the scope of the template ageing problem. The final category, which includes most of the improved matches, are fingerprints where the local structure is of low quality and the nonlinear anisotropic diffusion is able to suppress spurious information and enhance the coarser structure. In other words the structure is more stable at a larger scale than the details within the feature tile. Four

examples of this type of result are depicted in Figure 5-14. The two columns to the left are the

initial correlation pairs, and the two to the right are the scale-space enhanced versions. This type of improvement is what was anticipated when the nonlinear anisotropic scale-space

method was initially suggested. The examples in Figure 5-14 shows that this method is

capable of enhancing the correlation, and make possible an accurate match, for correlation pairs that are considerably affected by the template ageing.

**Figure 5-14: Four examples where the tensor driven diffusion made possible an accurate match; (left) initial correlation pairs, (right) scale-space filtered images. **

The nonlinear anisotropic scale-space, or tensor driven diffusion, method proposed in this section was not able to produce an overall better result than the initially calculated evaluation measures. Furthermore, when diffused extensively there is an apparent risk that low quality structure is smoothed in orientations deviating from the actual fingerprint pattern, due to inaccuracy in the analysis of the local structure. It has thus been shown that the template ageing problem may affect structure at larger scales than the ridge structure itself. For these cases the structure tensor of integration scale is not able to accurately analyse the fingerprint structure. The risk that features may be enhanced in incorrect orientations may be avoided by strongly limiting the maximum amount of diffusion allowed, as well as the maximum value of the integration scale. The most reasonable solution would be to decrease the number of iterations as a starting point, and then adapt the remaining parameters to produce valid results.

The tensor dependent diffusion proved to be capable of enhancing the correlation, and make possible an accurate match, for images that are considerably affected by the template ageing.

This result alone makes the method highly interesting and motivates further investigation on possibilities to enhance the overall result as well. For example some quality measure could be

65

used to determine whether diffusion enhancement is necessary or not, and in this way only perform smoothing for images that are less likely to initially perform an accurate match.

66

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