Quantitative Schlieren Image-Noise Reduction Using Inverse Process and Multi-Path Integration

This report deals with introducing two new techniques based on a novel concept of complex brightness gradient in quantitative schlieren images, “inverse process” and “multi-path integration” for image-noise reduction. Noise in schlieren images affects the projections (density thickness) images of computerized tomography (CT). One spot noise in the schlieren image appears in a line shape in the density thickness image. Noise effect like an infectious disease spreads from a noisy pixel to the next pixel in the direction of single-path integration. On the one hand, the noise in the schlieren image reduces the quality of the image and quantitative analysis and is undesirable; on the other it is unavoidable. Therefore, the importance of proper noise reduction techniques seems essential and tangible. In the present report, a novel technique “multi-path integration” is proposed for noise reduction in projections images of CT. Multi-path integration is required the schlieren brightness gradient in two orthogonal directions. The 20-directional quantitative schlieren optical system presents only images of schlieren brightness in the horizontal gradient and another 20-directional optical system seems necessary to obtain vertical schlieren brightness gradient, simultaneously. Using the “inverse process”, a new technique enables us to obtain vertical schlieren brightness gradient from horizontal experimental data without the necessity of a new optical system and can be used for obtaining any optional directions of schlieren brightness gradient.


Introduction
Schlieren imaging technique is a common tool in science and technology to visualize density gradients and investigate phenomena with non-uniform density flows in transparent media. Recently, this technique is reviewed in [1] and is shown that schlieren photography has been developed from a qualitative visualization method into a quantitative measurement method. In previous works [2]- [7], by employing and developing a non-scanning three-dimensional computerized tomography (3D-CT) technique using a delicate multi (20)-directional quantitative schlieren optical system with a flashlight source, measurement of the instantaneous density distributions of several types of flames in laminar and turbulent flows [2] [3] [4] [5] [6] and supersonic micro-jets [7] have been successfully obtained.
In the previous work [2], the target high-speed turbulent burner flames are investigated and reported. Figure 1 depicts the target flame, details dimensions of burner nozzle, coordinate system and 3D bird's-eye view of the CT-reconstructed density data. For more information related to the flow conditions, measured parameters of target flames and burner details refer to the previous work [2].
In the present report, two new techniques are introduced based on a novel concept of complex brightness gradient in the quantitative schlieren images, "inverse process" and "multi-path integration" for image-noise reduction. The new techniques partially are presented in international conferences [8] [9], in detail and entirely will be discussed here. Furthermore, it will be shown an important ability of complex schlieren brightness gradient, which is independence on the path and starting point in the integration process.
The schlieren images brightness change gradually (brightness gradient), while ( image noise is a random variation of brightness and is generally considered undesirable. Noise can deduct the quality of the image and quantitative analysis [10]. Noise in the schlieren images can arise from density gradients anywhere between the source slit and the knife-edge in the wind-tunnel research, and that is why a multiple-source schlieren system is introduced for noise reduction in [11]. Quantitative schlieren imaging requires low noise level images accordingly high data accuracy. Noise in the schlieren imaging is unavoidable.  [8], we presented the "inverse process" technique but in a different way than this time. In the previous work, the "inverse process" was performed after the CT-reconstruction, however, now is done before the CT-reconstruction procedure ( Figure 2). Indeed, in the previous work phantom (virtual) data of schlieren images are obtained by the "inverse process" and used to evaluate noise reduction technique while in the new "inverse process" technique actual experimental data is used for reproducing other directions of schlieren brightness gradient.
By using both white-light and monochromatic sources, color schlieren images have been produced from combining horizontal and vertical gradient brightness of common schlieren images by [12], but for this purpose, two independent knife-edges along two different beam paths are employed. About 70 years ago, a more complex arrangement is presented for obtaining two spatially separated images sensitive to orthogonal path deviations [13]. A good example of the same numerical supersonic jet-exit flow field is rendered as shadowgraph, bright-field schlieren (circular cutoff), vertical and horizontal knife-edge schlieren in [14]. Journal of Flow Control, Measurement & Visualization

Multi-Directional Quantitative Schlieren Optical System
Figures 3(a)-(c) illustrates the concept of a 20-directional schlieren camera. In the camera system, the target flame/non-uniform density field is observed from 180˚ direction using numerous schlieren optical systems simultaneously from θ = -85.5˚ to +85.5˚ at an interval of 9˚. Here, angle θ is defined as the horizontal angle from the x-axis. For pre-investigation and for time series observation (high-speed Schlieren movies) of the target flow, a high-speed camera (HSC) is used simultaneously with 20-direction Schlieren photographing apparatus (between cameras No. 13 and No. 14, Figure 3(b)).
A. Z. Nazari et al. The diagram of a single instance of the multi-directional quantitative schlieren camera system is depicted in Figure 3(d). This system is composed of two convex achromatic lenses of 50 mm diameter and 300 mm focal length, a flashlight source unit, a vertical knife-edge (for obtaining horizontal schlieren brightness gradient images), and digital camera.
The neutral-density (ND) filter (Fuji 1.5, exposure adjustment multiple 3.16) and stepped neutral-density filter is used for calibrating the cameras. The light unit is a xenon flashtube that emits full-spectrum white light with a uniform luminance rectangular area of 1 mm × 2 mm and a 35 μs duration.  Here, for brevity, the main points of Figure 4 are explained as follows.

Image Processing
The first goal is obtaining the density thickness of deviation density (Dt* (X(θ))) which shown in Figure 4(d) with a unit of (kg/m 3 )(m). The density thickness of deviation density Dt*(X(θ)) is obtained automatically from schlieren observation using spatial integration of deviation density Δρ*(X(θ),Y) along the line of sight. In practice for obtaining density thickness of deviation density ( Figure   4(i)), the image processing activity starts from Figure 4(e) by obtaining two sets of images, "with target" and "without target" (without any disturbance in the test section) with a horizontal brightness gradient in the x-direction. Two sets of images present by schlieren observation as B(X) and B n f (X) (brightness of schlieren where K is Gladstone-Dale constant for air (K = 2.26 × 10 −4 m 3 /kg), Δs (= 1 mm × 2 mm (Hor. × Ver.)) is the transparent width of the light source image on schlieren stop location and f (= 300 mm) is the focal length of the convergent lens. Deviation density thickness Dt(X(θ)) is, therefore, reproduced by transverse integration of d(Dt)/dX from schlieren images, as shown in Figure 4(h).
Finally, the "density thickness" Dt'(X(θ)) is obtained by adding the thickness of ( a ref ρ ρ * ) to the deviation density thickness Dt(X(θ)) on peripheral of the observed range of radius R (Figure 4(i)), as expressed by In the present study,

CT-Reconstruction
Density thickness images are used for CT-reconstruction by maximum likelihood-expectation maximization (ML-EM) [15] an appropriate CT algorithm to obtain the 3D reconstruction of density distribution, the ML-EM method [2]- [9] is employed for CT-reconstruction. The CT procedure is carried out in each horizontal plane of z-axis for the reconstruction of deviation density distribution Δρ(x,y) from a linear data set of density thickness Dt'(X(θ)) ( Figure 4(j)). Finally, 2D density distribution (Figure 4(k)) is obtained as follow.
The reconstruction was performed cross-section by cross-section and then the cross-sections were stacked to form a three-dimensional density distribution.

Inverse Process
As mentioned before, in the present study, using "inverse process" new ap-  Dt is expressed by Equation (2). Equation (2) can be rewritten in the following simple forms.
where α is a coefficient with parameters that explained in Equation (2). The deviation density thickness can be expressed as, Finally, in the inverse process, by combining the Equation (6) and (7), the deviation brightness of the quantitative schlieren image in each direction will be given by: The direction of the schlieren brightness gradient can be changed in the opposite direction (Figures 5(c2)-(f2)) only by applying one minus sign in the corresponding equation. Therefore, by using density thickness Dt in the inverse process, the schlieren brightness in horizontal, vertical and diagonal directions can be calculated (Equation (8)- (11)) without the necessity of a new optical system.
One of the benefits of using this approach is that now we can apply the "multi-path integration" noise reduction technique. The sample results for only camera number 10 are depicted for all mentioned directions in Figure 5.    with information on the gradient of density thickness in two orthogonal directions is called a "complex schlieren image".

Cauchy Integration Theorem
The Cauchy integration theorem in complex analysis is an important statement about line integrals for regular functions in the complex plane. Cauchy's integration theorem states that if f (z) be a regular function in the simply connected domain D, its contour integral on a path does not depend on the integration path and depends only on the beginning and ending points of the path. Therefore, if let C be a piecewise continuously differentiable path in D with start point a and end point b, and F is a complex antiderivative of f, then, If a = b, then C is a closed curve (loop) within D, accordingly, We applied the above theorem to develop the noise reduction technique. As shown in Figure 7 and Figure 10, for obtaining density thickness images "with decreased noise" the average of transverse integration from multi-path is employed.
It is remarkable, as a new finding, a new important concept in the complex schlieren image could be expressed as follow; The complex schlieren image for obtaining density thickness at an arbitrary position (ending point of the integration path) has one more potential power not depends on the starting point of the integration path. Therefore, the calculation of density thickness can be performed just by starting integration from any point on any path ( Figure 10).     initial noise spots (this may be a disadvantage of the proposed technique).

Numerical Simulation
Noise in schlieren imaging is unavoidable and can detract from the quality of the image and subsequent visual and quantitative analysis. Therefore, the need for proper noise reduction techniques is clear.

Multi-Path Integration Technique
As mentioned before, a new approach called "multi-path integration" novel technique against the former technique "single-path integration" is introduced for noise reduction in the projections (density thickness) images of CT. The new multi-path integration technique is required schlieren brightness gradients in both horizontal (x-directional) and vertical (z-directional) directions, or in other words, in two orthogonal directions (e.g. horizontal and vertical or two perpendicular diagonal directions) (Figure 2).
In the usual procedure, density thickness image reproduces by transverse integration of the schlieren image brightness in the single-path along schlieren brightness gradient (path No. 4 in Figure 10 (left)).

A. Z. Nazari et al. Journal of Flow Control, Measurement & Visualization
In the present study, for obtaining density thickness images "with decreased noise" the average of transverse integration from multi-path is employed. These paths are in four main horizontal and vertical directions (totally 28 paths, 7 paths in positive and 7 paths in negative of x and z directions) (Figure 10), as well as, the paths can be used from two perpendicular diagonal directions. The new procedure "multi-path integration" is one comprehensive technique that including the usual former "single-path integration" technique as well. In the former technique "single-path integration" (path No. 4, Figure 10), the density thickness for pixel (x, z) can be expressed from Equation (7) as follow.
On the right side of Equation (14), the first term is the density thickness data from prior adjacent pixel (the density thickness for the first-pixel column (1, z) is assumed zero) and the second term comes from the schlieren image brightness (Equation (2)). This second term for z (vertical) direction can be obtained by applying the inverse process technique. Calculation of two sample paths (No.  Figure 11 represents complete (Figure 11(a)) and incomplete (Figure 11(b)) observation of the schlieren brightness gradient for the laminar flame kernel. Hence, if the schlieren photography in one and more directions is not bounded to an ambient condition which means the schlieren photography is not observed complete target flow including the boundary condition, those directions could be omitted ( Figure 11(b) (Top direction)), because this treatment will give better results for noise reduction using multi-path integration. In the present study, paths from negative z-direction (7 paths,No. 22 to 28, Figure 10) are not considered in the calculation due to that top area of target flame is not captured by the schlieren photography system, because of the fact that lenses diameters are smaller than target flame height.

Projections Images of Density Thickness (Dt)
For better evaluation of the "multi-path integration" noise reduction technique,   different positions randomly (Figure 13(e)). Then, by using the former technique "single-path integration", in horizontal ( Figure 13(a1)), vertical ( Figure   13(b1)) and diagonal directions (Figure 13(c1) and Figure 13(d1)), the projections images of density thickness with the existence of noise effects obtained ( Figures 13(a2)-(d2)). Finally, the images of Figure 13(a1) (horizontal schlieren brightness gradient) and Figure 13(b1) (vertical schlieren brightness gradient) are processed by the above-mentioned conversion procedure "multi-path integration" to projection images of density thickness "with decreased noise level" ( Figure 13(f)). It is again noteworthy that the new set of projections images of Journal of Flow Control, Measurement & Visualization density thickness "with decreased noise level" can be obtained by using schlieren brightness gradient in horizontal and vertical or two perpendicular diagonal directions ( Figure 13(f)). As seen in Figure 7(i1), Figure 13(f) the noise effects are reduced to an acceptable level and slightly present around initial noise spots.

CT-Reconstruction Results
By employing the density thickness images, noisy (Figure 13(a2)) and with decreased noise level (Figure 13(f)) as projections for CT-reconstruction, the 3D instantaneous density distributions of target flame have been successfully obtained for z = 8 mm to 40 mm. Figure 14 shows a sample set of one horizontal and one vertical cross-section of 3D-CT distributions of density. The darker parts are related to lower density areas with higher temperature and the brighter parts are related to higher density points with lower temperature. For better comparing new and former techniques, the corresponding density contour diagrams of each cross-section are depicted in Figure 15, as well. The density distributions images from multi-path integration in Figure 14

Conclusions
We applied the Cauchy integration theorem to develop the noise reduction technique. Based on Cauchy's integration theorem statement, contour integral Journal of Flow Control, Measurement & Visualization depends only on the beginning and ending points of the paths and does not depend on the integration path. Therefore, the integration procedure in multi-path directions is employed. In the present study, a new technique "multi-path integration" is proposed for "noise reduction" in projections (density thickness) images of CT (computerized tomography). The schlieren brightness gradient information in two orthogonal directions (e.g. horizontal and vertical or two perpendicular diagonal directions) is required in the "multi-path integration" technique. The 20-directional quantitative schlieren optical system gives only images of schlieren brightness in the horizontal gradient and another 20-directional optical system seems necessary to obtain vertical schlieren brightness gradient, simultaneously. One suitable solution without the necessity of a new optical system is using the "inverse process". By employing the new "inverse process" technique, the actual experimental data is used for reproducing other directions of the schlieren brightness gradient. This new approach is a simple, efficient and cost-effective solution and can be used for obtaining any optional directions of schlieren brightness gradient.
In this study, a schlieren image with information on the brightness gradient in two orthogonal directions is called a "complex schlieren image". It has shown an important ability of complex schlieren brightness gradient, which is independence on the path and starting point in the integration process. The results validate that the "multi-path integration" novel technique has the capability of noise reduction in obtaining projections (density thickness) images of CT and the 3D density distributions of CT results.