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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.

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 [

In the previous work [

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 [

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 [

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 [

In the current investigation, the horizontal schlieren brightness gradient data of high-speed turbulent flame [

First, as shown in

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,

The diagram of a single instance of the multi-directional quantitative schlieren camera system is depicted in

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.

^{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 (_{nf}(X) (brightness of schlieren

image in no-flame condition). To obtain the density thickness of deviation density Dt(X(θ)) from B(X) and B_{nf}(X), both are processed in the following manner of

Δ B = B ( X ) − B n f ( X ) (1)

is scaled to d(Dt)/dX by

d ( D t ) / d X = − ( 1 / K ) ( Δ s / f ) [ Δ B ( X ) / B n f ( X ) ] (2)

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

D t ′ ( X ( θ ) ) = D t ( X ( θ ) ) + 2 ( R 2 − X 2 ( θ ) ) 1 / 2 ( ρ r e f − ρ a ∗ ) (3)

In the present study, ρ r e f = ρ a ∗ , accordingly, D t ′ ( X ( θ ) ) = D t ( X ( θ ) ) .

Density thickness images are used for CT-reconstruction by maximum likelihood-expectation maximization (ML-EM) [

ρ ( x , y ) = ρ a ∗ − Δ ρ ( x , y ) (4)

The reconstruction was performed cross-section by cross-section and then the cross-sections were stacked to form a three-dimensional density distribution. Therefore, a 2D distribution ρ(x,y) is accumulated in layers to form the 3D-CT distribution ρ(x,y,z). In the present study, the “density thickness” projections images of 400 (H) × 500 (V) pixel (32 mm × 40 mm) are used for CT-reconstruction to produce 3D data 400 (x) × 400 (y) × 500 (z) pixel (32 mm × 32 mm × 40 mm). The voxel size is 0.08 mm in each direction.

As mentioned before, in the present study, using “inverse process” new approach from density thickness (Dt) images (

d ( D t ) / d X ∝ Δ B ( X ) (5)

and

d ( D t ) / d X = α × Δ B ( X ) (6)

where α is a coefficient with parameters that explained in Equation (2). The deviation density thickness can be expressed as,

d ( D t ( x , z ) ) / d X = D t ( x , z ) − D t ( x − 1 , z ) (7)

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:

Δ B ( x , z ) H o r i z o n t a l , + x = ( 1 / α ) × [ D t ( x , z ) − D t ( x − 1 , z ) ] (8)

Δ B ( x , z ) V e r t i c a l , + z = ( 1 / α ) × [ D t ( x , z ) − D t ( x , z − 1 ) ] (9)

Δ B ( x , z ) D i a g o n a l , 45 ∘ = ( 1 / α ) × [ D t ( x , z ) − D t ( x − 1 , z − 1 ) ] (10)

Δ B ( x , z ) D i a g o n a l , 135 ∘ = ( 1 / α ) × [ D t ( x , z ) − D t ( x + 1 , z − 1 ) ] (11)

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

A key element in the schlieren setup is the knife-edge. In traditional schlieren imaging, gradients in the refractive index normal to the imaging plane are viewed by focusing the deflected light rays onto a knife-edge. The rays that are deflected above the knife-edge appear as bright regions in the image while rays that are deflected onto the knife-edge appear as dark regions in the image. The knife-edge serves as a cut-off filter for light intensity (brightness). The knife-edge orientation will reveal directional density gradients (brightness gradients). For example, vertical knife-edge orientation present horizontal

schlieren brightness gradient data and horizontal knife-edge orientation present vertical schlieren brightness gradient data. In the 20-directional quantitative schlieren optical system, a vertical knife-edge (

Therefore, by simultaneously considering the schlieren brightness gradient information in the horizontal direction (red, real part) and vertical direction (blue, imaginary part), the complex gradient of the density thickness distribution at any position can be obtained. The spatial integration to obtain the density thickness distributions can now be performed on any arbitrary path, and noise reduction now becomes possible. In this study, a schlieren image

with information on the gradient of density thickness in two orthogonal directions is called a “complex schlieren image”.

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,

∫ C f ( z ) = ∫ a b f ( z ) = F ( b ) − F ( a ) (12)

If a = b, then C is a closed curve (loop) within D, accordingly,

∮ C f ( z ) = 0 (13)

We applied the above theorem to develop the noise reduction technique. As shown in

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 (

First, as a preliminary investigation and for a deep understanding of complex schlieren brightness, a numerical simulation for schlieren brightness of a flame is performed. The results of the numerical simulation are depicted in

The noisy density thickness images obtained by single-path integration (former technique) are depicted underneath of corresponding each schlieren images (Figures 7(a2)-(h2)).

initial noise spots (this may be a disadvantage of the proposed technique).

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.

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) (

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

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) (

D t ( x , z ) = D t ( x − 1 , z ) + d ( D t ( x , z ) ) / d x (14)

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. 6 and No. 16) in the new technique “multi-path integration” is shown in the following equations.

D t ( x , z ) = D t ( x − 1 , z + 2 ) + d ( D t ( x , z + 2 ) ) d x − d ( D t ( x , z + 1 ) ) d z − d ( D t ( x , z ) ) d z (15)

D t ( x , z ) = D t ( x − 2 , z − 1 ) + d ( D t ( x − 2 , z ) ) d z + d ( D t ( x − 1 , z ) ) d x + d ( D t ( x , z ) ) d x (16)

First, the density thickness for pixel (x,y) is obtained from the multi-path (28 paths). Then, the average of all these paths is considered as the density thickness for the target pixel.

In the schlieren imaging system, brightness gradually increases (or decreases) from ambient condition (with constant brightness) to boundary condition and vice versa. This fact is effective whether schlieren photography is containing a

complete brightness gradient in the integrating direction or not.

For better evaluation of the “multi-path integration” noise reduction technique, in the new set of 20-directional schlieren images obtained from the inverse process, one thousand noises are distributed with various brightnesses and in the

different positions randomly (

By employing the density thickness images, noisy (

visible on the single-path integration results. Results demonstrate the importance of the new technique for noise reduction.

We applied the Cauchy integration theorem to develop the noise reduction technique. Based on Cauchy’s integration theorem statement, contour integral 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.

This research work was partly supported by The Nitto Foundation and JSPS Grants-in-Aid for Scientific Research (C)16K06118. The supports are gratefully acknowledged.

The authors declare no conflicts of interest regarding the publication of this paper.

Nazari, A.Z., Ishino, Y., Ito, F., Kondo, H., Yamada, R., Motohiro, T., Saiki, Y., Miyazato, Y. and Nakao, S. (2020) Quantitative Schlieren Image-Noise Reduction Using Inverse Process and Multi-Path Integration. Journal of Flow Control, Measurement & Visualization, 8, 25-44. https://doi.org/10.4236/jfcmv.2020.82002

B: Brightness of schlieren image [-]

B_{nf}: Brightness of schlieren image in no-flame (no disturbance) condition in the test section [-]

Dt: Density thickness of deviation density [(kg/m^{3})(m)]; derived value

Dt^{’}: Density thickness [(kg/m^{3})(m)]; derived value

Dt^{*}: Density thickness of deviation density [(kg/m^{3})(m)]; actual value

f: Focal length of convergent lens in schlieren system [m]

K: Gladstone-Dale constant of air [m^{3}/kg]

R: Radius of reconstruction area [m]

x, y, z: Cartesian coordinates system of reconstruction volume [m]

X(θ), Y(θ): Inclined coordinates by angle θ [m]

ΔB: Deviation brightness on schlieren image [-]

Δs: Transparent width of the light source [m]

Δρ: Deviation density [kg/m^{3}]; derived value

Δρ^{*}: Deviation density [kg/m^{3}]; actual value, ( Δ ρ * = ρ a * − ρ * )

θ: Angle of observation [° (degree)]

ρ: Density [kg/m^{3}]; derived value, ( ρ = ρ a * − Δ ρ )

ρ^{*}: Density [kg/m^{3}]; actual value

ρ a * : Ambient density of air [kg/m^{3}]; actual value

ρ_{ref}: Reference density [kg/m^{3}]

φ: Diameter [m]