Interactive plant model construction method based on Gaussian splashing

Abstract

The invention discloses an interactive plant model construction method based on Gaussian splashing, which is a high-resolution and interactive plant three-dimensional model construction method, and comprises the following steps: firstly, constructing a plant whole growth stage image data set, estimating a camera attitude, and obtaining initial point cloud data; further replacing the point cloud data with three-dimensional Gaussian, and constructing a high-resolution three-dimensional plant model; then, in combination with a Canny edge detection and color space analysis method, an SAM prompt point set is obtained from the plant image, after classification, the SAM prompt point set is sent to the SAM, and in combination with mask evaluation and mask post-processing means, a high-quality plant segmentation mask set is constructed; and finally, combining a three-dimensional Gaussian model, segmenting the mask set into Gaussian training feature vectors to represent semantic attributes, and finally obtaining a high-resolution interactive plant three-dimensional model, so that researchers can perform a series of operations such as phenotype analysis and plant growth model construction based on the model. And subsequent plant breeding and agricultural research work are supported.

Description

Technical Field

[0001] The present invention belongs to the field of combining computer vision with plant analysis, and provides a high-fidelity, interactive method for constructing a three-dimensional plant model, based on which subsequent tasks such as plant phenotypic analysis and growth model construction can be performed. Background Art

[0002] Plant phenotyping primarily encompasses structural, physiological, and temporal phenotypes. Growth status is assessed by measuring and analyzing external features such as plant morphology and structure, as well as internal characteristics such as water content and nutrient storage. Combining these structural and physiological phenotypic data with timelines to construct plant growth models is crucial for plant breeding and agricultural production. However, for plant structural phenotyping, traditional manual measurement methods are time-consuming and labor-intensive, relying on extensive manual effort, resulting in low efficiency and prone to human error. Compared to 2D images, directly reconstructing a 3D plant model offers greater practical value. By automating image acquisition and analysis and leveraging computer vision technology to generate high-resolution 3D plant models, we can facilitate subsequent plant phenotyping analysis. This significantly reduces manual intervention time, improves the accuracy and efficiency of phenotyping measurements, and provides an effective digital model for subsequent plant analysis.

[0003] Most existing three-dimensional plant data expressions use point cloud data (PCD). High-precision point cloud data can be acquired using laser scanning equipment, but this method is costly and difficult to operate. Point cloud data can also be obtained from multi-view images using the Structure from Motion (SfM) algorithm. However, the complex textures and thin structure of plant leaves are not taken into account, resulting in low accuracy in plant point cloud data. Furthermore, although point cloud models can capture the basic structure of plants, they still have significant deficiencies in visual representation. Sparse point cloud data makes it difficult to display the subtle morphology and complex surface textures of plants. The reconstructed model cannot achieve pixel-level visual effects, and is significantly different from the observed real plants.

[0004] In addition, existing methods adopt a global modeling strategy, focusing on the reconstruction and analysis of the overall structure of the plant. Although it can capture the data of the entire plant, it ignores the individual representation of each part of the plant, making subsequent fine-grained analysis of the plant very difficult. Although there are many segmentation networks (ScanNet, PointNet++, etc.) that can divide the overall point cloud data into specific parts such as leaves and stems, in order to obtain high-precision segmentation results, a large amount of data is required for training, and manual annotation of point cloud data is time-consuming and labor-intensive. At the same time, this type of plant model representation method lacks sufficient interactivity. This limitation reduces the flexibility and practicality of the model in practical applications, hinders researchers from efficiently focusing on the plant parts of interest and conducting in-depth phenotypic analysis or comparative studies, and limits the flexibility and practicality of plant models in practical applications.

[0005] Overall, the current three-dimensional digital models of plants have low fidelity, and their interactivity and flexibility are obviously insufficient, and are in urgent need of improvement. Summary of the Invention

[0006] This invention addresses the shortcomings of existing plant model construction methods and focuses on plants with distinct stem and leaf structures, numerous leaves, and complex surface vein textures (such as soybean plants). Utilizing Gaussian splatting technology, it aims to develop a method for constructing a three-dimensional plant digital model with high visual fidelity and strong interactivity. This allows researchers to perform a series of operations based on this model, including phenotypic analysis and plant growth model construction, to support subsequent plant breeding and agricultural research.

[0007] The present invention is based on the interactive plant model construction method of Gaussian splashing, and the specific steps are as follows:

[0008] S1: Use cameras to capture plant images from multiple perspectives at different plant growth stages to construct a dataset of images of all plant growth stages. image .

[0009] S2: For multi-view images of plants at the same growth stage, the AKAZE algorithm is used as a tool for detecting and describing image feature points in the SfM algorithm process. The SfM algorithm is used to restore the camera pose and obtain three-dimensional point cloud data. The camera pose and point cloud datasets at different shooting angles are obtained; the MVS framework is further used to expand the point cloud dataset.

[0010] S3: Expand each point in the point cloud dataset obtained in step S2 into a three-dimensional Gaussian function containing five parameters {μ, R, S, α, c} to obtain a three-dimensional Gaussian model of the plant.

[0011] S4: Using the multi-view plant image obtained in step S1 as supervision, the plant 3D Gaussian model obtained in step S3 is trained to obtain a high-fidelity plant 3D Gaussian model.

[0012] S5: For plant image set Data image For each image in the dataset, we combine Canny and color space analysis to obtain the cue point set; further, we use SAM to generate the segmentation mask and perform quality assessment to obtain the plant image dataset Data. image For each image in , a segmentation mask set Masks is constructed.

[0013] S6: Combine the high-fidelity plant three-dimensional Gaussian model obtained in step S4 with the plant image segmentation mask set Masks obtained in step S5 to train new feature vector attributes for the three-dimensional Gaussian to increase the interactivity of the model.

[0014] The advantages of the present invention are:

[0015] 1. The AKAZE feature extraction algorithm used in this paper utilizes nonlinear diffusion filtering to enhance edges and details, performing well when processing complex plant surface textures. Its multi-scale feature detection capability enables it to effectively capture the size and structure changes of plant leaves during different growth cycles, making it more suitable for feature extraction from plant image data. Combined with subsequent SfM and MVS processing, it can generate more accurate point cloud data for subsequent optimization.

[0016] 2. The present invention uses a three-dimensional Gaussian function instead of a point cloud as the data representation method for the model. Compared with traditional point clouds, the three-dimensional Gaussian model not only provides a more realistic visual effect, but also greatly improves the accuracy of the model, and can more accurately reproduce the geometric shape and surface characteristics of the plant. Through this high-fidelity digital model, researchers can intuitively and conveniently observe and analyze the growth and development of plants. At the same time, the three-dimensional Gaussian model of the plant inherits the coordinate properties of the point cloud model, making the analysis of phenotypic parameters more flexible and efficient;

[0017] 3. The present invention adopts the segmentation large model SAM to generate the segmentation mask of the plant image, avoiding the separate training of a dedicated plant image segmentation network and greatly saving the time of data annotation.

[0018] 4. To improve segmentation accuracy and reduce the workload of manually selecting cue points, the present invention introduces a method that combines Canny edge detection with multi-color space analysis. This method not only optimizes the SAM segmentation effect but also effectively reduces the generation of invalid masks. By designing rigorous mask evaluation criteria and post-processing methods, the present invention ensures that the segmentation results meet the requirements of plant image segmentation. Furthermore, the present invention extends the capabilities of SAM to the three-dimensional domain, improving the interactivity and practicality of 3D plant models. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] Figure 1 This is a flow chart of a method for constructing an interactive plant model based on Gaussian splashing according to the present invention;

[0020] Figure 2 Flowchart for training 3D Gaussian model of plants; Figure 3 Construct a flowchart for the segmentation mask set of plant images. DETAILED DESCRIPTION The present invention will be described in further detail below with reference to the accompanying drawings. The present invention is based on the construction method of the interactive plant model of Gaussian splashing, such as Figure 1 As shown, the specific steps are:

[0021] S1: Construct a plant full growth stage image dataset at different plant growth stages.

[0022] At different growth stages of plants, considering the mutual occlusion between plant leaves, the complex texture of leaves, as well as the low image resolution and missing perspectives that will affect the quality of the three-dimensional model. An RGB camera is used to acquire plant images from multiple perspectives; a consumer-grade Kinectv2 camera is used for image acquisition, which can provide high-definition images with a resolution of 1920*1080 to ensure that rich details of the plant surface are captured. At the same time, the overlap between adjacent perspectives is guaranteed to be more than 80% during the image acquisition process. After acquiring the plant images, image pre-processing such as denoising and contrast enhancement is performed to finally construct a high-quality, perspective-complete plant image dataset. image .

[0023] S2: Analyze multi-view images of plants at the same growth stage, and use the SfM algorithm to restore the camera pose and obtain 3D point cloud data for the multi-view images of plants at the same growth stage.

[0024] a. The AKAZE algorithm is used as a tool for detecting and describing image feature points in the SfM algorithm. Based on a nonlinear diffusion equation, the AKAZE algorithm constructs a multi-scale space for the input plant image. It detects feature points across a series of image versions, ensuring that the extracted plant image feature points are consistent across scales. For each detected feature point, AKAZE generates a descriptor that reflects the surrounding information of the feature point. It then calculates the Euclidean distance between these descriptors to identify the best matching feature point pairs across different plant images.

[0025] b. Using the best correspondences found between feature points, and following the SfM algorithm framework, we construct equations using the normalized coordinates between matching point pairs to solve for the essential matrix and fundamental matrix between the corresponding points. Further decomposing the fundamental and essential matrices yields the camera pose {P} at different viewing angles. This includes camera intrinsic parameters such as focal length, principal point coordinates, and distortion parameters, as well as two extrinsic parameters: the rotation matrix and translation matrix when converting from the world coordinate system to the camera coordinate system. This allows us to estimate the camera pose for each image.

[0026] c. Based on the camera's intrinsic and extrinsic parameters and detected image feature points acquired from multiple views, a multi-view triangulation method is used to estimate the 3D coordinates of each feature point. This creates a sparse point cloud dataset that initially depicts the plant's spatial structure. To further improve the accuracy of this point cloud dataset's representation of the plant's spatial structure, a global bundle adjustment is applied to minimize the reprojection error of the 3D points across all viewpoints. The previously estimated 3D point coordinates and multi-view camera parameters are then fine-tuned to ensure high accuracy and consistency in the reconstruction results.

[0027] Since the data obtained by the above SfM algorithm are all sparse point cloud data, the MVS framework is used to expand the plant sparse point cloud set to provide a more refined plant spatial structure expression point cloud dataset PCD.

[0028] S3: Expand the point cloud dataset PCD obtained in step S2 into a three-dimensional Gaussian model of the plant.

[0029] In the point cloud dataset PCD obtained in step S2, the valid attributes of each point include spatial coordinates, color, and normal vector, and only the spatial coordinate information (x, y, z) in the point cloud is retained as the center position μ of each three-dimensional Gaussian distribution.

[0030] Then, the random initialization method is used to introduce the rotation matrix R and the scaling matrix S to construct the covariance matrix ∑ = RSS of each Gaussian T R T The rotation matrix R represents the distribution direction of the three-dimensional Gaussian in space, and the scaling matrix S represents the scale information of the three-dimensional Gaussian in space. These two matrices can jointly characterize the shape characteristics of the three-dimensional Gaussian.

[0031] Furthermore, each 3D Gaussian is assigned an opacity parameter α and a coefficient vector c of spherical harmonics (SH) to represent its appearance. Third-order SHs are used here to well represent Gaussian color information, so the coefficient vector c has a dimension of 16.

[0032] Finally, through the above method, each point in PCD is expanded into a three-dimensional Gaussian function containing five parameters {μ, R, S, α, c}, and the three-dimensional Gaussian representation set of plants is obtained.

[0033] S4: Using the multi-view plant image obtained in step S1 as supervision, the plant 3D Gaussian model obtained in step S3 is trained to obtain a high-fidelity plant 3D Gaussian model.

[0034] like Figure 2 As shown, first, the multi-view camera pose set {P i} randomly select a camera pose P i , using Differentiable Rendering technology, the three-dimensional Gaussian set of the plant is projected onto the two-dimensional image plane in this posture, generating the rendered plant image I under the camera perspective. render .

[0035] Then, a real plant image I under the camera’s perspective is selected from the multi-view plant image dataset constructed in step S1. real , calculates the pixel-level difference loss between plant rendered images and real images.

[0036] Further calculate the gradient of the loss function to the Gaussian parameters, and update the parameters {μ, R, S, α, c} of all Gaussian in the plant Gaussian representation set by gradient backpropagation. Here we use L1 loss and L D-SSIM Loss joint construction of the loss function L used to train plant Gaussian parameters pixel (I render ,I real ).

[0037] L pixel (I render ,I real )=λL1+(1-λ)L D-SSIM

[0038] Among them, λ is an adjustable hyperparameter; L1 represents the L1 loss between two images, L D-SSIM Represents the L between the two images D-SSIM Loss, the two expressions are:

[0039]

[0040] Where H×W is the total number of image pixels, I1(x,y) and I2(x,y) represent the pixel values of the image at the pixel coordinates (x,y), μ is the mean of the image, σ is the variance of the image, and σ 12 is the covariance between the two images, and C1 and C2 are constants used to avoid the denominator being zero.

[0041] In the camera pose set {P i Repeat the above steps in [1], set an appropriate number of model training times (number of images x 100), and iterate through all camera poses. Through continuous iteration, a plant 3D Gaussian set is trained that can reconstruct the original plant with high fidelity, ultimately achieving a visual effect that is essentially indistinguishable from the real plant.

[0042] S5: Combine Canny and color space analysis to obtain the cue point set, use SAM to generate the segmentation mask, and construct the segmentation mask set Masks of the plant image after quality assessment.

[0043] For multi-view plant image set Data image For each image in , the classic Canny algorithm framework is used to detect image edge information, and the key parameters of the Canny algorithm are dynamically adjusted according to the characteristics of each image, such as Figure 3 As shown, specifically:

[0044] A. Use adaptive Gaussian filter to suppress noise on the image, where the size of the Gaussian filter is optimized according to the image resolution and expected edge width.

[0045] B. Calculate the gradient magnitude and direction between pixels through the Sobel operator to quantify the image brightness changes and determine the potential edge location.

[0046] C. Apply non-maximum suppression (NMS) to refine edges, retaining only local maxima as candidate edge points. Then, based on the grayscale distribution statistics of the image's pixel values, dynamically set the high and low thresholds for edge detection. The high threshold is used to identify strong edges, while the low threshold is used to identify weak edges. This ensures that background noise is effectively separated while sufficient edge detail is preserved.

[0047] D. Using the hysteresis thresholding method, the weak edge pixels connected to the strong edge are included in the final edge map, and the edge points are extracted from the edge of the image as the edge point set corresponding to each image.

[0048] E. Use the OpenCV image processing library to read the image's RGB data and map it to the HSV and Lab color spaces. Leveraging the increased sensitivity of these two color spaces to the green portion of the plant, gradient calculations are performed on the pixel values of each of the six converted channels (H, S, V, L, a, b), quantifying the intensity variation of each pixel in each channel. Based on the calculated gradient values, significant color or brightness changes are selected to construct a set of internal leaf detail points for each image. This overcomes the limitations of Canny edge detection in capturing subtle texture changes within plant images. Combined with the image edge point set obtained by the aforementioned Canny algorithm, a comprehensive cue point set is constructed for each image.

[0049] F. Divide the comprehensive cue point set for each image into plant parts (e.g., leaf 1, leaf 2, stem, etc.) and sequentially feed it into the SAM segmentation model. Leveraging the model's ability to accept multiple cue points for image segmentation, segmentation masks are generated for each plant image. Each segmentation mask is evaluated for boundary clarity, regional integrity, and alignment with the actual plant structure. The highest-quality mask is selected and added to the final plant image segmentation mask set.

[0050] According to the above steps, a segmentation mask set Masks is constructed for each image in the multi-view plant image dataset.

[0051] S6: Combining the high-fidelity plant three-dimensional Gaussian model obtained in step S4 with the plant image segmentation mask set obtained in step S5, new feature vector attributes are trained for the three-dimensional Gaussian to increase the interactivity of the model, and finally a high-fidelity, interactive plant three-dimensional digital model is obtained.

[0052] In the high-fidelity 3D plant Gaussian model obtained in step S4, each 3D Gaussian has attributes {μ, R, S, α, c}, which together constitute the Gaussian's shape and appearance properties. Based on this, a 10-dimensional feature vector f is introduced for each 3D Gaussian using a random initialization method to characterize the semantic attributes of the plant part it represents.

[0053] Then the feature vector is trained as follows:

[0054] First, randomly select a camera pose P from the camera pose set obtained in step S2 i , using the differentiable rendering method, the feature vector f in the three-dimensional Gaussian representation set is projected onto the two-dimensional image plane to obtain the rendering segmentation mask set under the camera perspective.

[0055] Furthermore, the loss between the segmentation mask set obtained by rendering and the real plant image segmentation mask set Masks obtained in S5 is calculated. The design of the loss function is consistent with step S4.

[0056] Finally, the gradient backpropagation method is used to update the feature vector parameters of each three-dimensional Gaussian.

[0057] In the camera pose set {P i}, set an appropriate number of model training times (number of images x 100), traverse all camera poses and repeatedly train the feature vectors of the plant 3D Gaussian, and finally obtain the plant 3D Gaussian representation set. The attributes of each 3D Gaussian are {μ, R, S, α, c, f}, and all these plant 3D Gaussian representations together constitute the 3D model of the plant.

[0058] In the final inference phase of the model, the 3D plant model receives query points (corresponding to point hints) in 3D space and generates a corresponding query vector Q. This query vector Q is matched with the eigenvectors f of all 3D Gaussians in the model, and similarities are calculated to locate and identify the specific plant part of interest to the researcher, thus achieving interactivity in the 3D model.

Claims

1. An interactive plant model construction method based on Gaussian splashing, characterized by: The specific steps are: S1: Use cameras to capture plant images from multiple perspectives at different plant growth stages to construct a dataset of images of all plant growth stages. image ; S2: For multi-view images of plants at the same growth stage, the AKAZE algorithm is used as a tool for detecting and describing image feature points in the SfM algorithm process. The SfM algorithm is used to recover the camera pose and obtain 3D point cloud data. The camera pose and point cloud datasets at different shooting angles are obtained. The MVS framework is further used to expand the point cloud dataset. S3: Expand each point in the point cloud dataset obtained in step S2 into a three-dimensional Gaussian function containing five parameters {μ, R, S, α, c} to obtain a three-dimensional Gaussian model of the plant; S4: Using the multi-view plant image obtained in step S1 as supervision, the plant 3D Gaussian model obtained in step S3 is trained to obtain a high-fidelity plant 3D Gaussian model; S5: For plant image set Data image For each image in the dataset, we combine Canny and color space analysis to obtain the cue point set; further, we use SAM to generate the segmentation mask and perform quality assessment to obtain the plant image dataset Data. image For each image in , a segmentation mask set Masks is constructed; S6: Combine the high-fidelity plant three-dimensional Gaussian model obtained in step S4 with the plant image segmentation mask set Masks obtained in step S5 to train new feature vector attributes for the three-dimensional Gaussian to increase the interactivity of the model.

2. The method for constructing an interactive plant model based on Gaussian splashing according to claim 1, characterized in that: In step 1, a consumer-grade Kinectv2 camera is used as the camera, and the overlap between adjacent view angles is ensured to be greater than 80% during image acquisition.

3. The method for constructing an interactive plant model based on Gaussian splashing according to claim 1, characterized in that: The specific method for step 2 is: a. Use the AKAZE algorithm to generate a descriptor that reflects the surrounding information of all detected feature points. Then, calculate the Euclidean distance between the descriptors to find the best matching feature point pairs on different plant images. b. Based on the SfM algorithm framework, the normalized coordinates between matching feature point pairs are used to construct equations to solve the essential matrix and fundamental matrix between the corresponding points. The camera pose under different shooting angles can be obtained by further decomposing the fundamental matrix and essential matrix. c. Based on the camera internal and external parameters obtained under multiple views and the detected image feature points, the multi-view triangulation method is used to estimate the spatial three-dimensional point coordinates corresponding to each feature point and construct a sparse point cloud dataset.

4. The method for constructing an interactive plant model based on Gaussian splashing according to claim 1, wherein: In step 2, after constructing the point cloud dataset, global bundle adjustment is applied to minimize the reprojection error of the 3D points under all view images, and the position coordinates of the previously estimated 3D points and the camera parameters under multiple view angles are adjusted.

5. The method for constructing an interactive plant model based on Gaussian splashing according to claim 1, wherein: The specific method of step 3 is as follows: in the point cloud dataset PCD obtained in step S2, only the spatial coordinate information (x, y, z) is retained in the valid attributes of each point as the center position μ of each three-dimensional Gaussian distribution; Then, the random initialization method is used to introduce the rotation matrix R and the scaling matrix S to construct the covariance matrix ∑ = RSS of each Gaussian T R T ; Further, an opacity parameter α and a coefficient vector c of the spherical harmonic function are assigned to each three-dimensional Gaussian to represent the appearance properties of the three-dimensional Gaussian.

6. The method for constructing an interactive plant model based on Gaussian splashing according to claim 5, characterized in that: The spherical harmonics are third-order spherical harmonics.

7. The method for constructing an interactive plant model based on Gaussian splashing according to claim 1, wherein: The specific method of step 4 is: first randomly select a camera pose P from the multi-view camera poses obtained in step S1 i , using differentiable rendering technology, the three-dimensional Gaussian set of plants is projected onto the two-dimensional image plane in this posture to generate the rendered plant image from the camera perspective; then, the real plant image from the camera perspective is selected from the multi-view plant image dataset constructed in step S1, and the pixel-level difference loss between the rendered plant image and the real image is calculated; further, the gradient of the loss function with respect to the Gaussian parameters is calculated, and the parameters {μ, R, S, α, c} of all Gaussians in the plant Gaussian representation set are updated by gradient backpropagation; through the above method, all camera postures are traversed and training is repeated.

8. The method for constructing an interactive plant model based on Gaussian splashing according to claim 1, wherein: The specific method of step S5 is: A. Use adaptive Gaussian filter to suppress noise on the image; B. Calculate the gradient magnitude and direction between pixels using the Sobel operator; C. Apply non-maximum suppression to refine the edge, retaining only the local maximum value as the candidate edge point; then dynamically set the high and low thresholds of edge detection based on the grayscale distribution statistics of the pixel values of the image; D. Using the edge connection method, the weak edge pixels connected to the strong edge are included in the final edge map, and the edge points are extracted from the edges of the image as the edge point set corresponding to each image; E. Use the OpenCV image processing library to read the image's RGB data and map it to HSV and Lab color spaces. Taking advantage of the higher sensitivity of these two color spaces to the green portion of the plant, perform gradient calculations on the converted six-channel pixel values. Select points with significant color or brightness changes to construct a leaf detail point set for each image. Combined with the image edge point set obtained using the Canny algorithm, a comprehensive cue point set is constructed for each image. F. Divide the comprehensive cue point set of each image according to the various parts of the plant and feed them into the SAM segmentation model in sequence. Utilizing the model's ability to accept multi-point cues for image segmentation, segmentation masks for each part of each plant image are obtained. Each segmentation mask is evaluated for boundary clarity, regional integrity, and matching degree with the actual plant structure, and the mask with the highest quality is selected and added to the final plant image segmentation mask set.

9. The method for constructing an interactive plant model based on Gaussian splashing according to claim 1, wherein: The specific method of step S6 is: using the random initialization method, a 10-dimensional feature vector f is introduced for each 3D Gaussian to characterize the semantic attributes of the plant parts it represents; then the feature vector is trained as follows: First, randomly select a camera pose P from the camera pose set obtained in step S2 i , using the differentiable rendering method, the feature vector f in the three-dimensional Gaussian representation set is projected onto the two-dimensional image plane to obtain the rendered segmentation mask set under the camera perspective; further, the loss between the segmentation mask set obtained by rendering and the real plant image segmentation mask set Masks obtained by S5 is calculated; finally, the gradient backpropagation method is used to update the feature vector parameters of each three-dimensional Gaussian; through the above method, all camera postures are traversed and trained repeatedly.

10. The method for constructing an interactive plant model based on Gaussian splashing according to claim 7 or 9, characterized in that: Using L1 loss and L D-SSIM The loss function L used in training is constructed jointly pixel (I render ,I real ); L pixel (AND render ,AND real )=λL1+(1-λ)L D-SSIM Among them, I render This is the plant image rendered from the camera perspective; I real Real plant images from the camera’s perspective; λ is an adjustable hyperparameter; L1 represents the L1 loss between two images, L D-SSIM Represents the L between the two images D-SSIM Loss, the two expressions are: Where H×W is the total number of image pixels, I1(x,y) and I2(x,y) represent the pixel values of the image at the pixel coordinates (x,y), μ is the mean of the image, σ is the variance of the image, and σ 12 is the covariance between the two images, and C1 and C2 are constants.

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