## Abstract

Pixel resolution polarization-sensitive second harmonic generation (PSHG) imaging has been recently shown as a promising imaging modality, by largely enhancing the capabilities of conventional intensity-based SHG microscopy. PSHG is able to obtain structural information from the elementary SHG active structures, which play an important role in many biological processes. Although the technique is of major interest, acquiring such information requires long offline processing, even with current computers. In this paper, we present an approach based on Fourier analysis of the anisotropy signature that allows processing the PSHG images in less than a second in standard single core computers. This represents a temporal improvement of several orders of magnitude compared to conventional fitting algorithms. This opens up the possibility for fast PSHG information with the subsequent benefit of potential use in medical applications.

©2010 Optical Society of America

## 1. Introduction

Second harmonic generation (SHG) laser scanning microscopy is considered nowadays one of the most promising minimally invasive, high-resolution optical techniques for clinical applications [1]. Because of recent technological advantages in micro-endoscopes/fibers [2–4] and laser sources [5], SHG imaging shows a great potential for clinical usage as an optical biopsy tool [6].

Earlier, numerous studies on the SHG contrast have demonstrated that collagen, myosin, and microtubules are effective SHG converters in tissues [1]. Consequently, biological structures that are consisting of the above endogenous SHG sources can be imaged using intensity-based SHG microscopy. Despite the fact that the SHG intensity is the only contrast mechanism for generating the images, several morphological parameters can be obtained [7]. For example, by comparing the forward and epi-detected SHG signals, conclusions on the dimensions of collagen fibrils can be obtained [8]. Other methodologies take advantage of the characteristic striation pattern for quantitative interpretation and extraction of information in muscles [9]. In such cases, the analysis of the sarcomere pattern was used for the study of rare diseases, including muscular dystrophy [10] and osteogenesis imperfecta [11]. More recently, the use of image processing based on spatial Fourier analysis has been used to infer properties of tissue and molecules of the intensity-based SHG imaging [12]. Specifically, bidimensional (2D) Fourier transform (FT) was used to spatially quantify the disorganization of collagen fibres due to photo-thermal damage in porcine corneas [13]. It was found that the regularities in fibres organization leads to an elliptical distribution in the bidimensional (2D-FT) transformed space, whereas randomness leads to a more circular distribution [14]. Also very recently, it was presented that additional information on collagen fibre orientation and maximum spatial frequency can be obtained using 2D-FT in an SHG image [15]. Likewise, it was shown that the 2D-FT can also be performed in the epi-detection [16].

In addition to the above approaches, the polarization dependency of the SHG signal (PSHG) can also be used to generate contrast [17, 18]. In particular, due to the geometric characteristics of local hyperpolarizability tensor *β ^{(}*

^{2)}arrangement, by rotating the incoming linear polarization (or the sample), the detected SHG signal intensity from every point provides a characteristic modulation, often called the anisotropy or signature/fingerprint curve [19, 20]. Such PSHG modulation can be exploited to obtain information unreachable by intensity-based only SHG imaging [21]. For example it provides intrinsic tissue characterization without any labeling and without using an analyzer in the detection [22–24]. In particular, in collagen, PSHG has allowed obtaining the orientation of the

*β*

^{(}^{2)}dominant axis with respect the long axis of the supramolecule, which is related to the helical pitch angle of the polypeptide chain of the collagen triple-helix [19]. In muscle such orientation is attributed to the α-helix of the myosin’s coiled coil (myosin tail) of thick filaments [19]. In cultured cortical neurons this orientation coincides with the architecture of the tubulin heterodimmers forming the axons’ microtubules [25] and in starch granules it indicates amylopectin as the SHG active molecule [26]. To obtain such information an iteration code is usually used [19–25]. This iteration code can be based in a fitting algorithm able to fit the PSHG images to a biophysical model in a pixel by pixel fashion [21]. However, because of the pixel resolution examination, a typical fitting procedure takes many hours to process one image (~6 hrs for a 512x512 pixels image). This slow processing time drastically hampers the use of the of PSHG methodology for clinical applications. Having a fast routine able to obtain such parameters in a robust way might open up many new and exciting applications into life sciences.

In this work, we present a polarization 1-D FT analysis of the anisotropy curve to retrieve the biophysical parameters of the proposed model, referred to us as Fast Fourier Polarization SHG (FF-PSHG) analysis, to obtain the same information as with the iteration fitting-based procedure, but instead of hours, in a few hundreds of milliseconds using a regular processor. Considering a PSHG image of a three dimensional data set, I(x,y,α), where x-y refer to the spatial axis of the image and α refers to the polarization dependency of the SHG image (anisotropy curve), this FF-PSHG analysis is performed only on the polarization axis, *α,* of every pixel. This is in contrast to the spatial image processing methods using 2D-FT in the (x,y) axis discussed above [15, 16].

To show the feasibility of this method, in section 2 we describe the theory and physical implications of the FF-PSHG analysis. This is followed in section 3 by the experimental demonstration of the method in biological samples, showing the capability to retrieve the orientation of the supramolecule assembly and the hyperpolarizability tensor *β ^{(}*

^{2)}dominant axis orientation (normally related to the helical pitch angle). In this section, the capability of the method to perform discrimination between different tissues is also demonstrated using two different strategies. Finally, these results are followed by the conclusions.

## 2. Theory

The biophysical model used here (see refs [20,21,23–25]) refers to an SHG active supramolecular assembly with cylindrical symmetry. The SHG signal dependency of such structures on the input polarization of the fundamental beam can be written as:

*ϕ*and α are the orientation of the long axis of the cylinder, which we assume coincides with the supramolecular assembly alignment, and the angle of the fundamental beam polarization, respectively, defined with respect the lab x-axis [21].

*A = I*,

_{0}d_{31}*B = I*and

_{0}d_{33}*C = I*, where

_{0}d_{15}*I*is proportional to the intensity of the excitation fundamental field, and

_{0}*d*,

_{31}*d*and

_{33}*d*are the non-zero elements of the nonlinear susceptibility tensor characterizing the tissue under cylindrical symmetry assumption [21].

_{31}The free parameters *A*, *B*, *C* and *ϕ* are usually obtained by fitting the experimental SHG images for every polarization α to Eq. (1). To do that, an iterative nonlinear algorithm is usually utilized. However, this is a lengthy task as, depending on the size of the image, this fitting procedure may take several hours. To speed up such process, Eq. (1) can be rewritten in a more convenient form as a sum of cosine frequency components as follows:

*a*Note that the parameters

_{0}=C^{2}/2 +3/8 (A^{2}+B^{2}) +AB/4, a_{2}= B^{2}/2-A^{2}/2 and a_{4}= (A-B)^{2}/8 – C^{2}/2.*ϕ*,

*a*,

_{0}*a*and

_{2}*a*contain now the whole information relative to our biophysical model (tensor elements). In what follows, we show that these components can be readily obtained by our FF-PSHG analysis in an efficient manner.

_{4}As commented in the introduction, in this work we are going to perform the 1D-FT only on the polarization axis, *α* as *i(x,y,Ω)=F _{α}{I(x,y,α)}.* By doing so, the Fourier transform of Eq. (2) in a pixel, determined by (x,y), with a polarization sampling between 0° and 180°, results in

*δ*(2−Ω) and δ(4−Ω) respectively, with the advantage that an immediate averaging of the two set of results (from 0° to 180° and from 180° to 360°) is obtained. In the rest of the document, for simplicity we assume the sampling is in the range from 0° to 180°. Note that the quadratic nature of PSHG response [see Eq. (1)] generates a symmetric polarization response in the polarization intervals

*α*∈ [0,

*π*], therefore

*ϕ*has the same periodicity, which for convenience we chose the range

*ϕ*∈ [-

*π*/2,

*π*/2].

Before go further, it is worth to note that Eq. (1) possesses a mathematical intrinsic ambiguity that affects any PSHG experiment. This ambiguity is apparent when the same result is obtained by exchanging *A* and *B* and adding *π*/2 phase to the orientation *ϕ* [21]. From a physical point of view, this ambiguity appears because the model is build in a manner that assumes a minimum SHG signal when the incident polarization is perpendicular to the cylinder’s long axis. Experimentally, this has been reported to occur in several biosamples such as microtubulin of axons [25], collagen [18] or starch [26], and results in *B/C* >*A/C ≈1*. However, muscle [20] shows the minimum SHG signal when the incident polarization is parallel to the thick filaments orientation (assumed to posses the cylindrical symmetry), with *B/C <A/C ≈1*. Since the ambiguity cannot be solved using mathematical criteria, *a priori* knowledge on the different sample PSHG response was needed. When using the fitting algorithm, the ambiguity is solved in every pixel by assigning the value closer to the unity to *A/C.* Then if this value is associated to the sinus in Eq. (1) the orientation is directly the retrieved angle *ϕ*. On the contrary, if *A/C* is associated to the cosine in Eq. (1), the actual orientation is*ϕ+π/2*.

#### 2.1. Determining the orientation of the supramolecular assembly ϕ

The above ambiguity also affects our FF-PSHG analysis, particularly in the orientation *ϕ*. Direct observation of Eq. (2) shows that *ϕ* only affects the cosine argument. Therefore, when performing Fourier transform to Eq. (3), it will appear as a phase in the first and second (second and four) coefficients when performing the sampling between 0° and 180° (0° and 360°). Then, the extraction of the orientation *ϕ* consists in computing the complex argument of the second coefficient as

In Eq. (4), the ambiguity is apparent in the fact that the obtained angle *ϕ'* and the orientation *ϕ* can be different, since *ϕ'* also include information on the sign of *a _{2}*. This is because

*a*can either be positive (|

_{2}= B^{2}/2-A^{2}/2*B|*>|

*A|*) or negative (|

*B|*<|

*A|*). This unknown sign is transformed in the intrinsic ambiguity of

*π*/2 in calculating the angle orientation

*ϕ.*This is totally equivalent to the ambiguity in Eq. (1) by exchanging

*A*and

*B*and adding

*π*/2 to the orientation

*ϕ*[21]. Similarly, the condition used with iterative fitting algorithms, |

*A/C|<|B/C|,*results in

*a*

_{2}>

*a*

_{4}, characteristic of collagen and starch, while |

*A/C|>|B/C|*results in

*a*

_{2}<

*a*

_{4}, which is typical in myosin. Therefore, by comparing

*a*

_{2}and

*a*

_{4}it is possible to solve the indetermination as follows:

For other tissues it will be possible to design different strategies and define specific criteria. Also note that since the extraction of *ϕ.*is based on the phase of the polarization-spectral components, it is, in principle, independent of possible errors affecting the amplitudes *a _{0}*,

*a*and

_{2}*a*, adding robustness to the method.

_{4}#### 2.2. Extraction of the biophysical parameters

The rest of the parameters, *A*, *B* and *C* can be then extracted analytically by combining Eqs. (1) and (2) as:

*ϕ*. Once these three parameters have been computed, the tensor element ratios can be calculated as

*d*/

_{31}*d*and

_{15}=A/C*d*/

_{33}*d*. This can be performed in an almost instantaneous way (considering the current speed of modern computers) with no constrains.

_{15}=B/COnce this is done, it is possible to go a step forward and calculate the angle *θ _{e}* between the hyperpolarizability tensor

*β*

^{(}^{2)}dominant axis (the nonlinear SHG-dipole, normally related to the helical pitch angle) and the long axis of the supramolecular assemble . To do that, a series of restrictions, related with or in addition to the “single-axis molecule” approach, are normally imposed: (1) there is only one major orientation

*ϕ*in each pixel, (2) the long axis of the supramolecular assemble is parallel to the imaging plane (2D), (3) both Kleinman and cylindrical symmetries hold. All these conditions imply that

*A=C,*or equivalently,

*d*/

_{31}*d*. However, any deviation from the above restriction, experimental errors and detection noise level, resulted in a certain distribution around

_{15}=1*d*/

_{31}*d*[21, 23, 25, 26]. In this situation, the algorithm can be forced to fulfill

_{15}=1*A=C*by considering

*a*and

_{0}=A/2 +3/8 (A^{2}+B^{2}) +AB/4, a_{2}= B^{2}/2-A^{2}/2*a*and use only one of the two equations in (6) to determine

_{4}= (A + B)^{2}/8 – A/,*A*and

*B.*Then, the results of

*d*/

_{33}*d*or alternatively

_{15},*A*and

*B*can be used to obtain

*θ*as [21–23,25,26]:

_{e}#### 2.3. Pixels with erroneous results

In previous works, we have shown the capability of the iterative algorithms to remove pixels with erroneous results by filtering them out by setting a threshold on the fitting coefficient of determination, r^{2} (usually keeping pixels presented r^{2} > 90%) [21,23,25,26]. Here we propose a different strategy that is based on the analysis of the spectral components for the PSHG modulation response.

The source of noise in the PSHG images includes experimental errors like anisotropy of the sample, depolarization introduced by the optical components, optical misalignment, non-exact determination of the polarization, saturation and poor signal to noise ratio (SNR). This means that the noise within a pixel can be considered being equally distributed among all the spectral components obtained after performing the 1D-FF in the polarization axis. Among all these components, only those with *Ω* ≤ 2 have biophysical meaning according to current model [see. Eq. (3)]. Therefore, the origin of any signal appearing in polarization frequency components with *Ω* > 2 can be associated to noise. As result, since FF-PSHG analysis only uses the spectral components *Ω* ≤ 2, noise associated to the components with *Ω* > 2 is intrinsically filtered in determining all the previous parameters.

In addition to this filtering, the noise at components *Ω* > 2 can be used to estimate the total amount of error in the coefficients *a _{0}*,

*a*and

_{2}*a*. To do that, we assume that the error in the frequency components at

_{4}*Ω*= 0, 1 and 2 (related with

*a*,

_{0}*a*and

_{2}*a*) is affected in a similar way as those components with

_{4}*Ω*> 2. Therefore the experimental error in determining

*a*,

_{0}*a*and

_{2}*a*in a pixel can be estimated comparing the spectral components as

_{4}For example, the signal detected in pixels in areas outside any SHG active tissue is noise in nature and therefore results in a value of *e* ≈1. However, pixels in areas with a good signal to noise ratio will result in *e* ≈0. Then, since *e* quantifies the error in a pixel, it can be used to filter out pixels with *e* above a certain threshold value, *e*
_{th}, which are considered erroneous [i.e., do not match Eqs. (1)–(3)]. Typical values for the threshold are in the range *e*
_{th} ≈0.02-0.1.

## 3 Results-discussion

In this section, we show the capability of determining the orientation of the supramolecular assembly *ϕ*, also referred as fiber orientation, discussing the ambiguity described in subsection 2.1, and the determination of *θ _{e}*. This is followed by two methods to perform discrimination among tissues.

#### 3.1 Single SHG-active structure images

To show the ability of the algorithm to locally determine the orientation of the supramolecular assembly we analyze a representative case: a granule of starch. A granule of starch has been previously reported to possess a radial molecular orientation [26]. This sample is ideal to show the performance of the method since it allows obtaining data within the whole orientation range, from 0° to 180°. The multiphoton microscope used to acquire the PSHG images has been thoroughly described in Refs [21,23,25,26]. The linear polarization at the sample plane exhibited an extinction ratio of 25:1. By comparing the summed reconstruction of linear polarization images with the image created using circular polarization, we found this ratio adequate. Figure 1
shows the results, measuring the angle *ϕ* using both the FF-PSHG analysis, which is obtained in 100 *ms*, and a fitting algorithm, which lasted ~6 hours with 400 iterations per pixel. The results are very similar, clearly retrieving the radial-like structure. The small deviation of the radial symmetry in Fig. 1 might be attributed to the imperfections on the starch granule. We can also observe that a smooth change from pixel to pixel is obtained with FF-PSHG analysis, without the need of pixel averaging, as is the case in the image obtained with the iterative algorithm. We attribute this smooth variation to the filtering process intrinsic in the FF-PSHG analysis. In addition to this filtering procedure, pixels with *e* > 0.05 have been removed from the image (black color). Notice that only points near the external surface of the starch granule disappears, denoting the quality of the measurement. In the case of the fitting algorithm, pixels with coefficient of determination r^{2}<90%, has been filtered out. Finally, when using a fitting algorithm, final results slight change depending on the initial conditions and number of iterations. This is not the case in our FF-PSHG analysis, since its analytical nature always provides the same results. This adds robustness and consistency to the analysis.

We next analyze the capability of the method to clearly determine changes in the *ϕ* orientation by using a more sophisticated sample. Figure 2(a)
shows a detail on the orientation for a collagen fiber shown later in Fig. 4
. In this figure we have manually delineated the orientation of the fiber (white line), computing the angle at every point of the line. This result, shown in Fig. 2(b), is then compared with the angle retrieved at selected pixels of the curve using our FF-PSHG analysis. A nice agreement is observed, showing consistency of the method that is able to calculate angular deformations. This result entails us to use this method to analyze complex fiber situations as shown in Fig. 2(c). Again, an image showing a smooth variation of the fiber orientations is obtained.

With the above results, the reliability of the method to determine any fiber orientation change is clearly demonstrated. This allows analyze complicated situations and go a step forward to obtain the helical pitch angle in a sample, with pixel resolution, and compare the results obtained with the fitting algorithm and the FF-PSHG analysis. The results, corresponding to bundle of collagen fibers oriented in different directions, are showed in Fig. 3
(the corresponding fiber orientation is shown in Fig. 2c). Figure 3(a) shows the superposition of the SHG intensity images for all the polarizations. This figure show the difficulties to obtain SHG signal in some areas, specifically in most of the points in the top part of the collagen bundle that will result in a poor noise to signal ratio. As a consequence, the analysis performed using the iterative algorithm, shown in Fig. 3(b), lacks important parts of the image, which has been filtered out due to the low quality of fitting in the top part of the image (points with coefficient of determination, r^{2}<85% where removed). In spite of the decrease of useful pixel, the fitting algorithm is able to correctly retrieve the helical pitch angle, whose distribution is shown in Fig. 3(c), with the maximum frequency at *θ*
_{e} = 44.4° and a distribution width of Δ*θ*
_{e} = 5.4° (the helical pitch angle obtained with X-ray diffraction measurements is ~45°). On the contrary, the FF-PSHG analysis shown in figure Fig. 3(d) is able to map *θ*
_{e} in the entire sample, even for those areas with low SHG signal quality (notice the top part of the bundle of collagen fibers). This is possible to the noise filtering intrinsic to the method. The image shows smooth changes that give the impression of volume, which correlates well with the contour of Fig. 3(a). This makes us suspect that the variation in *θ*
_{e} can be attributed to be mainly produced by out of plane fiber axis orientations. The *θ*
_{e} frequency distribution obtained with the FF-PSHG analysis is shown in Fig. 3(e), showing a displacement of the maximum, at *θ*
_{e} = 42.3° and a distribution width of Δ*θ*
_{e} = 4.9°. This displacement of the maximum is attributed to major number of pixels with *θ*
_{e} ≈40°, appearing in the top part of the bundle of collagen fibers, which are filtered out by the fitting algorithm in Fig. 3(c).

Regarding the time required to compute the above Figs. 2-3 (500 x 500 pixels), the FF-PSHG analysis lasted around 100 *ms* to compute the fiber orientation, while the calculus of *θ*
_{e} required less than 300 *ms.* Figures obtained with the fitting algorithm, where obtained with 400 iterations per pixel and lasted ~6 hours.

#### 3.2 Multiple SHG-active structure images

PSHG offers the unique characteristic of identifying and discriminating different SHG active molecules, with pixel resolution, in the same image [23]. In this section we show that our FF-PSHG analysis can also be used with discrimination purposes by computing *B/A* parameter and *θ*
_{e} in every pixel [using Eq. (7)]. The results for unstained temporalis muscle from rat are shown in Fig. 4(a), where it is possible to observe a clear discrimination between two tissues, orange corresponding to muscle and blue to collagen. In this case, the time required to compute Fig. 4(a) was less than 300 *ms*.

In addition to the discrimination method described above, the FF-PSHG analysis offers a simple discrimination alternative based on directly mapping the cosine frequency components *a _{0}*,

*a*and

_{2}*a*into RGB images. Since the values of

_{4}*a*,

_{0}*a*and

_{2}*a*depend on the actual SHG molecule, the weight for every RGB channel is different for different tissues. Therefore, different tissues appear with different pseudocolor in the same image. The results are shown in Fig. 4(b). We can observe that both tissues are clearly differentiated. In order to identify what are the actual tissues displayed, the typical relation among values

_{4}*a*,

_{0}*a*and

_{2}*a*must be characterized. In the case of Fig. 4(b), and by comparing with Fig. 4(a), in the RGB representation yellow corresponds to collagen and purple to myosin. This provides a simple method to discriminate among different tissues, getting an instantaneous perception of the image, with the advantage that the time required to compute Fig. 4(b) was less than 100 ms, in a single core computer, after acquiring the corresponding PSHG data.

_{4}Comparing Figs. 4(a) and 4(b), we see that both methods provide similar discrimination capabilities, the main difference being the image appearance. This differences in the images appears because the cosine frequency components *a _{0}*,

*a*and

_{2}*a*contains information on the ratio among

_{4}*a*,

_{0}*a*and

_{2}*a*, which is always the same for a tissue (providing the discrimination capability), and on the intensity, providing smoother changes in the image. This results in an apparent better quality of Fig. 4(b) when compared with Fig. 4(a). However, although the RGB representation provides the fastest method to discriminate between different SHG sources, the standard method has the advantage that the tissue identification is based on a known intrinsic characteristic of the SHG active molecule, as is the helical pitch angle, providing a clear criterion in case of ambiguous situations. This is clearly apparent when plotting the frequency distribution for the helical pitch angle as showed in Fig. 4(c). The two well separated, not overlapping peaks centered at 43° and 64° for collagen and muscle, respectively, show the ability of the method to unambiguously distinguish between tissues in the same image.

_{4}## 4. Conclusions

Polarization-sensitive Second Harmonic Generation is a promising imaging modality that enables statistically studying the orientational distribution of the *β ^{(}*

^{2)}dominant axis (related to the helical pitch angle) of a number of molecules which play a role in many biological processes: collagen, microtubulin and myosin and additional structures, like starch, which has been also used to probe polarization state in a microscope [27]. Especially when optical clearing is used, this information can be acquired several hundreds of microns deep in tissues [28]. Therefore, many applications can be enhanced by the development of new and faster algorithms than the current ones, which are executed “offline”, requiring from minutes to hours to process an image of 500 by 500 pixels, even with multi-core computers.

In this paper, we have presented for the first time an approach that allows processing in few milliseconds an image based on 1D Fourier analysis of the PSHG modulation response obtaining a temporal improvement of near five orders of magnitude. This opens the possibility for PSHG imaging to penetrate new fields in medicine at video rates, acting for example as an instantaneous diagnostic supporting method in surgery. The results are in total agreement of those obtained by conventional fitting algorithms, where the intrinsic noise filtering results in a smother response and a better contrast, while its analytical nature provides robustness and consistency to the analysis. In conclusion, we have presented a sub-second method to process PSHG images to extract full biophysical meaning and straight visualization methods that can be useful for many fields in microscopy and biomedicine that possess additional advantages that do not possess its prior competitors.

## Acknowledgments

This work is supported by the Generalitat de Catalunya grant 2009-SGR-159 and by the Spanish government grant TEC2009-09698 Authors also acknowledge the Laserlab-Europe Cont (JRA4: Optobio212025) and the Photonics4Life networks of excellence. This research has been partially supported by Fundació Cellex Barcelona.

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