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Light intensity independence during dynamic laser speckle analysis
Optics Communications 366 (2016) 185–193
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Invited Paper
Light intensity independence during dynamic laser speckle analysis Renan Oliveira Reis a, Hector J. Rabal b, Roberto A. Braga a,n a
Universidade Federal de Lavras, Departamento de Engenharia, CP3037, Lavras, MG, Brazil Centro de Investigaciones Ópticas (CCT CONICET La Plata-CIC) and UID Optimo, Departamento de Ciencias Básicas, Facultad de Ingeniería, Universidad Nacional de la Plata, P.O. Box 3, Gonnet, La Plata 1897, Argentina
b
art ic l e i nf o
a b s t r a c t
Article history: Received 31 August 2015 Received in revised form 22 December 2015 Accepted 23 December 2015
We explore some different normalizations of current dynamic laser speckle activity measures searching for their performance with respect to the illumination inhomogeneity of the samples. Inertia Moment and Average Value of Differences of the co-occurrence matrix are compared using a paint-drying case study on a uniform sample where attenuation in a portion of the illuminated area is introduced using a neutral density filter. In this way, all environmental conditions being equal but non-uniform illumination permits the comparison on a better approximation to objectivity. The results presented show that it is possible to mitigate the effects of the illumination in the activities measured by the dynamic laser speckle. & 2015 Elsevier B.V. All rights reserved.
Keywords: Biospeckle Paint drying Co-occurrence matrix
1. Introduction Dynamic laser speckle analysis is conducted by means of image processing of the speckle patterns in order to evaluate their level of changes with time, which can be expressed by means of graphical and numerical outcomes. Graphical outcomes are normally adopted where the aim is to create a map of activity displaying regions with different activities. In turn, numerical outcomes are adopted when the observed area is considered homogeneous, thus without significant changes of activity. In numerical approaches, some dynamic laser speckle applications use second order statistics such as autocorrelation analysis [1,2], or the Inertia Moment and its variation Absolute Value of the Differences [3,4]. In the Inertia Moment (IM) and in the Absolute Value of the Differences (AVD), analysis of the data is done by means of the cooccurrence matrix obtained from the time history of the speckle pattern [5], originally proposed through a series of texture evaluation operations [6]. The final values of the IM and AVD are presented with respect to the grey level distribution of the points in the co-occurrence matrix around its principal diagonal, defined (using a mechanical analogy) as Eq. (1).
n
Corresponding author. E-mail addresses: [email protected]fla.br, [email protected] (R.A. Braga).
http://dx.doi.org/10.1016/j.optcom.2015.12.062 0030-4018/& 2015 Elsevier B.V. All rights reserved.
⎛ Mij ⎞ IM= ∑ ∑ ⎜ ⎟ (i − j )2 ⎝ Norm ⎠
(1)
where Mij are the entries in the co-occurrence matrix defined as the number of times that a grey level i is followed in time by a grey level j and Norm is a normalization constant of the data, which is the sum of each row of the Mij so that the sum is then forced to be 1 [3]. Then, the coefficient of the squared difference can be interpreted as an estimation of the conditional probability of the intensity transition i-j given that the first occurrence was i. So, the result of the calculation is the contribution of that row to the IM, which is the mean value of the magnitude (i j)2, also given that the first occurrence was i. In order to compare alternatives we will call this normalization local line normalization (LLN). In the AVD, the difference is the substitution of the square operation by the absolute value of the differences between i and j. In [7] a different normalization was proposed in order to enhance the robustness of the procedure, thereby reducing the effect of the variations of the intensity in some areas of the image, such as when one illuminates a round object or even an irregular object that should have the same activity in all regions of the image. We will call this normalization whole image normalization (WIN), defined by dividing each matrix entry by the total number of occurrences. Then this weight factor becomes an estimation of the probability of the difference (i j) independently of the value of i. The IM or AVD defined with this normalization is the mean value of (i j)2, or |i j| respectively, that is, the intensity change for any value of the first intensity i. In the IM, those occurrences that are far from the principal
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diagonal are weighted by the square or absolute operation. Occurrences on the diagonal do not add to the result, as they represent no changes between consecutive frames in the time history. It can be seen that the results rely heavily on intensity measurements that are affected by the registering medium. Digital cameras, which use charge coupled devices (CCD) arrays to register the scene, as occur with CMOS sensors, in digital cameras suffer with the nonlinearities during the assembling and storage [8]. One reason for the nonlinearity is due to the saturation point, where pixels with irradiance above the limit are set to the maximum value represented by the limit. In addition, the variation of the light in the sample can be attributed to other factors, among them the distribution of the intensity of the light in the laser beam, that can spread over the surface as a Gaussian function, i.e., more light in the center with a reduction in the surrounding circles. The two normalizations give different results, as for the first to become equal to the second, each row should be multiplied by the probability of i appearing as the first element, that is, by the first image histogram. The second normalization [7] is not dependent on the histogram of the first image, and it is somewhat immune to offsets subtraction. That is, when the observed phenomenon is the same in the whole illuminated area, two THSPs where some rows are more illuminated (by stray light or by some irregularities in the sample surface) should give similar results. The search for a greater independence of the biospeckle laser results concerned with the level of illumination in the sample led to a proposal [9] modifying the Fujii algorithm [10] (Eq. (1)), and another that proposed a pre-processing approach of the image using normal vectors [11]. Both works dealt with the enhancement of the equalization of the contrast of different areas within an image. However, they did not investigate the results numerically and subjective judgment of graphical outcomes cannot express the actual efficiency. In this work we test the influence of the illumination inhomogeneity in dynamic laser speckle patterns using numerical analysis on actual experimental data. This is done using a similar measurement to that proposed by Fujii [10], described by:
F (x, y)= ∑ k
Ik (x, y) − Ik +1 (x, y) Ik (x, y) + Ik +1 (x, y)
Fig. 1. Experimental configuration of the data acquisition with a neutral density filter creating a dark area in the speckle pattern of paint drying on a plate.
(2)
where Ik (x, y ) is the intensity in pixel (x,y) and frame k of a stack of images of dynamic speckle and the bars indicate an absolute value.
2. Methodology We tested the traditional approaches for numerical analysis of the dynamic laser speckle and some new variations in regions differing only in their illumination conditions. A local change in illumination should be of no consequence to the results, provided that the linearity of the CCD detector can be assumed. Eqs. (3) and (4) present the Inertia Moment (IM) and Average Value of Differences (AVD) algorithms using a mimic concept adopted from the Fujii algorithm (1), including a denominator in the traditional method required in order to obtain a measure independent of the illumination units.
⎛ Mij ⎞ (i − j )2 New IM= ∑ ∑ ⎜ ⎟ ⎝ Norm ⎠ (i + j )2
(3)
Fig. 2. Construction of the two Time History Speckle Patterns (THSP) using one column from the dark region in one and one column from the light one in the other.
⎛ Mij ⎞ i − j New AVD= ∑ ∑ ⎜ ⎟ ⎝ Norm ⎠ i + j
(4)
By using these expressions, constant attenuation factors due to inhomogeneous illumination on otherwise equal activity regions should be canceled out. One additional change, in Eqs. (3) and (4), was analyzed and it consisted in raising to power p the values i and j before their use in
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⎛ M ⎞ (ip − j )2 ij p ⎟⎟ New IMp = ΣΣ ⎜⎜ 2 ⎝ Norm ⎠ (ip + jp )
(5)
⎛ Mij ⎞ ip − jp New AVDp= ∑ ∑ ⎜ ⎟ ⎝ Norm ⎠ ip + jp
(6)
where ip and jp are i and j raised to power values (0.1, 0.2, 0.3, 0.4, 0.5, 0.8 and 1). i.e. ip ¼ ip and jp ¼jp. 2.1. Dynamic speckle data from a drying paint sample
Fig. 3. Composite image built by using lines from the THSPs in the light and in the dark areas.
the calculations, where p is a parameter. This test was performed based on the improvements found before in some measuring methods by using non-extensive statistics [12].
Data from a drying paint sample were tested, where the amount of light illuminating its surface was different, as shown in Fig. 1. A glass plate, covered with paint during the drying process, was illuminated by helium neon laser light and its speckle images recorded and processed as usual. A neutral density filter (D¼ 0.5) was introduced to cover a part of the illuminated region. In this way, both parts, undergoing the same dynamic process, should lead to similar measurements but under different illuminations. There was some inhomogeneity in the illumination due to the use of a spherical wave, but it was small in comparison with that introduced by the filter. The time history of the speckle pattern was constructed for 11 drying time steps (0, 10, 20, 30, 40, 50, 60, 75, 105, 120, 150 min).
Fig. 4. Comparison of the traditional (a) IM and (b) AVD, as well as the proposed changes in the (c) IM and (d) AVD. The local line normalization (dotted line, in area 1, light; and dashed line, in area 2, dark) and whole image normalization (continuous normal line, in area 1, light, and continuous bold line, in area 2, dark) of drying paint illuminated by the laser.
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Fig. 5. Comparison of the traditional (a) IM and (b) AVD, as well as the proposed changes in the (c) IM and (d) AVD with the local line normalization (dotted and dashed lines) and whole image normalization (continuous lines) in the combined THSP. The light THSP with 512 lines, without any combination, was analyzed and is represented by the dotted and continuous lines with “x” marks. The combination of 100 dark columns, over the first 100 columns of the light THSP, is represented by the dashed and bold continuous lines of drying paint illuminated by the laser.
Fig. 6. Errors of the IM and AVD with different normalizations related to one bright area of the THSP of the painting at minute zero with the inclusion of rows from the dark area, where the continuous line with “x” marks represents the IM using the traditional local line normalization, the dashed line represents the AVD using the traditional local line normalization, the dotted line represents the IM using the local line normalization associated with the proposed normalization, and the bold continuous line represents the AVD using the local line normalization associated with the proposed normalization.
2.2. Analysis of the data The areas of the same activity but different illumination obtained as in Fig. 1 were analyzed both separately and in combination. The
Fig. 7. Errors of the IM and AVD with different normalizations related to one bright area of the THSP of the paint at minute zero with the inclusion of rows from the dark area, where the continuous line with “x” marks represents the IM using the traditional whole image normalization, the dashed line represents the AVD using the traditional whole image normalization, the dotted line represents the IM using the whole image normalization associated with the proposed normalization, and the bold continuous line represents the AVD using the whole image normalization associated with the proposed normalization.
analysis of the separated areas used the THSP matrices constructed as presented in Fig. 2 for each step of drying, and the numerical outcomes were calculated using the different normalizations. The local line normalization [3] and the whole image normalization [4] were tested with and without the changes proposed by Eqs. (3) and
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Fig. 8. Analysis of each column of a THSP with dark and light areas using the IM and AVD with the local line and whole image normalizations. The mean values of the light (left) and dark (right) regarding lines 231–254 (transition) and the standard deviation are presented separately: (a) the IM using the local line normalization, (b) the IM using the whole image normalization, (c) the AVD using the local line normalization and (d) the AVD using the whole image normalization.
(4). The testing was also performed with the change proposed in Eq. (5), regarding raising to a power the intensities associated with all the normalizations tested before. The second approach used the combinations of the dark and light areas. This was done by picking a number of rows from the dark area THSP and introducing them into the THSP of the light area (Fig. 3), starting with one line and increasing the number to 100 lines. An error index was created to compare the approaches. It is defined in Eq. (7)
error =1 −
IMLight THSP IMMixed THSP
(7)
where IMLight THSP is the value of the Inertia Moment applied to a THSP when the light was not damped by the filter, and IMMixed THSP is the value of the Inertia Moment applied to a THSP with light lines associated with an increasing number of lines from 1 to 100 taken from the other (dark) THSP, as described in Fig. 3. Additional analyses of the results were conducted by means of the evaluation using the co-occurrence matrix of each column of Fig. 1 with time and the comparison of their results obtained with the different approaches tested in this work. These analyses and comparisons were carried out in the light and dark areas including the transition region, and the mean and standard deviation were used to evaluate their differences.
3. Results and discussions 3.1. Results of paint drying with two different non-connected strips in the THSP In Fig. 4a and b we show the results of the different normalizations (local line normalization and whole image normalization) on the AVD and IM methods applied to two areas of the paint drying separately (as obtained in Fig. 1), where label 1 means the light area and label 2 means the dark area. The results were parametrized in order to adjust them to the same scale, so the first point, related to time zero, is always 1, and the actual offset between them is observed in the next points. The same was done using the proposed normalization (Eqs. (3) and (4)) and is represented in Fig. 4c and d. It is clear from Fig. 4a and b that the activity observed by IM and AVD in the two areas of paint drying showed the same behavior apart from minor differences between the activities in the separate areas 1 and 2, i.e., light and dark areas respectively. Once the two different areas were analyzed without any crossed information about each other, the results seem reasonable, and some effort should be made in order to create a procedure that is completely independent of the illumination level without any reference. The adoption of the proposed normalization (Eqs. (3) and (4)) presented the phenomenon with the same behavior, where a clear improvement in the outcomes was not observed. The improvement
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Fig. 9. Analysis of each line of a THSP with a dark and light area using the IM and AVD with the local line and whole image normalizations associated with the proposed normalization. The mean values of the light (left) and dark (right) regarding lines 231–254 (transition) and the standard deviation are presented separately: (a) the IM using the local line normalization, (b) the IM using the whole image normalization, (c) the AVD using the local line normalization and (d) the AVD using the whole image normalization.
expected would be the reduction of the offset between the outcomes, i.e., the dotted line closer to the dashed line, and the continuous line closer to the bold continuous line. Therefore, one can conclude that the additional efforts proposed by Eqs. (3) and (4) are not necessary at first glance. 3.2. Results of combined areas in the THSP The IM and AVD on the connected areas, which means the insertion of dark lines on the light THSP (Fig. 3), is presented in Fig. 5 with the adoption of the traditional (local line and whole image) normalizations and of the proposed normalization (Eqs. (3) and (4)). One hundred lines taken from the dark THSP were inserted to replace the first 100 lines of the light THSP. Whilst the analysis of the separate areas (dark and light) observed in Fig. 4 is relevant to guarantee the same results under distinct illuminations of the same material, the results presented in Fig. 5 are most common in samples where the light is not homogeneous. In Fig. 5 it is possible to see that a band with 100 dark lines inserted on a light THSP did not present visible differences when IM and AVD were applied using the traditional normalizations [3,4]. This was an important result, since the applications of the dynamic laser speckle in homogeneous samples (i.e. they should have the same activity) should present the same results comparing a THSP with changes in the illumination of its lines. That occurrence is common in most applications of biological material such as seeds [13], fruits [2], kefir [14,15], etc. and is related to the great roughness of the surface or to its spherical form. The proposed normalization (Fig. 5c and d) was able to follow
the phenomenon, although with more ripple mainly when the local line normalization [3] was adopted. The calculations using intensities raised to power p ¼0.3 (Eqs. (5) and (6)) were also tested, with roughly the same outcomes observed as in the approaches before (not presented). The approach using IM and AVD with the local line normalization biased by the proposed normalization associated with the power 0.3 presented more ripple than the others. The ripple occurs when the activity is considered stabilized; therefore, the proposed normalization (Fig. 4c and d and Fig. 5c and d) is more sensitive to the noise presented in those stabilized regions, which cannot be considered an important drawback of the proposed technique, since the major applications are concerned with the transition from one activity state to another. Thus, from time zero to 30 min, when the paint drying process showed stabilization, the proposed normalization, as well as the power introduced in the equations, could follow the phenomenon in the same way as the traditional methods. Therefore, the additional tests conducted were done to investigate in detail the differences, sensitivities and influence of illumination in the results. They were carried out, for instance, to evaluate the effect of light variation on the results of dynamic laser speckle, regarding the offset observed in Figs. 4 and 5, as well as the variations with time. These additional tests were done in two ways: defining an error index related to the insertion of dark lines in a light THSP from 1 to 100, and comparing the results of using the co-occurrence matrix obtained from each column of Fig. 2 analyzed by the tested methods.
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Fig. 10. Analysis of each line of a THSP with a dark and light area using the IM and AVD with the local line and whole image normalizations using power 0.3. The mean values of the light (left) and dark (right) regarding lines 231–254 (transition) and the standard deviation are presented separately: (a) the IM using the local line normalization, (b) the IM using the whole image normalization, (c) the AVD using the local line normalization, (d) the AVD using the whole image normalization.
3.3. Additional tests – inserting dark lines in a light THSP Fig. 6 presents the errors (Eq. (7)) produced by the gradual inclusion of dark lines on the head of a THSP with bright lines, both related to the same dynamic laser speckle phenomenon (drying paint), in order to evaluate the sensitivity of the methods. The errors were disclosed first the local line normalization adopted for IM and AVD. The errors were small, not more than 2.5%, and they increased with the number of lines inserted. In addition, it is possible to observe some jumps in the tendency, which should be analyzed in detail. The jumps are linked to the local line normalization, which varies in accordance with each line of the THSP, which varies and creates the co-occurrence matrix with some unbalanced regions [7]. This can be proved by the stability observed in Fig. 7 when the whole image normalization was adopted as expected [7]. Fig. 7 shows the errors related to the gradual inclusion of dark lines in the THSP with bright lines, both related to the same dynamic laser speckle phenomenon, in this case using the whole image normalization [7]. In contrast with the local line normalization [3], the whole image normalization associated with the changes proposed in this work presented the best results (bold line). The errors behaved in a similar way when the paint was allowed to dry for different times. The errors regarding the insertion of lines, such as in the results of Figs. 6 and 7 when the power 0.3 was used (Eqs. (5) and (6)), showed similar outcomes (not presented). We could observe that the traditional methods are relatively independent of the illumination inhomogeneity on the surfaces analyzed by the dynamic laser speckle (Fig. 4). However, the independence can be improved by means of some adjustments such
as the use of a denominator in the traditional equations (Fig. 5), which can be observed in detail with the additional tests. 3.4. Additional tests – comparing each column of the sample In Fig. 8, the IM and AVD from the THSPs, formed by using each column of the speckle patterns (Fig. 2), are presented and they show the outcomes in the light and dark THSP areas over time. In Fig. 8 the traditional AVD and IM methods using the local line normalization and the whole image normalization are presented and it is clear that the level of illumination drives the results, with the mean values of the dark portion different from the light one. The high variation of the local line normalization was expected and it confirms the enhancement provided by the whole image normalization [7] despite the observed offset presented when the traditional normalizations are adopted. Fig. 9 presents a reduction in the differences between the mean values of the right side (light THSP) and the mean values of the left side (dark THSP) in this case using the proposed normalization (Eqs. (3) and (4)). The AVD method using local line and whole image normalization presented the best results. The same comparions of IM and AVD with variations were conducted using the power 0.3 (Eqs. (5) and (6)) at the zero instant are presented in Fig. 10. Figs. 8–11 present the mean values of the IM and AVD using all the tested approaches, as well as the standard deviation of the values on each side of the transition point (columns 231–254). It is clear that the step presented by the traditional methods (Fig. 8) is predominant. It is also observed that the adoption of whole image normalization presented outcomes with less variation, as expected [7].
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Fig. 11. Analysis of each column of a THSP with a dark and light area using the IM and AVD with the local line and whole image normalizations associated with the proposed normalization and power 0.3. The mean values of the light (left) and dark (right) regarding lines 231–254 (transition) and the standard deviation are presented separately: (a) the IM using the local line normalization, (b) the IM using the whole image normalization, (c) the AVD using the local line normalization, (d) the AVD using the whole image normalization.
Table 1 Local line normalization, whole image normalization and proposed normalization with the percentage of the differences between the results on the light (left) and dark (right) sides of the transition columns 231–254 of the THSP. Difference between the results on the light and dark sides of the THSP (%)
IM local line normalization AVD local line normalization IM whole image normalization AVD whole image normalization
Traditional normalization
Proposed normalization
23.9
4.8
19.0
3.1
53.2
3.3
25.3
1.9
In Table 1 we show the percentage difference between the mean values of each side of the transition point (lines 231–254) using the methods on each column with and without the power, in this case the power 0.3. In the results presented in Table 1, the outcomes are related only to the zero instant after painting and with power p ¼1. Other powers were also tested and the results showed that the best approach regarding all drying times is the AVD using the whole image normalization with the proposed denominator associated. Additionally, the adoption of powers does not change the best results observed with power equal to 1, particularly when the AVD was used associated with the proposed normalization (introduction of a denominator). The percentage of difference between the
results in the light and dark sides using the powers is presented in Fig. 12a, all of them related to the zero instant after painting. It is possible to see that the approaches related to the proposed normalization (Eqs. (5) and (6)) presented the best results at p ¼1, although the adoption of other powers (0.1–0.9) did not compromise the results. However, when the traditional methods were used, the adoption of a small power (0.1) enhanced the results, i.e., the differences between the two sides tended to be lowest. Fig. 12b presents the evolution of the percentage difference with respect to the powers at 30 min of drying time. At this point, the drying process had stabilized (see Fig. 4), and the results are different from the zero instant. The best result can be attributed to the AVD using local line normalization associated with the power 1 and with the proposed normalization, despite its larger variation provided by the adoption of the whole image normalization. Therefore, as a first choice we can select the AVD using local line normalization associated with the power 1 as the one that best fulfils a lower sensitivity to illumination inhomogeneity. However, if we consider only the analysis when the activities are not stabilized, it is possible also to consider the usage of the AVD with the whole image normalization, since the outcomes are more stable and with reduced variations. The usage of more than one outcome can be one alternative, since biological phenomena are very complex, with great peculiarities, thus offering the opportunity to bias the best outcome to a particular phenomenon. The independence of the light's uneven illumination during the dynamic laser speckle analysis is an important achievement of this work.
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Fig. 12. Evolution of the percentage difference regarding the powers at two times of drying (a) zero and (b) 30 min; associated with local line normalization (LLN), whole image normalization (WIN) and proposed normalization (New) with the percentage of the difference between the mean values of the activity on the light (left) and dark (right) sides of the transition columns 231–254 of the THSP. The value p means the adopted power.
4. Conclusions
References
We tested the traditional methods to analyze dynamic laser speckle when the illumination of a sample with uniform activity is not homogeneous. The traditional methods presented reliable results. However, when observed in detail, we can see that the methods proposed here can improve the independence with respect to the illumination. In particular, we observe that the AVD method using the local normalization associated with the inclusion of the denominator in the traditional method (proposed normalization) presented the best results. Particularly, we elected the AVD using local line normalization associated with power 1 as the one that best fulfils a lower sensitivity to illumination inhomogeneity. When the activities were not stabilized, it was possible also to consider the usage of the AVD with the whole image normalization. The combination of more than one outcome should be considered to choose the best outcome.
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Acknowledgments This work was supported by Grant PICT 2008-1430 from the Agencia Nacional de Promoción en Ciencia y Tecnología (ANPCyT), Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Comisión de Investigaciones Científicas de la Provincia de Buenos Aires (CIC) and Universidad Nacional de La Plata (UNLP), Argentina. We also thank the Federal University of Lavras, and CNPq, Fapemig, Capes and Finep who partially financed the project.
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Invited Paper
Light intensity independence during dynamic laser speckle analysis Renan Oliveira Reis a, Hector J. Rabal b, Roberto A. Braga a,n a
Universidade Federal de Lavras, Departamento de Engenharia, CP3037, Lavras, MG, Brazil Centro de Investigaciones Ópticas (CCT CONICET La Plata-CIC) and UID Optimo, Departamento de Ciencias Básicas, Facultad de Ingeniería, Universidad Nacional de la Plata, P.O. Box 3, Gonnet, La Plata 1897, Argentina
b
art ic l e i nf o
a b s t r a c t
Article history: Received 31 August 2015 Received in revised form 22 December 2015 Accepted 23 December 2015
We explore some different normalizations of current dynamic laser speckle activity measures searching for their performance with respect to the illumination inhomogeneity of the samples. Inertia Moment and Average Value of Differences of the co-occurrence matrix are compared using a paint-drying case study on a uniform sample where attenuation in a portion of the illuminated area is introduced using a neutral density filter. In this way, all environmental conditions being equal but non-uniform illumination permits the comparison on a better approximation to objectivity. The results presented show that it is possible to mitigate the effects of the illumination in the activities measured by the dynamic laser speckle. & 2015 Elsevier B.V. All rights reserved.
Keywords: Biospeckle Paint drying Co-occurrence matrix
1. Introduction Dynamic laser speckle analysis is conducted by means of image processing of the speckle patterns in order to evaluate their level of changes with time, which can be expressed by means of graphical and numerical outcomes. Graphical outcomes are normally adopted where the aim is to create a map of activity displaying regions with different activities. In turn, numerical outcomes are adopted when the observed area is considered homogeneous, thus without significant changes of activity. In numerical approaches, some dynamic laser speckle applications use second order statistics such as autocorrelation analysis [1,2], or the Inertia Moment and its variation Absolute Value of the Differences [3,4]. In the Inertia Moment (IM) and in the Absolute Value of the Differences (AVD), analysis of the data is done by means of the cooccurrence matrix obtained from the time history of the speckle pattern [5], originally proposed through a series of texture evaluation operations [6]. The final values of the IM and AVD are presented with respect to the grey level distribution of the points in the co-occurrence matrix around its principal diagonal, defined (using a mechanical analogy) as Eq. (1).
n
Corresponding author. E-mail addresses: [email protected]fla.br, [email protected] (R.A. Braga).
http://dx.doi.org/10.1016/j.optcom.2015.12.062 0030-4018/& 2015 Elsevier B.V. All rights reserved.
⎛ Mij ⎞ IM= ∑ ∑ ⎜ ⎟ (i − j )2 ⎝ Norm ⎠
(1)
where Mij are the entries in the co-occurrence matrix defined as the number of times that a grey level i is followed in time by a grey level j and Norm is a normalization constant of the data, which is the sum of each row of the Mij so that the sum is then forced to be 1 [3]. Then, the coefficient of the squared difference can be interpreted as an estimation of the conditional probability of the intensity transition i-j given that the first occurrence was i. So, the result of the calculation is the contribution of that row to the IM, which is the mean value of the magnitude (i j)2, also given that the first occurrence was i. In order to compare alternatives we will call this normalization local line normalization (LLN). In the AVD, the difference is the substitution of the square operation by the absolute value of the differences between i and j. In [7] a different normalization was proposed in order to enhance the robustness of the procedure, thereby reducing the effect of the variations of the intensity in some areas of the image, such as when one illuminates a round object or even an irregular object that should have the same activity in all regions of the image. We will call this normalization whole image normalization (WIN), defined by dividing each matrix entry by the total number of occurrences. Then this weight factor becomes an estimation of the probability of the difference (i j) independently of the value of i. The IM or AVD defined with this normalization is the mean value of (i j)2, or |i j| respectively, that is, the intensity change for any value of the first intensity i. In the IM, those occurrences that are far from the principal
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diagonal are weighted by the square or absolute operation. Occurrences on the diagonal do not add to the result, as they represent no changes between consecutive frames in the time history. It can be seen that the results rely heavily on intensity measurements that are affected by the registering medium. Digital cameras, which use charge coupled devices (CCD) arrays to register the scene, as occur with CMOS sensors, in digital cameras suffer with the nonlinearities during the assembling and storage [8]. One reason for the nonlinearity is due to the saturation point, where pixels with irradiance above the limit are set to the maximum value represented by the limit. In addition, the variation of the light in the sample can be attributed to other factors, among them the distribution of the intensity of the light in the laser beam, that can spread over the surface as a Gaussian function, i.e., more light in the center with a reduction in the surrounding circles. The two normalizations give different results, as for the first to become equal to the second, each row should be multiplied by the probability of i appearing as the first element, that is, by the first image histogram. The second normalization [7] is not dependent on the histogram of the first image, and it is somewhat immune to offsets subtraction. That is, when the observed phenomenon is the same in the whole illuminated area, two THSPs where some rows are more illuminated (by stray light or by some irregularities in the sample surface) should give similar results. The search for a greater independence of the biospeckle laser results concerned with the level of illumination in the sample led to a proposal [9] modifying the Fujii algorithm [10] (Eq. (1)), and another that proposed a pre-processing approach of the image using normal vectors [11]. Both works dealt with the enhancement of the equalization of the contrast of different areas within an image. However, they did not investigate the results numerically and subjective judgment of graphical outcomes cannot express the actual efficiency. In this work we test the influence of the illumination inhomogeneity in dynamic laser speckle patterns using numerical analysis on actual experimental data. This is done using a similar measurement to that proposed by Fujii [10], described by:
F (x, y)= ∑ k
Ik (x, y) − Ik +1 (x, y) Ik (x, y) + Ik +1 (x, y)
Fig. 1. Experimental configuration of the data acquisition with a neutral density filter creating a dark area in the speckle pattern of paint drying on a plate.
(2)
where Ik (x, y ) is the intensity in pixel (x,y) and frame k of a stack of images of dynamic speckle and the bars indicate an absolute value.
2. Methodology We tested the traditional approaches for numerical analysis of the dynamic laser speckle and some new variations in regions differing only in their illumination conditions. A local change in illumination should be of no consequence to the results, provided that the linearity of the CCD detector can be assumed. Eqs. (3) and (4) present the Inertia Moment (IM) and Average Value of Differences (AVD) algorithms using a mimic concept adopted from the Fujii algorithm (1), including a denominator in the traditional method required in order to obtain a measure independent of the illumination units.
⎛ Mij ⎞ (i − j )2 New IM= ∑ ∑ ⎜ ⎟ ⎝ Norm ⎠ (i + j )2
(3)
Fig. 2. Construction of the two Time History Speckle Patterns (THSP) using one column from the dark region in one and one column from the light one in the other.
⎛ Mij ⎞ i − j New AVD= ∑ ∑ ⎜ ⎟ ⎝ Norm ⎠ i + j
(4)
By using these expressions, constant attenuation factors due to inhomogeneous illumination on otherwise equal activity regions should be canceled out. One additional change, in Eqs. (3) and (4), was analyzed and it consisted in raising to power p the values i and j before their use in
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⎛ M ⎞ (ip − j )2 ij p ⎟⎟ New IMp = ΣΣ ⎜⎜ 2 ⎝ Norm ⎠ (ip + jp )
(5)
⎛ Mij ⎞ ip − jp New AVDp= ∑ ∑ ⎜ ⎟ ⎝ Norm ⎠ ip + jp
(6)
where ip and jp are i and j raised to power values (0.1, 0.2, 0.3, 0.4, 0.5, 0.8 and 1). i.e. ip ¼ ip and jp ¼jp. 2.1. Dynamic speckle data from a drying paint sample
Fig. 3. Composite image built by using lines from the THSPs in the light and in the dark areas.
the calculations, where p is a parameter. This test was performed based on the improvements found before in some measuring methods by using non-extensive statistics [12].
Data from a drying paint sample were tested, where the amount of light illuminating its surface was different, as shown in Fig. 1. A glass plate, covered with paint during the drying process, was illuminated by helium neon laser light and its speckle images recorded and processed as usual. A neutral density filter (D¼ 0.5) was introduced to cover a part of the illuminated region. In this way, both parts, undergoing the same dynamic process, should lead to similar measurements but under different illuminations. There was some inhomogeneity in the illumination due to the use of a spherical wave, but it was small in comparison with that introduced by the filter. The time history of the speckle pattern was constructed for 11 drying time steps (0, 10, 20, 30, 40, 50, 60, 75, 105, 120, 150 min).
Fig. 4. Comparison of the traditional (a) IM and (b) AVD, as well as the proposed changes in the (c) IM and (d) AVD. The local line normalization (dotted line, in area 1, light; and dashed line, in area 2, dark) and whole image normalization (continuous normal line, in area 1, light, and continuous bold line, in area 2, dark) of drying paint illuminated by the laser.
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Fig. 5. Comparison of the traditional (a) IM and (b) AVD, as well as the proposed changes in the (c) IM and (d) AVD with the local line normalization (dotted and dashed lines) and whole image normalization (continuous lines) in the combined THSP. The light THSP with 512 lines, without any combination, was analyzed and is represented by the dotted and continuous lines with “x” marks. The combination of 100 dark columns, over the first 100 columns of the light THSP, is represented by the dashed and bold continuous lines of drying paint illuminated by the laser.
Fig. 6. Errors of the IM and AVD with different normalizations related to one bright area of the THSP of the painting at minute zero with the inclusion of rows from the dark area, where the continuous line with “x” marks represents the IM using the traditional local line normalization, the dashed line represents the AVD using the traditional local line normalization, the dotted line represents the IM using the local line normalization associated with the proposed normalization, and the bold continuous line represents the AVD using the local line normalization associated with the proposed normalization.
2.2. Analysis of the data The areas of the same activity but different illumination obtained as in Fig. 1 were analyzed both separately and in combination. The
Fig. 7. Errors of the IM and AVD with different normalizations related to one bright area of the THSP of the paint at minute zero with the inclusion of rows from the dark area, where the continuous line with “x” marks represents the IM using the traditional whole image normalization, the dashed line represents the AVD using the traditional whole image normalization, the dotted line represents the IM using the whole image normalization associated with the proposed normalization, and the bold continuous line represents the AVD using the whole image normalization associated with the proposed normalization.
analysis of the separated areas used the THSP matrices constructed as presented in Fig. 2 for each step of drying, and the numerical outcomes were calculated using the different normalizations. The local line normalization [3] and the whole image normalization [4] were tested with and without the changes proposed by Eqs. (3) and
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Fig. 8. Analysis of each column of a THSP with dark and light areas using the IM and AVD with the local line and whole image normalizations. The mean values of the light (left) and dark (right) regarding lines 231–254 (transition) and the standard deviation are presented separately: (a) the IM using the local line normalization, (b) the IM using the whole image normalization, (c) the AVD using the local line normalization and (d) the AVD using the whole image normalization.
(4). The testing was also performed with the change proposed in Eq. (5), regarding raising to a power the intensities associated with all the normalizations tested before. The second approach used the combinations of the dark and light areas. This was done by picking a number of rows from the dark area THSP and introducing them into the THSP of the light area (Fig. 3), starting with one line and increasing the number to 100 lines. An error index was created to compare the approaches. It is defined in Eq. (7)
error =1 −
IMLight THSP IMMixed THSP
(7)
where IMLight THSP is the value of the Inertia Moment applied to a THSP when the light was not damped by the filter, and IMMixed THSP is the value of the Inertia Moment applied to a THSP with light lines associated with an increasing number of lines from 1 to 100 taken from the other (dark) THSP, as described in Fig. 3. Additional analyses of the results were conducted by means of the evaluation using the co-occurrence matrix of each column of Fig. 1 with time and the comparison of their results obtained with the different approaches tested in this work. These analyses and comparisons were carried out in the light and dark areas including the transition region, and the mean and standard deviation were used to evaluate their differences.
3. Results and discussions 3.1. Results of paint drying with two different non-connected strips in the THSP In Fig. 4a and b we show the results of the different normalizations (local line normalization and whole image normalization) on the AVD and IM methods applied to two areas of the paint drying separately (as obtained in Fig. 1), where label 1 means the light area and label 2 means the dark area. The results were parametrized in order to adjust them to the same scale, so the first point, related to time zero, is always 1, and the actual offset between them is observed in the next points. The same was done using the proposed normalization (Eqs. (3) and (4)) and is represented in Fig. 4c and d. It is clear from Fig. 4a and b that the activity observed by IM and AVD in the two areas of paint drying showed the same behavior apart from minor differences between the activities in the separate areas 1 and 2, i.e., light and dark areas respectively. Once the two different areas were analyzed without any crossed information about each other, the results seem reasonable, and some effort should be made in order to create a procedure that is completely independent of the illumination level without any reference. The adoption of the proposed normalization (Eqs. (3) and (4)) presented the phenomenon with the same behavior, where a clear improvement in the outcomes was not observed. The improvement
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Fig. 9. Analysis of each line of a THSP with a dark and light area using the IM and AVD with the local line and whole image normalizations associated with the proposed normalization. The mean values of the light (left) and dark (right) regarding lines 231–254 (transition) and the standard deviation are presented separately: (a) the IM using the local line normalization, (b) the IM using the whole image normalization, (c) the AVD using the local line normalization and (d) the AVD using the whole image normalization.
expected would be the reduction of the offset between the outcomes, i.e., the dotted line closer to the dashed line, and the continuous line closer to the bold continuous line. Therefore, one can conclude that the additional efforts proposed by Eqs. (3) and (4) are not necessary at first glance. 3.2. Results of combined areas in the THSP The IM and AVD on the connected areas, which means the insertion of dark lines on the light THSP (Fig. 3), is presented in Fig. 5 with the adoption of the traditional (local line and whole image) normalizations and of the proposed normalization (Eqs. (3) and (4)). One hundred lines taken from the dark THSP were inserted to replace the first 100 lines of the light THSP. Whilst the analysis of the separate areas (dark and light) observed in Fig. 4 is relevant to guarantee the same results under distinct illuminations of the same material, the results presented in Fig. 5 are most common in samples where the light is not homogeneous. In Fig. 5 it is possible to see that a band with 100 dark lines inserted on a light THSP did not present visible differences when IM and AVD were applied using the traditional normalizations [3,4]. This was an important result, since the applications of the dynamic laser speckle in homogeneous samples (i.e. they should have the same activity) should present the same results comparing a THSP with changes in the illumination of its lines. That occurrence is common in most applications of biological material such as seeds [13], fruits [2], kefir [14,15], etc. and is related to the great roughness of the surface or to its spherical form. The proposed normalization (Fig. 5c and d) was able to follow
the phenomenon, although with more ripple mainly when the local line normalization [3] was adopted. The calculations using intensities raised to power p ¼0.3 (Eqs. (5) and (6)) were also tested, with roughly the same outcomes observed as in the approaches before (not presented). The approach using IM and AVD with the local line normalization biased by the proposed normalization associated with the power 0.3 presented more ripple than the others. The ripple occurs when the activity is considered stabilized; therefore, the proposed normalization (Fig. 4c and d and Fig. 5c and d) is more sensitive to the noise presented in those stabilized regions, which cannot be considered an important drawback of the proposed technique, since the major applications are concerned with the transition from one activity state to another. Thus, from time zero to 30 min, when the paint drying process showed stabilization, the proposed normalization, as well as the power introduced in the equations, could follow the phenomenon in the same way as the traditional methods. Therefore, the additional tests conducted were done to investigate in detail the differences, sensitivities and influence of illumination in the results. They were carried out, for instance, to evaluate the effect of light variation on the results of dynamic laser speckle, regarding the offset observed in Figs. 4 and 5, as well as the variations with time. These additional tests were done in two ways: defining an error index related to the insertion of dark lines in a light THSP from 1 to 100, and comparing the results of using the co-occurrence matrix obtained from each column of Fig. 2 analyzed by the tested methods.
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Fig. 10. Analysis of each line of a THSP with a dark and light area using the IM and AVD with the local line and whole image normalizations using power 0.3. The mean values of the light (left) and dark (right) regarding lines 231–254 (transition) and the standard deviation are presented separately: (a) the IM using the local line normalization, (b) the IM using the whole image normalization, (c) the AVD using the local line normalization, (d) the AVD using the whole image normalization.
3.3. Additional tests – inserting dark lines in a light THSP Fig. 6 presents the errors (Eq. (7)) produced by the gradual inclusion of dark lines on the head of a THSP with bright lines, both related to the same dynamic laser speckle phenomenon (drying paint), in order to evaluate the sensitivity of the methods. The errors were disclosed first the local line normalization adopted for IM and AVD. The errors were small, not more than 2.5%, and they increased with the number of lines inserted. In addition, it is possible to observe some jumps in the tendency, which should be analyzed in detail. The jumps are linked to the local line normalization, which varies in accordance with each line of the THSP, which varies and creates the co-occurrence matrix with some unbalanced regions [7]. This can be proved by the stability observed in Fig. 7 when the whole image normalization was adopted as expected [7]. Fig. 7 shows the errors related to the gradual inclusion of dark lines in the THSP with bright lines, both related to the same dynamic laser speckle phenomenon, in this case using the whole image normalization [7]. In contrast with the local line normalization [3], the whole image normalization associated with the changes proposed in this work presented the best results (bold line). The errors behaved in a similar way when the paint was allowed to dry for different times. The errors regarding the insertion of lines, such as in the results of Figs. 6 and 7 when the power 0.3 was used (Eqs. (5) and (6)), showed similar outcomes (not presented). We could observe that the traditional methods are relatively independent of the illumination inhomogeneity on the surfaces analyzed by the dynamic laser speckle (Fig. 4). However, the independence can be improved by means of some adjustments such
as the use of a denominator in the traditional equations (Fig. 5), which can be observed in detail with the additional tests. 3.4. Additional tests – comparing each column of the sample In Fig. 8, the IM and AVD from the THSPs, formed by using each column of the speckle patterns (Fig. 2), are presented and they show the outcomes in the light and dark THSP areas over time. In Fig. 8 the traditional AVD and IM methods using the local line normalization and the whole image normalization are presented and it is clear that the level of illumination drives the results, with the mean values of the dark portion different from the light one. The high variation of the local line normalization was expected and it confirms the enhancement provided by the whole image normalization [7] despite the observed offset presented when the traditional normalizations are adopted. Fig. 9 presents a reduction in the differences between the mean values of the right side (light THSP) and the mean values of the left side (dark THSP) in this case using the proposed normalization (Eqs. (3) and (4)). The AVD method using local line and whole image normalization presented the best results. The same comparions of IM and AVD with variations were conducted using the power 0.3 (Eqs. (5) and (6)) at the zero instant are presented in Fig. 10. Figs. 8–11 present the mean values of the IM and AVD using all the tested approaches, as well as the standard deviation of the values on each side of the transition point (columns 231–254). It is clear that the step presented by the traditional methods (Fig. 8) is predominant. It is also observed that the adoption of whole image normalization presented outcomes with less variation, as expected [7].
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Fig. 11. Analysis of each column of a THSP with a dark and light area using the IM and AVD with the local line and whole image normalizations associated with the proposed normalization and power 0.3. The mean values of the light (left) and dark (right) regarding lines 231–254 (transition) and the standard deviation are presented separately: (a) the IM using the local line normalization, (b) the IM using the whole image normalization, (c) the AVD using the local line normalization, (d) the AVD using the whole image normalization.
Table 1 Local line normalization, whole image normalization and proposed normalization with the percentage of the differences between the results on the light (left) and dark (right) sides of the transition columns 231–254 of the THSP. Difference between the results on the light and dark sides of the THSP (%)
IM local line normalization AVD local line normalization IM whole image normalization AVD whole image normalization
Traditional normalization
Proposed normalization
23.9
4.8
19.0
3.1
53.2
3.3
25.3
1.9
In Table 1 we show the percentage difference between the mean values of each side of the transition point (lines 231–254) using the methods on each column with and without the power, in this case the power 0.3. In the results presented in Table 1, the outcomes are related only to the zero instant after painting and with power p ¼1. Other powers were also tested and the results showed that the best approach regarding all drying times is the AVD using the whole image normalization with the proposed denominator associated. Additionally, the adoption of powers does not change the best results observed with power equal to 1, particularly when the AVD was used associated with the proposed normalization (introduction of a denominator). The percentage of difference between the
results in the light and dark sides using the powers is presented in Fig. 12a, all of them related to the zero instant after painting. It is possible to see that the approaches related to the proposed normalization (Eqs. (5) and (6)) presented the best results at p ¼1, although the adoption of other powers (0.1–0.9) did not compromise the results. However, when the traditional methods were used, the adoption of a small power (0.1) enhanced the results, i.e., the differences between the two sides tended to be lowest. Fig. 12b presents the evolution of the percentage difference with respect to the powers at 30 min of drying time. At this point, the drying process had stabilized (see Fig. 4), and the results are different from the zero instant. The best result can be attributed to the AVD using local line normalization associated with the power 1 and with the proposed normalization, despite its larger variation provided by the adoption of the whole image normalization. Therefore, as a first choice we can select the AVD using local line normalization associated with the power 1 as the one that best fulfils a lower sensitivity to illumination inhomogeneity. However, if we consider only the analysis when the activities are not stabilized, it is possible also to consider the usage of the AVD with the whole image normalization, since the outcomes are more stable and with reduced variations. The usage of more than one outcome can be one alternative, since biological phenomena are very complex, with great peculiarities, thus offering the opportunity to bias the best outcome to a particular phenomenon. The independence of the light's uneven illumination during the dynamic laser speckle analysis is an important achievement of this work.
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Fig. 12. Evolution of the percentage difference regarding the powers at two times of drying (a) zero and (b) 30 min; associated with local line normalization (LLN), whole image normalization (WIN) and proposed normalization (New) with the percentage of the difference between the mean values of the activity on the light (left) and dark (right) sides of the transition columns 231–254 of the THSP. The value p means the adopted power.
4. Conclusions
References
We tested the traditional methods to analyze dynamic laser speckle when the illumination of a sample with uniform activity is not homogeneous. The traditional methods presented reliable results. However, when observed in detail, we can see that the methods proposed here can improve the independence with respect to the illumination. In particular, we observe that the AVD method using the local normalization associated with the inclusion of the denominator in the traditional method (proposed normalization) presented the best results. Particularly, we elected the AVD using local line normalization associated with power 1 as the one that best fulfils a lower sensitivity to illumination inhomogeneity. When the activities were not stabilized, it was possible also to consider the usage of the AVD with the whole image normalization. The combination of more than one outcome should be considered to choose the best outcome.
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Acknowledgments This work was supported by Grant PICT 2008-1430 from the Agencia Nacional de Promoción en Ciencia y Tecnología (ANPCyT), Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Comisión de Investigaciones Científicas de la Provincia de Buenos Aires (CIC) and Universidad Nacional de La Plata (UNLP), Argentina. We also thank the Federal University of Lavras, and CNPq, Fapemig, Capes and Finep who partially financed the project.