Slow-moving small target trajectory detection method based on multi-frame coloring in reverberation background

Abstract

This invention proposes a slow, small target trajectory detection method based on multi-frame coloring in a reverberant background. It employs the GMM or RPCA method, and obtains a spliced ​​tensor through direct or selective splicing. The spliced ​​tensor is then colorized to obtain a colored trajectory detection result. This invention utilizes the inter-frame positional migration continuity of slow, small targets, the inter-frame positional randomness of dynamic reverberation, and the inter-frame positional invariance of static reverberation to detect continuous target trajectories. In the detection results, static and dynamic reverberation are colored with a single color, and the continuous target trajectory is represented by color gradient stripes. The basic principles and implementation scheme of this invention have been verified through synthetic data processing results. The results show that, compared with traditional methods, the proposed colored trajectory detection method fully utilizes the frame-domain differences between slow, small target trajectories and dynamic and static reverberation, resulting in better continuous trajectory detection performance.

Description

Technical Field

[0001] This invention belongs to the field of target detection, and specifically relates to a method for detecting slow small target trajectories based on multi-frame coloring in a reverberant background. Background Technology

[0002] In the field of target detection, the Gaussian Mixture Model (GMM) method is one of the effective methods for active sonar to detect slow, small targets under reverberation-limited conditions. However, the GMM-based method uses a direct summation process to accumulate the image, ignoring the frame-domain difference between the continuous trajectory of the slow, small target and the dynamic reverberation, resulting in severe confusion between the target trajectory and the dynamic reverberation. Although the post-processed Radon transform can filter out some dynamic reverberation, it cannot filter out strip-shaped dynamic reverberation (Fan Wei, Zhu Daizhu, Zhang Deze, Zeng Sai, "Gaussian Mixture Model and Radon Transform for Background Suppression of Sonar Images", Journal of Underwater Unmanned Systems, 2018, 26(05):492-497.).

[0003] Robust Principal Component Analysis (RPCA) is also an effective method for detecting slow, small targets under reverberation constraints, but it cannot effectively suppress dynamic reverberation. Furthermore, this method uses a convex kernel norm as a substitute, resulting in strong static reverberation residue (Ge F, Chen Y and Li W, “Target detection and tracking via structured convex optimization”, 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), New Orleans, LA, USA, 2017, pp. 426-430; CN113050098A-Anti-Frogman Sonar Reverberation Suppression Method Based on Block Sparse Robust Principal Component Analysis-Disclosure). Therefore, the target trajectory in the accumulated history image obtained by this method is severely confused with both dynamic and static reverberation.

[0004] Therefore, the problem of detecting continuous trajectories of slow, small targets under dynamic and static reverberation constraints remains to be solved. Summary of the Invention

[0005] The technical problem solved by this invention is: in order to solve the problem of difficulty in distinguishing between reverberation and slow small target trajectories, this invention proposes a slow small target trajectory detection method based on multi-frame coloring in the background of reverberation.

[0006] The technical solution of this invention is: a method for detecting slow small target trajectories under reverberant background based on multi-frame coloring, characterized by comprising the following steps:

[0007] Step 1: Perform conventional beamforming and matched filtering on the pairwise domain data, take the absolute value, and obtain multiple frames of images;

[0008] Step 2: Using the GMM or RPCA method, obtain the spliced ​​tensor through direct splicing or selective splicing.

[0009] Step 3: Design the color palette matrix and calculate the color value matrix of the three primary colors, including the following sub-steps:

[0010] Step 3.1: Based on the tensor To determine the number of frames, design an N×3 dimensional color palette matrix C. The nth row of the color palette matrix contains the color values ​​of the three primary colors, Cn. n,: =[r n ,g n ,b n ]. r is the column vector of red color values.

[0011]

[0012] T stands for transpose. Indicates rounding down, r n This represents the nth element of vector r. n ,b n Similarly, g is a column vector of green color values.

[0013] When N is odd

[0014]

[0015] When N is even

[0016]

[0017] b is a column vector of blue color values.

[0018]

[0019] Step 3.2: Based on the palette matrix C and tensor Calculate the color value tensors of the three primary colors

[0020]

[0021]

[0022]

[0023] in, Tensor Page n;

[0024] Step 3.3: Calculation The intensity proportion of each element in the corresponding frame-dimensional sequence is used to obtain the weight tensor.

[0025]

[0026] Step 3.4: Utilizing the weight tensor By performing a page-weighted summation of the three primary color value tensors and taking the absolute value, we obtain the three primary color value matrix:

[0027]

[0028]

[0029]

[0030] The three primary color value matrices R, G, B are used as the color trajectory detection results, and true color display is adopted to ultimately realize the trajectory detection of slow-moving small targets.

[0031] Furthermore, the feature is that, in step 1, the sound source emits pulses to the detection area at a period of T, and the receiving array samples to obtain multiple frames of array element domain data; the array element domain data is processed using conventional beamforming and matched filtering, and the absolute value is taken to obtain N frames of angle-distance images; each frame of the image consists of target echo, static reverberation, and dynamic reverberation, and the nth (n = 1, 2, ..., N) frame of the image is denoted as... Where P is the number of sampling points in the distance dimension and Q is the number of sampling points in the angle dimension.

[0032] Furthermore, the characteristic is that, in step 2, the splicing tensor Can be adopted

[0033] Using splicing tensors Or splice tensors

[0034] Furthermore, the splicing tensor is characterized by... Obtained through direct splicing, the calculation formula is:

[0035]

[0036] In the formula, Tensor Page n.

[0037] Furthermore, the splicing tensor is characterized by... The result is obtained by selecting the largest element and combining them; the calculation formula is as follows:

[0038]

[0039] in, Tensor The element in row p, column q, and page n, where w1 = 1 and w0 = 0. for The index of the maximum value in the middle.

[0040] Invention Effects

[0041] The technical advantages of this invention are as follows: This method first obtains multiple frames of images for the same detection scene, then uses the GMM or RPCA method to obtain multi-frame reverberation suppression results, then obtains a spliced ​​tensor through direct splicing or selective splicing, and finally performs color processing on the spliced ​​tensor along the frame dimension. The output of the proposed method is displayed in true color, which can express the frame domain differences between continuous trajectories and static and dynamic reverberation, achieving trajectory detection performance superior to traditional methods.

[0042] This invention employs the GMM or RPCA method, and then obtains a spliced ​​tensor through direct splicing or selective splicing. The spliced ​​tensor is then colored to obtain a colored trajectory detection result. This invention utilizes the inter-frame position migration continuity of slow, small targets, the inter-frame position randomness of dynamic reverberation, and the inter-frame position invariance of static reverberation to detect continuous target trajectories. In the detection results of this invention, static and dynamic reverberation are marked with a single color, and the continuous target trajectory is represented by color gradient stripes.

[0043] The basic principles and implementation schemes of this invention have been verified by the results of synthetic data processing. The results show that, compared with traditional methods, the color trajectory detection method proposed in this invention makes full use of the frame domain differences between slow small target trajectories and dynamic and static reverberation, and has better continuous trajectory detection performance. Attached Figure Description

[0044] Figure 1 A schematic diagram illustrating the sources of target trajectory and reverberation in the cumulative image of the process;

[0045] Figure 2 This is the complete flowchart of the present invention;

[0046] Figure 3 The first frame of the implementation example is the angle-distance image;

[0047] Figure 4 For the angle-distance image of frame 50 in the implementation example;

[0048] Figure 5 For the angle-distance image of frame 100 in the implementation example;

[0049] Figure 6 This is a pseudo-color history accumulation image obtained by summing and accumulating using the GMM method in the implementation example;

[0050] Figure 7 The example uses the GMM method to obtain a pseudo-color history accumulation image through the maximum accumulation method.

[0051] Figure 8 This is a pseudo-color history accumulation image obtained by summing and accumulating using the RPCA method in the implementation example;

[0052] Figure 9 The example uses the RPCA method to obtain a pseudo-color history accumulation image through the maximum accumulation method.

[0053] Figure 10 The example uses the GMM method to obtain the color trajectory detection results through direct stitching.

[0054] Figure 11 The example uses the GMM method to obtain the color trajectory detection results through a large-scale stitching method.

[0055] Figure 12 The example uses the RPCA method to obtain the color trajectory detection results through direct stitching.

[0056] Figure 13 The color trajectory detection results are obtained by using the RPCA method in the implementation example through the large-scale stitching method. Detailed Implementation

[0057] In the description of this invention, it should be understood that the terms "center," "longitudinal," "lateral," "length," "width," "thickness," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "top," "bottom," "inner," "outer," "clockwise," and "counterclockwise," etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are only for the convenience of describing this invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on this invention.

[0058] See Figures 1-13 The main contents of this invention are as follows:

[0059] (1) Multi-pulse array element domain data is processed using conventional beamforming and matched filtering, and the absolute value is taken to obtain multi-frame angle-range images. Referring to the literature (Fan Wei, Zhu Daizhu, Zhang Deze, Zeng Sai, "Gaussian mixture model and Radon transform for background suppression of sonar images", Journal of Underwater Unmanned Systems, 2018, 26(05):492-497.) or the literature (Ge F, Chen Y and LiW, "Target detection and tracking via structured convex optimization", 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), New Orleans, LA, USA, 2017, pp.426-430.), the GMM method or RPCA method is used to process multi-frame images to obtain multi-frame reverberation suppression results. If the direct stitching method is used, the multi-frame reverberation suppression results are stitched page by page to obtain the stitched tensor. If the large-scale splicing method is adopted, the reverberation suppression results of multiple frames are spliced ​​page by page, and then the non-maximum values ​​in the zero-frame dimensional sequence are set to obtain the weighted splicing tensor.

[0060] (2) Design a color palette matrix based on the frame number and calculate the color value tensors of the three primary colors. Calculate the intensity ratio of each element in the spliced ​​tensor in the corresponding frame dimension sequence, use it as a weight, and perform weighted summation of the three primary color value tensors page by page to obtain the three primary color value matrix, i.e., the color trajectory detection result. Finally, display in true color.

[0061] (3) The processing results of the traditional method and the proposed method are presented using synthetic data. The results show that, compared with the traditional method, under the condition of static reverberation and dynamic reverberation, the proposed method uses color gradient trajectory to indicate slow small targets, which is different from single-color dynamic reverberation and static reverberation, and has better continuous trajectory detection performance.

[0062] The technical solution adopted by this invention to solve the existing problems can be divided into the following three steps:

[0063] 1) Obtain multiple frames of images using conventional beamforming and matched filtering.

[0064] 2) Use the GMM method or RPCA method to obtain the spliced ​​tensor through direct splicing or selective splicing.

[0065] 3) Design a color palette matrix and calculate the color value tensors of the three primary colors. Using the element intensity ratio of the spliced ​​tensor as weights, sum the color value tensors of the three primary colors page-by-page to obtain the color value matrix of the three primary colors. Use the color value matrix of the three primary colors as the trajectory detection result of this method.

[0066] 4) Each step of the present invention will be described in detail below:

[0067] Step 1) mainly involves the acquisition of multiple frames of images, and its relevant theories and specific content are as follows:

[0068] Consider a scenario where an active sonar is used to detect a fixed area. The sound source emits pulses into the detection area at periods T, and the receiver samples these pulses to obtain multiple frames of element-domain data. Conventional beamforming and matched filtering are used to process the element-domain data, and the absolute values ​​are taken to obtain N frames of angle-range images. Each frame consists of target echo, static reverberation, and dynamic reverberation. Let the nth (n = 1, 2, ..., N) frame be denoted as ______. Where P is the number of sampling points in the distance dimension and Q is the number of sampling points in the angle dimension.

[0069] Step 2) mainly concerns the acquisition of the splicing tensor, and the relevant theories and specific content are as follows:

[0070] Assume that static reverberation is frame-invariant, dynamic reverberation is frame-rapidly variable, and the echo of a slow, small target exhibits continuous positional migration across frames. Traditional GMM methods effectively suppress static reverberation but struggle to suppress dynamic reverberation. In the history accumulation image obtained by this method, slow, small targets exhibit continuous trajectories, dynamic reverberation appears as random spots or stripes, and static reverberation is not represented. Traditional RPCA methods can suppress most static reverberation, but static reverberation residuals still exist in the output. Furthermore, this method is difficult to effectively suppress dynamic reverberation. Therefore, in the history accumulation image, the trajectory of a slow, small target, dynamic reverberation, and static reverberation are all represented. There are multiple ways to obtain the history accumulation image. Here, we focus on summation accumulation and maximum-selection accumulation. Let the nth frame image D be... (n) The reverberation suppression result is E (n) If the summation and accumulation method is used, then the process accumulation image H s The calculation formula is:

[0071]

[0072] This formula uses direct summation. The superscript S indicates that a summation and accumulation method is used.

[0073] Generally, slow-moving small targets exhibit continuous inter-frame positional migration; therefore, consecutive pixels of the target trajectory in the cumulative image originate from consecutive frames, such as... Figure 1As shown in the diagram, solid circles mark the target position in the current frame, dashed circles mark the target positions in other frames, solid rectangles and triangles represent different forms of dynamic reverberation, and solid pentagrams represent static reverberation. Dynamic reverberation exhibits rapid inter-frame variation, so the dynamic reverberation in the history accumulation image randomly originates from different frames. Static reverberation exhibits inter-frame invariance, so the static reverberation in the history accumulation image originates from all frames. Therefore, there is a significant frame-domain difference between the slow small target trajectory and the reverberation, and this information is lost in the direct summation process of the history accumulation image. The history accumulation image is displayed in pseudo-color, which severely confuses the slow small target trajectory, static reverberation, and dynamic reverberation.

[0074] To preserve frame-domain differences, the method proposed in this invention first avoids direct summation and then concatenates the reverberation suppression results of multiple frames according to equation (3) to obtain the concatenated tensor.

[0075]

[0076] In the formula, Tensor Page n. For ease of analysis, equation (3) is referred to as the direct splicing method.

[0077] If the maximum accumulation method is used, then the process accumulation image H m The calculation formula is

[0078]

[0079] In the formula, For tensor The (p,q)th frame dimension sequence, Representation matrix H m The (p,q)th element, max(·) represents taking the maximum value, and the superscript m indicates that the maximum cumulative method is used. If we denote... exist The index in is Equation (4) is equivalent to:

[0080]

[0081] In the formula, w1 = 1, w0 = 0. Therefore, formula (5) essentially still uses direct summation.

[0082] The method proposed in this invention avoids direct summation, and therefore obtains the weighted splicing tensor according to equation (6).

[0083]

[0084] For ease of analysis, equation (6) is referred to as the "selective large-size splicing method". For the two splicing methods mentioned above, only one needs to be selected, and then the resulting material is further processed with coloring. To simplify the mathematical description, the results obtained through either the direct splicing method or the selective large-size splicing method will be described below. or Recorded as These are collectively referred to as spliced ​​tensors.

[0085] Step 3) mainly involves the coloring processing of the spliced ​​tensors, and the relevant theories and specific contents are as follows:

[0086] First, according to tensors Given the number of pages, i.e., the number of frames, design an N×3 dimensional color palette matrix C. The nth row of the color palette matrix is ​​the group of the three primary color values ​​C. n,: =[r n ,g n ,b n Where r is the column vector of red color values,

[0087]

[0088] T stands for transpose. Indicates rounding down, r n This represents the nth element of vector r. n ,b n Similarly, when g is a column vector of green color values ​​and N is an odd number,

[0089]

[0090] When N is even

[0091]

[0092] b is a column vector of blue color values.

[0093]

[0094] Secondly, based on the palette matrix C and the tensor Calculate the color value tensors of the three primary colors tensor The formula for calculating page n is:

[0095]

[0096]

[0097]

[0098] Then, calculate The intensity proportion of each element in the corresponding frame-dimensional sequence is used to obtain the weight tensor. The calculation formula is

[0099]

[0100] Finally, using the weight tensor The primary color value matrix is ​​obtained by weighting and summing the tensors of the three primary colors page by page, and then taking the absolute value. The calculation formula is as follows:

[0101]

[0102]

[0103]

[0104] The color value matrix R, G, B of the three primary colors is used as the color trajectory detection result of the method proposed in this invention, and true color display is employed. The entire implementation process is as follows: Figure 2 As shown.

[0105] The following example, using a monostatic active sonar multi-period detection process, illustrates an implementation of this invention. The implementation example uses synthetic data processing results to verify that the proposed color trajectory detection method exhibits superior detection performance compared to traditional methods.

[0106] 1) Obtain multi-frame angle-distance images:

[0107] Conventional beamforming and matched filtering are performed on multi-period array element domain data, and the absolute values ​​are taken to obtain multi-frame angle-range images. The number of frames is 100. Figure 3 This is the first frame image. Figure 4 This is the 50th frame image. Figure 5 This is the 100th frame of the image. The reverberation consists of static reverberation and dynamic reverberation. Figure 3 , Figure 4 and Figure 5 In the model, static reverberation is marked with arrows to maintain inter-frame invariance; dynamic reverberation is marked with ellipses, exhibiting rapid inter-frame variation. Dynamic reverberation exists in various forms, often appearing as speckled or striped patterns. To fully demonstrate the performance of the proposed method, different forms of dynamic reverberation are added to the measured data to attempt to obfuscate them from the target trajectory. It is assumed that the small target moves slowly, exhibiting continuous inter-frame positional migration, and shows significant frame-domain differences from rapidly varying dynamic reverberation. For ease of analysis, four targets are added to the measured data. Figure 3 , Figure 4 and Figure 5 In the diagram, targets are marked and numbered with boxes. The trajectories of target 1 and target 2 intersect, while the trajectories of target 3 and target 4 are consecutive.

[0108] 2) Calculate the reverberation suppression results across multiple frames and perform colorization processing:

[0109] To demonstrate the universality and superiority of the proposed method, multi-frame reverberation suppression results were obtained using the GMM or RPCA method. Then, using either a summation-accumulation method or a maximum-selection-accumulation method, four pseudo-color history accumulated images were obtained, as shown below. Figures 6-9 As shown in the figure. The target trajectory is marked with a box, and the static reverberation is marked with an ellipse. Compared to... Figures 3-5 , Figures 6-9 The static reverberation was effectively suppressed. However, Figure 8 and Figure 9 Static reverberation residue still exists. Furthermore, pseudo-color images can only express intensity information, not frame-domain information. Therefore, static reverberation, dynamic reverberation, and target trajectory are severely confused, making target trajectory detection impossible. Assuming a target can only generate one trajectory, then box 1 contains two targets, and box 2 contains one target. This is clearly inconsistent with reality. Moreover, the true trajectories of the two targets in box 1 cannot be determined.

[0110] However, the method proposed in this invention can effectively solve the above problems and has better target trajectory detection performance. Using the GMM or RPCA method, multi-frame reverberation suppression results are obtained, and then combined in pairs using either direct stitching or selective stitching to obtain four stitched tensors. The four stitched tensors are then colored to obtain four color trajectory detection results, such as... Figures 10-13 As shown.

[0111] exist Figures 10-13 In this context, the target trajectory is marked with a box, and the static reverberation is marked with an ellipse. Multi-frame colorization processing can preserve frame-domain information, expressing frame-domain differences through color differences. Therefore, in Figures 10-13 In this context, color differences clearly distinguish the target trajectory from static and dynamic reverberation. Because slow-moving, small targets exhibit continuous inter-frame positional migration, the consecutive pixels of the target trajectory originate from consecutive frames. Since consecutive frames correspond to consecutive color values, therefore... Figures 10-13 The trajectory of small targets moving at medium to slow speeds appears as color gradient stripes. Dynamic reverberation lacks inter-frame positional continuity and appears randomly at a certain location within a frame. Figures 10-13 The dynamic reverb in the image contains only the color value information of a single frame. Static reverb has inter-frame position invariance. Because a direct splicing method is used, Figure 12 The static reverb in the image contains color values ​​from all frames, which are ultimately blended into white. Due to the use of a large-scale splicing method, Figure 13 The static reverberation in the image contains the individual color value information of the frame containing the maximum value in the frame-dimensional sequence. Therefore, in the color trajectory detection results, the reverberation appears as monochrome value spots or stripes, exhibiting a significant color difference from the target trajectory with its color gradient. Thus, the proposed method achieves superior target trajectory detection performance compared to traditional methods.

[0112] Furthermore, since the frame-domain information of the trajectory pixels is effectively expressed, the color gradient direction of the colored trajectory can indicate the direction of the target's movement. According to the color palette matrix, the red end of the trajectory indicates the starting position of the moving target, and the blue end indicates the ending position of the moving target. Thanks to the color continuity and directional indication of the colored trajectory, the true trajectory of the two targets in box 1 can be uniquely determined. Based on this characteristic, the color of the third point can be predicted from the colors of the two points on the trajectory, making it easy to determine which trajectory a pixel on the intersecting trajectory comes from. Observing the colored intersecting trajectory in box 1, it can be seen that the two targets move from the upper left to the lower right and from the lower left to the upper right, respectively, which is consistent with the movement characteristics of target 1 and target 2. Observing the target trajectory in box 2, it can be seen that the trajectory gradually changes from red to blue, then abruptly changes back to red, and then gradually changes from red to blue again. It is known that a target can only occupy one position in space, so the single trajectory in box 2 is actually composed of two trajectories spliced ​​together, so there are two targets in box 2. At the same time, both targets move from top to bottom, which is consistent with Figures 3-5 The motion characteristics of targets 3 and 4.

[0113] In summary, by using color differences or frame domain differences, the target trajectory can be effectively distinguished from dynamic reverberation and static reverberation, while the differences in motion characteristics between target trajectories are effectively expressed.

[0114] Based on the implementation examples, it can be concluded that the method proposed in this invention has a better detection effect on slow, small target trajectories compared with traditional methods.

Claims

1. A method for detecting slow small target trajectories in a reverberant background based on multi-frame coloring, characterized in that, Includes the following steps: Step 1: Perform conventional beamforming and matched filtering on the pairwise domain data, take the absolute value, and obtain multiple frames of images; Step 2: Using the GMM or RPCA method, obtain the spliced ​​tensor through direct splicing or selective splicing. ; Step 3: Design the color palette matrix and calculate the color value matrix of the three primary colors, including the following sub-steps: Step 3.1: Based on the tensor Frame rate, design Color Palette Matrix ; where the palette matrix is ​​the first Behavioral primary color value group ; This is a column vector of red color values. Indicates transpose. Indicates rounding down. Representing vectors The One element; Similarly; A column vector of green color values; when When it is an odd number, when When it is even, This is a column vector of blue color values. ； Step 3.2: Based on the palette matrix and tensor Calculate the color tensors of the three primary colors : in, Tensor The Page; Step 3.3: Calculation The intensity proportion of each element in the corresponding frame-dimensional sequence is used to obtain the weight tensor. : ； The number of sampling points in the distance dimension. The number of sampling points in the angular dimension; Step 3.4: Utilizing the weight tensor By performing a weighted summation of the three primary color value tensors page by page and taking the absolute value, we obtain the three primary color value matrix: The color value matrix of the three primary colors As the result of color trajectory detection, true color display is used to ultimately achieve trajectory detection of slow-moving small targets.

2. The method for detecting slow small target trajectories in a reverberant background based on multi-frame coloring as described in claim 1, characterized in that, In step 1, the sound source periodically... Pulses are transmitted to the detection area, and array sampling is used to obtain multiple frames of array element domain data. Conventional beamforming and matched filtering are used to process the array element domain data, and the absolute value is taken to obtain... Frame angle-distance image; Each frame of the image consists of target echo, static reverberation, and dynamic reverberation, denoted as the [missing information]. Frame image is ,in ;in, The number of sampling points in the distance dimension. The number of sampling points in the angular dimension.

3. The method for detecting slow small target trajectories in a reverberant background based on multi-frame coloring as described in claim 1, characterized in that, In step 2, tensors are spliced. splicing tensors can be used Or splice tensors ,in Defined as a spliced ​​tensor obtained through direct splicing; Defined as a spliced ​​tensor obtained by selecting the largest value.

4. The method for detecting slow small target trajectories in a reverberant background based on multi-frame coloring as described in claim 3, characterized in that, The splicing tensor Obtained through direct splicing, the calculation formula is: In the formula, Tensor The Page.

5. The method for detecting slow small target trajectories in a reverberant background based on multi-frame coloring as described in claim 3, characterized in that, The splicing tensor The result is obtained by selecting the largest element and combining them; the calculation formula is as follows: in, Tensor The Okay, number Column, number Page elements, , for The index of the maximum value in the middle.

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