Curved surface light intensity uniformizing method based on turbine blade film holes

Abstract

The invention provides a curved surface light intensity uniformizing method based on turbine blade air film holes, and aims to solve the problem of non-uniform imaging illumination caused by characteristics of curved surfaces, discontinuity and the like in special objects such as turbine blades. The technology comprises three parts of self-adaptive curvature calculation, dynamic illumination modeling and self-adaptive Gaussian double logarithm Retinex brightness averaging. Firstly, bilateral filtering is applied in an HSV space to extract image brightness parameters, and a Lambertian reflection model is adopted to simulate illumination. Secondly, the light source direction is dynamically adjusted according to the image scene and the size, and curved surface curvature information is calculated to assist in establishment of an illumination model; and meanwhile, illumination calculation is completed according to a Cook-Torrent model and a DiseyBRDF (Bidirectional Reflection Distribution Function) model. Image edges are extracted by combining Canny edge detection, a Hessian matrix and gas film hole features, and then the relation between the reflection intensity and the curvature is estimated. And finally, the brightness is adjusted by using an adaptive double logarithm Retinex algorithm, and the image quality is improved through iterative optimization. The method can effectively enhance the image quality in various environments, is especially suitable for application scenes with high-precision illumination adjustment requirements such as turbine blade non-contact strain measurement, and has the advantages of high robustness and good image enhancement effect.

Description

Technical Field

[0001] The present invention relates to the field of non-contact strain measurement of aero-engine turbine blades, and designs a curved surface light intensity uniformity technology based on the air film holes of the turbine blades. Background Art

[0002] In the field of strain measurement of aero-engine blades, the digital image correlation method (DIC), as an important strain measurement means, its measurement accuracy is directly affected by the image quality. However, due to the influence of the complex working conditions of the engine and the characteristics of the blade curved surface, the collected strain images generally have the problem of uneven illumination, specifically manifested as local overexposure and underexposure phenomena. This problem not only exists in the field of aero-engines, but also commonly exists in application scenarios such as turbine strain measurement in water supply systems, strain detection of pipeline inner walls, and health monitoring of civil building structures. Especially in the measurement of aero-engine blades, in order to meet the shooting requirements of high-speed moving objects, high-power light sources need to be used to ensure image clarity and reduce motion blur, which further exacerbates the problem of uneven image illumination. Existing image enhancement algorithms often exhibit over-enhancement phenomena when processing such non-uniform illumination images, resulting in the loss of image detail information and seriously affecting the accuracy of DIC strain analysis. Therefore, developing a special image enhancement algorithm suitable for DIC calculation of aero-engine strain has important theoretical significance and engineering application value for improving strain measurement accuracy and ensuring the safe operation of the engine. Summary of the Invention

[0003] In view of the deficiencies of traditional image processing technologies, the present invention proposes a curved surface light intensity uniformity technology based on the air film holes of turbine blades. This technology improves the light intensity uniformity by improving the image processing algorithm and has strong application prospects.

[0004] The technical solution of the present invention is a curved surface light intensity uniformity method based on the air film holes of turbine blades, and this method includes:

[0005] Step 1: Convert the turbine blade image into the HSV color domain, and decompose and extract the V component for its three channels based on the HSV color space;

[0006] Step 2: Filter the V component using a bilateral Gaussian filter for space and intensity;

[0007] Step 3: Perform dynamic range compression on the filtered V component;

[0008] Step 4: Calculate the brightness value of the region after dynamic range compression, and compare it with a preset threshold. If it is greater than the threshold, return to Step 3; if it is less than the threshold, enter Step 5:

[0009] Step 5: Calculate the curvature of the photographed subject in the image;

[0010] The following two methods are used to calculate the curvature. When the number of stripes, cracks, and speckles larger than the set size threshold in the main body of the image is less than the set number threshold, Method 1 is adopted; otherwise, Method 2 is adopted.

[0011] Method 1:

[0012] Step 5.1: The image gradient represents the change rate of the image brightness in the horizontal and vertical directions. The change rate in the horizontal direction is Gx, and the change rate in the vertical direction is Gy. Use the gradient function to obtain the brightness change of each pixel point in the horizontal and vertical directions.

[0013] Step 5.2: First, obtain the second-order gradient of the image brightness channel. By recalculating the first-order gradients Gx and Gy, obtain the second-order derivative Hx2 in the horizontal direction and the second-order derivative Hy2 in the vertical direction.

[0014] Calculate to obtain the matrix (x, y) represents the pixel coordinates, and v represents the V component.

[0015] Step 5.3: Calculate the curvature:

[0016] Method 2:

[0017] Step 5.1: Binarize the image into a grayscale image.

[0018] Step 5.2: Remove the areas within the set range around the image.

[0019] Step 5.3: Accurately find the positions of the air film holes based on the characteristics of the shape, size, and distribution law of the air film holes.

[0020] Step 5.4: The calculation formula for the curvature is: y′(x) represents the first-order derivative, and y″(x) represents the second-order derivative.

[0021] Step 5.5: Calculate the curvature of each qualified hole and output the position and curvature value of the hole.

[0022] Step 6: Dynamically adjust the illumination intensity using the curvature information calculated in Step 5.

[0023] Furthermore, the filtering function of the bilateral Gaussian filter in Step 2 is:

[0024]

[0025] Among them, I out(i, j) represents the output image, W(i, j) represents the normalization factor, which is used to ensure that the range of the output value is the same as that of the input value. Ω represents the range of the neighborhood window centered on (i, j), I(k, l) represents the Gaussian weight in the intensity domain, f spatial (i, j, k, l) represents the Gaussian weight in the spatial domain, f intensity (i, j, k, l) represents the Gaussian weight in the intensity domain. (i, j) represents the pixel coordinates of the input image, and (k, l) represents the pixel coordinates within a certain neighborhood window in the image; Iout(i, j), fspatial(i, j, k, l) are the Gaussian weights in the spatial domain, representing the distance weights of pixel positions, and fintensity(i, j, k, l) is the Gaussian weight in the intensity domain, representing the similarity weights of pixel values; W(i, j) is the normalization factor, which is used to ensure that the range of the output value is the same as that of the input value.

[0026] Further, the specific method of step 3 is as follows:

[0027] V″ = log(1 + V ′ ) where V ′ = (V - Vmin) / (Vmax - Vmean);

[0028] V represents the optical parameter of the V component, Vmin represents the minimum value, Vmax represents the maximum value, Vmean represents the average value, and V″ is the latest optical parameter; the original optical parameter is replaced with the latest optical parameter.

[0029] Further, the specific method of step 3 is as follows:

[0030] The optical parameter V of the input image is adaptively adjusted through the following formula:

[0031] O(x, y) = (0.5)log(1 + log b (1 + (V - Vmin) / (Vmax - Vmean))) / Vmax

[0032] O(x, y) is the corrected image brightness parameter, and b is a constant value.

[0033] Further, the specific method of step 6 is as follows:

[0034] Based on the curvature information, the bidirectional reflectance distribution model BRDF is used to dynamically adjust the illumination intensity. By comparing the curvature with the set threshold, the geometric shadowing function of the bidirectional reflectance distribution function is adjusted, thereby adjusting the light intensity.

[0035]

[0036] where, f rThe bidirectional reflectance distribution function is denoted as, the normal distribution function is denoted as D(h), the Fresnel term is denoted as F(h), and the geometric shadowing function is denoted as G(l, v, h). n, h, v, and l represent the surface normal, the half-angle vector, the viewing direction, and the light direction, respectively.

[0037] The present invention has significant advantages over the prior art: First, through two curvature fitting methods, the curvature of the curved surface blade can be calculated more accurately, thereby optimizing the light intensity distribution; Second, the design of the double-logarithmic compression Gaussian algorithm effectively retains the image details and reduces the influence of noise. Brief Description of the Drawings

[0038] Figure 1 is the flowchart of the curved surface light intensity uniformity algorithm in the curved surface light intensity uniformity technology based on the film cooling holes of the turbine blade of the present invention.

[0039] Figure 2 is the simulation diagram of the algorithm brightness averaging and normalization in the curved surface light intensity uniformity technology based on the film cooling holes of the turbine blade of the present invention.

[0040] Figure 3 is the original image (left) and enhanced image (right) of the planar image enhancement contrast in the curved surface light intensity uniformity technology based on the film cooling holes of the turbine blade of the present invention obtained by the treatment of Example 1.

[0041] Figure 4 is the original image (left) and enhanced image (right) of the curved surface image enhancement contrast in the curved surface light intensity uniformity technology based on the film cooling holes of the turbine blade of the present invention obtained by the treatment of Example 2.

[0042] Figure 5 is the original film cooling hole image (left) and the film cooling hole after binary extraction (right) of the binary extraction effect diagram of the film cooling holes in the curved surface light intensity uniformity technology based on the film cooling holes of the turbine blade of the present invention. Detailed Embodiments

[0043] Appendix Figure 1 is the flowchart of the curved surface light intensity uniformity algorithm in the curved surface light intensity uniformity technology based on the film cooling holes of the turbine blade of the present invention: The technical implementation solution of the present invention is to first convert the image into the HSV color domain, decompose and extract the V component for its three channels based on the HSV color space. Aiming at the problems such as poor edge extraction using the traditional Gaussian filter, the present invention uses bilateral filtering. By using an enhanced bilateral filter, Gaussian filtering based on space and intensity is adopted. The overall bilateral filter function is designed as follows:

[0044]

[0045] Further design a dynamic range compression module. The implementation idea of the dynamic range compression is: V″″ = log(1 + V′), where V′ =(V - Vmin) / (Vmax - Vmean). That is, first find the maximum value, minimum value, and average value of the light parameters of the image, and then use the dynamic range compression formula to calculate V'. Replace the previous light parameter component with the data after taking the logarithm of it. After the image undergoes logarithmic transformation, the contrast of the darker area will be improved, and the light intensity will be controlled within a certain range, avoiding overflows and underflows.

[0046] Furthermore, change log(1 + v) of the logarithmic transformation to log(1 + log b(1 + c * v)).

[0047] And add the b and c constant adjustment factors. According to experiments, for b, select the range of 2 - 8; for c, select the range of 8 - 20. The final double - logarithm algorithm formula is as follows:

[0048]

[0049] I double_log represents the image data after double - logarithmic transformation and normalization, which is used for subsequent adaptive illumination adjustment. max(log(1 + log(1 + )) represents the maximum value after double - logarithmic transformation in the entire image, and min(log(1 + log(1 + )) represents the minimum value after double - logarithmic transformation in the entire image; finally, apply this formula to the following adaptive adjustment formula. Utilize the characteristics of the image illumination distribution, and adaptively adjust the light parameters through the Gaussian function to correct the uneven - illumination image, reduce the brightness value of the over - illuminated area, and increase the brightness value of the under - illuminated area; to achieve the above - mentioned goal, the light parameter V of the input image F(x, y) is adaptively adjusted through the following formula:

[0050] O(x,y)=(v)y,y=(0.5)log(1 + log b (1+(V - Vmin) / (Vmax - Vmean))) / Vmax

[0051] O(x,y) is the corrected image brightness parameter, and y is the exponential value used for brightness adjustment.

[0052] There are two methods for calculating the curvature of the curved surface blade. First, approximately calculate the curvature through the overall shape of the curved surface. The image gradients (Gx and Gy) represent the change rates of the image brightness in the horizontal and vertical directions. Using the gradient function, the brightness change of each pixel point in the horizontal direction (Gx) and vertical direction (Gy) can be obtained. Further calculate the Hessian matrix. Its formula is: During specific calculations, first obtain the second-order gradient of the image luminance channel. By recalculating the first-order gradients (Gx and Gy), obtain the second-order derivative in the horizontal direction (Hx2) and the second-order derivative in the vertical direction (Hy2): The Gaussian curvature (K) describes the degree of curvature of the image surface. It can be approximately calculated through the elements of the Hessian matrix. This formula can effectively calculate the Gaussian curvature at each pixel point of the image. The specific formula for the Gaussian curvature is as follows: Extract the edges of the image through the Canny edge detection algorithm, and combine the calculated curvature information to extract the curvature values of the edge regions.

[0053] Another way to calculate curvature is that the imaging shape and size of the film holes are different under different curvatures, so the curvature of each region is fitted, and then the average value is calculated. The specific implementation is as follows: First, binarize the grayscale image. Use a custom threshold, usually extracting information according to the standard image library. This threshold determines which pixels are regarded as the foreground (white) or the background (black). Since there may be noise or irrelevant regions at the edges of the image, we remove the regions within a certain range (20 pixels) around the image. And accurately find the positions of the film holes through the characteristics of the shape, size, and distribution law of the film holes. Figure 5 is the effect diagram of binarization extraction of film holes; Use the regionprops function to calculate the properties of each connected region in the binary image, such as area, centroid, bounding box, major axis length, minor axis length, and pixel list, etc. Calculate the first derivative (dy) and the second derivative (d2y) of the polynomial. The first derivative is the slope of the tangent line, and the second derivative is the key to curvature. The formula for curvature is: For each qualified hole, calculate the curvature and output the position and curvature value of the hole. At the same time, store the curvature in an array for convenient subsequent calculation of the average curvature. Finally, the code calculates the average curvature of all effective film holes.

[0054] Furthermore, the curvature information can be used to dynamically adjust the light intensity, thereby improving the visual effect of the image. The BRDF model is used to describe the reflection characteristics when light interacts with the surface. By adjusting the BRDF parameters, the lighting effects of different curvature regions can be simulated:

[0055] Diffuse reflection coefficient (rho_d): Reduce diffuse reflection when the curvature is large.

[0056] Specular reflection coefficient (rho_s): Enhance specular reflection when the curvature is large.

[0057] The Cook-Torrance BRDF is a physically based reflection model, mainly for modeling metals and plastics, and is very suitable for turbine blades and other industrial fields. The Cook-Torrance BRDF consists of three main terms: the normal distribution function D(h), the Fresnel reflection F(h), and the geometric occlusion term G(l, v, h). The Cook-Torrance BRDF formula is: The light intensity can be adjusted by adjusting the parameters of the BRDF to make the lighting effect more in line with the actual situation.

[0058] By adjusting the parameters of the above reflection model and lighting model, combined with the Gaussian curvature, the lighting effect of the image can be optimized, especially in high-curvature areas. To simulate the lighting effect more realistically, the Fresnel reflection model (Schlick approximation) is also applied, which describes the reflection coefficient when light interacts with the surface. At a specific angle, the reflectivity of light changes with the incident angle, so by dynamically adjusting the lighting direction, a more realistic lighting effect can be simulated.

[0059] Finally, the image is sent to an adaptive filter for noise reduction and the overall image is sharpened and enhanced to obtain an image with uniform brightness after processing.

[0060] The processed image has higher lighting uniformity and is suitable for the lighting optimization requirements of complex curved surface objects.

[0061] Figure 2 Shows the simulation diagram of the brightness normalization algorithm in the curved surface light intensity uniformization technology based on the turbine blade film cooling holes of the present invention. The present invention adjusts the lighting distribution through exponential function transformation to achieve the uniformization of the overall brightness. Specifically, after setting the target average brightness parameter, the light intensity parameters in the image are normalized. When a certain light parameter component is lower than the average value, the calculated difference is negative. According to the characteristics of the exponential function, the negative exponential term tends to be positive, thus enhancing the brightness of this area. On the contrary, when the light parameter component is higher than the average value, the calculated difference is positive, and the exponential term tends to be negative, thus reducing the brightness of this area. This exponential transformation mechanism can ensure that the overall light parameters gradually converge to the set target mean value, thus achieving the effect of lighting uniformization. Further, by inputting images with different brightness distributions for simulation verification, as Figure 2 shown, when the deviation between the brightness value at a certain point and the overall average brightness value is greater than 0.1797, the double logarithmic transformation function will enhance the brightness value of this point; on the contrary, when the lighting value at a certain point (x, y) is higher than the global lighting mean value, the double logarithmic function dynamically adjusts the brightness value of this point according to the exponential decay mechanism to reduce the phenomenon of local overexposure of lighting.

[0062] Figure 3The figure shows the enhanced contrast planar image under the action of the curved surface light intensity homogenization technology based on the film cooling holes of the turbine blade in the present invention. First, the image to be processed is subjected to bilateral filtering to extract the illumination parameter V. Since it is a planar object, a relatively simple gradient function is used to calculate the image curvature. Then, the curvature information is sent to the illumination model for more perfect illumination modeling to obtain a more realistic V parameter of the image that conforms to the actual environment. Further, the parameter b of the double logarithm is set to 5, the parameter c is set to 15, and the luminance parameter V is subjected to double logarithm compression. Further, the threshold is set to 0.27 for iterative optimization. Finally, the above light parameters are averaged in luminance by a Gaussian function adaptive function, and adaptive noise reduction processing is performed before output to eliminate the influence of noise and detail enhancement to obtain richer detail information.

[0063] In the original image, there are phenomena of overexposure or underexposure in some areas, resulting in uneven illumination distribution, which affects the accuracy of visual measurement and subsequent processing. The present invention optimizes the light intensity distribution to balance the image brightness globally. Compared with the original image, the enhanced image has been significantly improved in terms of brightness uniformity and overall visual consistency, effectively suppressing local overexposure or underexposure phenomena and improving the accuracy of optical measurement and visual inspection.

[0064] Figure 4 It is the enhanced image contrast of the curved surface image in the wave processing of the curved surface light intensity uniformity technology based on the film cooling holes of the turbine blade in the present invention. First, the image to be processed is subjected to bilateral filtering to extract the illumination parameter V. Since it is a curved surface object and is a blade image containing film cooling holes, the edge information is calculated by using Canny and then the curvature information is fitted. Then, the local curvature is calculated by comparing after extracting the film cooling holes, and finally the overall image curvature is synthesized. Then, the curvature information is sent to the illumination model for more perfect illumination modeling to obtain a more realistic V parameter of the image that conforms to the actual environment. Further, because the illumination distribution of the curved surface image is more uneven, the parameter b of the double logarithm is set to 6, the parameter c is set to 17, and the luminance parameter V is subjected to double logarithm compression. The threshold is set to 0.27 for iterative optimization. Finally, the above light parameters are averaged in luminance by a Gaussian function adaptive function, and adaptive noise reduction processing is performed before output to eliminate the influence of noise and detail enhancement to obtain richer detail information.

[0065] Regarding the problem of uneven exposure caused by different curvatures in the curved surface object, the enhanced image can still ensure the same brightness at positions with different curvatures. At the same time, this method can also compensate for the defocus effect caused by curvature changes, improve the image clarity, and ensure high-quality visual information can still be obtained under complex curved surface structures.

[0066] Figure 5This is the effect diagram of the binarization extraction of the film holes in the curved surface light intensity uniformity technology based on the turbine blade film holes of the present invention. By setting and adjusting the binarization threshold and performing Sobel edge detection, most of the shaped objects on the turbine blade are extracted. Then, through shape recognition and rule recognition filtering, useless information such as speckle noise is filtered out, and only the film holes are retained. Compared with the original film hole photo, the extraction effect can cover all the film holes without extracting extra useless information.

[0067] Table 1 shows the results of six groups of test images of the curved surface light intensity uniformity system based on the turbine blade film holes of the present invention and common brightness uniformity enhancement algorithms in the industry under different illuminance conditions. By extracting and comparing the PSNR (peak signal-to-noise ratio) values of each algorithm, the results show that the algorithm of the present invention has significant advantages in terms of information entropy and average gradient compared with other algorithms. Specifically, the algorithm of the present invention demonstrates excellent performance in terms of brightness uniformity and image detail restoration, proving that the algorithm can not only effectively improve the uniformity of image brightness but also provide significant improvements in detail restoration and image quality maintenance.

[0068] Compared with the prior art, the significant advantage of the present invention lies in its precise calculation method for the curvature of the curved surface blade of an aeroengine. By adopting two curvature fitting methods and a complete lighting modeling technique, the present invention can more accurately reflect the curved surface geometric features as well as the material and light reflection characteristics, thus significantly improving the accuracy of curvature calculation. Based on the accurate curvature values, the present invention further optimizes the light intensity adjustment algorithm, especially introducing the Gaussian algorithm with double logarithmic compression. When adjusting the light intensity distribution, this algorithm can significantly improve the lighting uniformity. In addition, the Gaussian algorithm with double logarithmic compression effectively retains the image detail information while achieving efficient noise suppression, providing a clearer basis for subsequent image processing. This innovative design not only improves the effect of image enhancement but also provides strong support for more complex image processing algorithms. Any modifications, equivalent replacements, etc. made within the spirit and principles of the present invention shall be included within the protection scope of the present invention.

[0069] Table 1 Comparison of PSNR between the algorithm of the present invention and common image enhancement algorithms;

[0070]

Claims

1. A method for uniform surface light intensity based on the film cooling holes of a turbine blade, the method comprising: Step 1: Convert the turbine blade image into the HSV color domain, and decompose and extract the V component from its three channels based on the HSV color space; Step 2: Filter the V component using a bilateral Gaussian filter for space and intensity; Step 3: Perform dynamic range compression on the filtered V component; Step 4: Calculate the brightness value of the region after dynamic range compression and compare it with a preset threshold. If it is greater than the threshold, return to Step 3; if it is less than the threshold, enter Step 5: Step 5: Calculate the curvature of the captured subject in the image; The curvature is calculated using the following two methods. When the number of stripes, cracks, and speckles in the image subject that are larger than the set size threshold is less than the set number threshold, Method 1 is used; otherwise, Method 2 is used; Method 1: Step 5.1: The image gradient represents the change rate of the image brightness in the horizontal and vertical directions. The change rate in the horizontal direction is Gx, and the change rate in the vertical direction is Gy; use the gradient function to obtain the brightness change of each pixel point in the horizontal and vertical directions; Step 5.2: First, obtain the second-order gradient of the image luminance channel. By recalculating the first-order gradients Gx and Gy, obtain the second-order derivative Hx2 in the horizontal direction and the second-order derivative Hy2 in the vertical direction; The calculated matrix (x, y) represents the pixel coordinates, and v represents the V component; Step 5.3: Calculate the curvature: Method 2: Step 5.1: Binarize the image into a grayscale image; Step 5.2: Remove the regions within a set range around the image; Step 5.3: Accurately find the positions of the film cooling holes based on the characteristics of the shape, size, and distribution law of the film cooling holes; Step 5.4: The calculation formula for curvature is as follows: y′(x) represents the first derivative, and y″(x) represents the second derivative; Step 5.5: Calculate the curvature of each qualified hole and output the position and curvature value of the hole; Step 6: Dynamically adjust the illumination intensity using the curvature information calculated in Step 5.

2. The method for uniform surface light intensity based on the film cooling holes of a turbine blade according to claim 1, wherein, The filtering function of the bilateral Gaussian filter in Step 2 is: Among them, I out (i,j) represents the output image, W(i,j) represents the normalization factor, which is used to ensure that the range of the output value is the same as that of the input value, Ω represents the neighborhood window range centered on (i,j), I(k,l) represents the Gaussian weight in the intensity domain, f spatial (i,j,k,l) represents the Gaussian weight in the spatial domain, f intensity (i,j,k,l) represents the Gaussian weight in the intensity domain, (i,j) represents the pixel coordinates of the input image, (k,l) represents the pixel coordinates within a certain neighborhood window in the image; Iout(i,j), fspatial(i,j,k,l) are the Gaussian weights in the spatial domain, representing the distance weight of the pixel position, fintensity(i,j,k,l) is the Gaussian weight in the intensity domain, representing the similarity weight of the pixel value; W(i,j) is the normalization factor, which is used to ensure that the range of the output value is the same as that of the input value.

3. A method for uniform surface light intensity based on the film holes of a turbine blade according to claim 1, characterized in that, The specific method of Step 3 is: V″ = log(1 + V′) where V′ = (V - Vmin) / (Vmax - Vmean); V represents the light parameter of the V component, Vmin represents the minimum value, Vmax represents the maximum value, Vmean represents the average value, and V″ is the latest light parameter; replace the original light parameter with the latest light parameter.

4. A method for uniform surface light intensity based on turbine blade film holes as claimed in claim 1, characterized in that The specific method of Step 3 is: The light parameter V of the input image is adaptively adjusted through the following formula: O(x,y) = (0.5)log(1 + log b (1 + (V - Vmin) / (Vmax - Vmean))) / Vmax O(x, y) is the corrected image brightness parameter, and b is a constant value.

5. A method for uniform surface light intensity based on film holes of a turbine blade, as described in claim 1, wherein The specific method of Step 6 is: Dynamically adjust the illumination intensity based on the curvature information using the bidirectional reflectance distribution model BRDF. Compare the curvature with the set threshold, and adjust the geometric shadowing function of the bidirectional reflectance distribution function to adjust the light intensity. where f r represents the bidirectional reflectance distribution function, D(h) represents the normal distribution function, F(h) represents the Fresnel term, G(l, v, h) represents the geometric shadowing function, and n, h, v, l represent the surface normal, half-angle vector, viewing direction, and light direction, respectively.

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