La derivación judicial en el contexto español y mexicano

Traditional light microscopy relies on variations of light intensities reflected from or transmitted through the sample. Thus, cells or thin tissues with high transparency result in only slight intensity changes of the light. With any photodetector, this low intensity change gives extremely low contrast images. However, the phase changes of light that are induced by the sample are abundant but cannot be directly detected. This reality has triggered researchers to explore how to implement the interference and diffraction in a microscope to reveal the phase information of the samples. Optical microscopy that quantitatively retrieves both phase and intensity of the light is named quantitative phase microscopy (QPM). The basic theory of them can date back to Thomas Young's 1801 discovery of double-slit interference 1. In this chapter, one of the QPM techniques, digital holographic microscopy, is adapted. Also, the automated operation and imaging process was built for the proposed microscopy system.

5.1.1 Theory of holography

The essential principle of digital holographic microscopy (DHM) is the measurement of microscopic phase shift through classical interferometry and modern Fourier optical transforms. The small transparent structures of the object do not introduce obvious intensity change but do introduce the phase change of light. The phase information can be encoded through an interferometry

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configuration, which is intensity information and can be recorded easily. These phase changes from the sample can be displayed as disturbances on the interference fringes. The equations (1) and (2) represent the reference wave going through unobstructed and object wave going through the sample. R₀and O₀ indicate the amplitude of reference wave and the object wave. The ω₁ indicates the angular frequency of both waves. The

k r! "∗

indicates the dot product of wave vector and optical length. The ϕ in equation (2) is the phase changes produced by the sample.

( )

(

)

0 1 R(r , t)" =R exp j[ −ωt+ k r ]!∗" (1)

( )

(

)

0 1 O(r, t)" =O exp j[ −ωt+ k r!∗" +ϕ] (2)

According to wave equation, the hologram pattern (IH) in original in-line setup can be written below:

2 * *

H o R

I = +| O R | =(I +I ) OR+ +RO (3)

Combined with wave equations of reference wave (1) and object wave (2), the hologram pattern can be rewritten in equation (4):

( ) ( )

jφ j φ

2 2

H 0 0 0 0 0 0

I =O +R + O R e + R O e − (4)

As shown from above equations, the phase change ϕ produced by the sample can be located at both j( )φ j( φ)

0 0 0 0

O R *e and R O *e − . Those two phases are inverted and overlap in the frequency domain. These are called twin images in an in-line arrangement. The overlapping twin images cause blurring and reduce the image quality which complicates the phase reconstruction process. However, when the two waves interference at a particular angle, the object wave O_off and reference wave R_off have a slightly different incident angle, which causes the different

k!

in two waves. Therefore, the reference wave and object wave can be rewritten as below:

( )

(

)

0 1

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(

)

(

)

0 1

O (r, t)_off # =O ∗exp j[ −ωt+ k!!"_o∗r# +ϕ)] (6) Then the hologram can be rewritten as:

( ) ( ) ( ( ) ) jφ r j φ r 2 2 H_off 0 0 0 0 0 0 I O R O R e +ko−kr ∗ R O e − +k kr−o∗ = + + ∗ !!" !!" # + ∗ !!" !!" # (7)

As shown, two factors O R *e_{0 0} j(φ+(k!!" !!" #o−kr)∗r) _and j(φ ( )r)

0 0

O R ∗e +k k!!" !!" #r−o∗ _{are in} symmetrical locations in spatial frequency domain, while the zero order

O

2₀

+R

₀2

is located in the origin point (center point) of spatial frequency domain. Therefore, it’s easy to separate them and retrieve the phase j in the numerical reconstruction. 5.1.2 Retrieving phase in DHM

In digital holographic microscopy (DHM) system, numerical methods hold the key to accurate decoding of the phase information from the interference pattern or called digital hologram. The numerical steps used in off-axis DHM technique includes: discrete 2D Fourier transform, spatial frequency filtering 2, numerical propagation (digital refocusing if required), phase unwrapping, aberration correction 3. As mentioned above, the slight tilt angle q in an off-axis arrangement can spatially separate virtual image and the true image in frequency domain, shown in figure 5.1, (fast Fourier transfer) FFT image. The mathematical description of this is equation 15. The first non-constant term describes the positive complex amplitude information whilst the second non-constant term describes the virtual imaging information. Since these two terms are complex conjugates and the base of them is mathematical constant e. After Fourier transformation, these two terms appear in two symmetrical positions in the frequency domain.

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Figure 5.1 (a) Phase retrieval process; FFT,Fast Fourier transformation; iFFT,

inverse Fast Fourier transformation; Angle calculation (Angle); Absolute value calculation (Abs). (b) Background removal process.

Figure 5.1 (a) shows the flow chart of the phase retrieval process. Firstly, the Fast Fourier transformation (FFT) is applied to the recorded hologram to get the frequency domain image (FFT image). Next, a digital mask is applied to the FFT image to filter out the first order and move it to the centre as the filtered FFT image. Later the inverse Fourier Transform is applied to the centred FFT image and a complex matrix is obtained. The intensity image (amplitude information) can be obtained by taking the absolute value of the complex matrix, while the phase map can be acquired by angle calculation of the complex value of the matrix. The acquired phase map is folded or wrapped within 0 to 2π. To get the continuous phase value corresponding to the optical thickness of samples, a 2D unwrapping process needs to be operated to get final unwrapped phase map. However, the phase value we get from the hologram is non-simulated data, which contains noise from the stray light and the CCD. So the simple unwrap function in Matlab is not powerful enough for the noise sensitive unwrapping processing. We implemented the 2D-SRNCP phase unwrapping method 4 to achieve credible phase map. After unwrapping, the next operation is to calculate the optical thickness of the sample. In this process, the below equation will be needed, in which φ(x,y) is the phase value we get from above steps, λ₀is the wavelength of

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the laser light and the ∆n is the refractive index difference between the sample and the media.

( )

_, ( , ) 0 2 x y h x y n ϕ λ π = Δ

The height map of the sample can be plotted using the h x y

(

,

)

value. The obtained height map is accompanied by a background curve if the reference wave goes through slightly different optics with the object wave. For example, there is no objective lens in the reference wave. In order to remove the background, an aberration mask (negative background in figure 5.1 (b)) is calculated from the Zernike polynomials fitting of the original height map to get the final corrected height map.

To achieve high throughput cells investigation or diseases diagnosis, these numerical calculations should be programmed to be automated and adaptive to different sample and imaging conditions. The complexity lies in the spatial frequency filtering process. Since the distribution of spatial frequencies of different samples varies significantly, the filter window needs to be adjusted correspondingly for the precise extraction of the desired frequency orders (first orders). One straightforward filtering method for precise extraction is manually selecting the first order 3, 5 in the spatial frequency domain. However, the manual spatial filtering is time-consuming and depends on subjective judgment, which is not suitable for dynamic reconstruction. One of the automatic filtering methods requires setting the threshold to obtain the spectrum and extract its boundary as the boundary of the final filtering window 6. This raises difficulties in defining the spatial filter because of the blurry boundary and non-regular distribution of the spectrum. Obviously, this approach requires manual input for each hologram, which is therefore not optimal for automation. While there are some histogram analysis techniques 6-7 aiming to provide adaptive filtering, unfortunately, those histogram analysis processes still require manual intervention (e.g. initial windowing parameter) to calculate the appropriate threshold. Hence, these techniques are still unable to provide sufficient flexibility or simplicity for automated selection of the appropriate spatial filter for a wide variety of samples especially confluent live cell cultures with highly scattering backgrounds.

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Therefore, we developed an adaptive spatial filtering for phase reconstruction, which will be introduced thoroughly in chapter 5.3.

In document Mediación intrajudicial en los juicios familiares de los primeros cuatro distritos judiciales en Nuevo León (página 137-144)