CN101079151A - 360 degree around panorama generation method based on serial static image - Google Patents

360 degree around panorama generation method based on serial static image Download PDF

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CN101079151A
CN101079151A CN 200610053842 CN200610053842A CN101079151A CN 101079151 A CN101079151 A CN 101079151A CN 200610053842 CN200610053842 CN 200610053842 CN 200610053842 A CN200610053842 A CN 200610053842A CN 101079151 A CN101079151 A CN 101079151A
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image
images
pixel
sequence
panoramic
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CN100485720C (en
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朱信忠
赵建民
徐慧英
杨琳
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浙江师范大学
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Abstract

The invention discloses a full view generating method of a sequence static picture for splicing a group of static pictures to a cylindrical full view picture. A group of static pictures are twelve or more than twelve pictures that the digital camera takes the object or the environment of the needed generating full view picture from the different angle. The method comprises the following steps: taking the sequence pictures; preprocessing the taken pictures; splicing the pictures. The method needs the low hardware device and doesn' t need the expensive hardware investment, which provides the quick resultant velocity, the high intellectual degree, and can display the full view pictures.

Description

一种基于序列静态图像的360°环视全景生成方法 One kind of panorama generating method based on the 360 ​​° sequence still image surveying

技术领域 FIELD

本发明涉及360°环视全景图像的生成方法。 The present invention relates to a method for generating 360 ° Panoramic image.

背景技术 Background technique

全景图是虚拟现实和计算视觉中一种重要的场景表示方法,它指的是在固定的视点,在垂直方向180°和水平方向360°的图像视图,简单的形式可以是固定视平面上的360°环视视图。 Panorama virtual reality and computer vision in an important scene representation, it refers to a fixed viewpoint, at 180 ° and the image view in the horizontal direction of 360 ° in the vertical direction, a simple form can be a fixation on a plane 360 ° around a view. 通常有两种方式来获得全景图:直接方式和图像拼接方式。 There are generally two ways to obtain the panorama: direct mode and image splicing. 前一种方式简单易行,但它需要使用专业全景相机、全景摄像机等特殊器材,这些器材通常价格昂贵且使用复杂,实际应用面较窄,难以普及。 The former approach is simple, but it requires the use of professional panoramic camera, panoramic camera and other special equipment, these devices are often expensive and complex to use practical application surface is narrow, difficult to spread. 因此基于图像拼接思想的全景视图生成方法的应用非常广泛。 Therefore, application-based method of generating image stitching panoramic view of the idea of ​​a very wide range. 图像拼接就是利用若干离散局部图像作为基础数据,经过一系列的图像分析处理后生成全景。 Image stitching is the use of a number of discrete partial image as basic data, after a series of image analysis processing to generate panoramic.

全景图一般有三种形式:立方体全景图、球面全景图和柱面全景图。 Panorama generally three forms: cube panorama spherical panorama and cylindrical panoramic image. 立方体全景图在图像的获取和校正过程中会遇到很大困难。 Cube panorama will encounter great difficulties in access and correction process image. 球面全景图拼接过程中的图像求交、定位较困难,且很难找到一种与球面相对应、易于计算机存储访问的数据结构来存放球面图像数据。 Spherical panorama stitching process image intersection, positioning more difficult, and it is difficult to find a spherical correspond easy data structures, computer-storage access to store the spherical image data. 柱面全景图在单幅图像获取比立方体形式和球面形式简单,而且容易展开为一个矩形图像,可直接用计算机常用的图像格式存储和访问。 Cylindrical panoramic image acquisition simpler than cubic form and spherical form a single image, and easy to expand to a rectangular image, can be directly computer common image format storage and access.

如专利申请号为200410015828.0的中国发明专利申请,公开了一种180°大视场全景凝视成像方法。 Patent application number 200410015828.0 of the Chinese patent application, discloses a 180 ° wide field of view panorama staring imaging method. 它采用圆柱平面投影法,用二次反射的环形透镜作第一次成像,实现大视角全景凝视成像,用中继透镜作第二次成像,获得实像,用平面光电成像器件接收並显示三维空间,主要应用于机器人全景视觉、管道内壁检测、医学内窥成像等领域,虽然提出了一种直接获取全景图像的方案,但该方案中全景图像的生成方法对硬件条件及环境要求较高,且不能生成完整的360°环视全景。 It employs a cylindrical plane projection method, with the annular lens secondary reflected for the first time imaging, to achieve a large angle of view panoramic staring imaging, using the relay lens for a second imaging to obtain a real image receiving plane optical imaging device and displaying three-dimensional space , mainly used in the field of robotics panoramic vision, the pipe wall inspection, the medical endoscopic imaging, although proposed a direct access to the panoramic image of the program, but the higher the embodiment method of generating a panoramic image on the hardware condition and the environmental requirements, and You can not generate a full 360 ° panoramic.

如申请号为03115149.3的中国发明专利,公开了一种基于两张鱼眼图像的全景生成方法。 The Application No. 03115149.3 Chinese invention patent, discloses a method based on a panoramic two fish-eye image generating method. 该方法包括鱼眼图像预处理、建立空间模型、拼合参数寻优、生成全景图像四个部分。 The method includes the fisheye image pre-established spatial model, split parameter optimization, generating a panoramic image into four parts. 但在实际应用中并不能自动、快速的找到模型的最佳拼合参数,需要人工调整拼合参数。 But it does not automatically and quickly find the best split parameters of the model in practical application, the need to manually adjust the split parameters. 尤其关键的是,由于本发明要求待拼合的鱼眼图像必须具备理论上完备的空间模型,只能采用两张鱼眼图像,需要相机装配价格昂贵的鱼眼镜头,普通平面图像不能使用,难以得到平民化普及应用。 Especially critical is that, since the present invention requires the fish-eye image to be split must have theoretically complete space model, only use two fish-eye image, it is necessary the camera assembly expensive fisheye lens, an ordinary planar image can not be used, it is difficult to get civilian universal application.

再如申请号为03137660.6的中国发明专利,公开了一种平面图像全景重建立体图像的方法,其过程包括,选择平面图像,针对平面图像的空间分布,建立每个象素的深度列表,根据深度列表对平面图像的每个象素进行视差位移处理,重建全部视差序列图像,将视差序列图像立体合成。 Another example application No. 03137660.6 Chinese invention patent, discloses a method of flat image panorama reconstructed stereoscopic image, the process comprising selecting a planar image, the spatial plane distribution of the image, establishing a depth of a list of each pixel, the depth list for each pixel planar image is parallax displacement process, reconstruction of the entire parallax image sequence, the parallax image sequence binaural synthesis. 其中,采用了三维建模、图像几何变换、图像视差变换技术来实现立体图像的重建。 Wherein, using a three-dimensional modeling, image geometric transformation, image parallax transform reconstruct the shape of the stereoscopic image. 但该方法在平面图像的获取、处理和定位、校正过程中会遇到很多实际困难,且难以处理实景大图和实现可控立体景深。 However, this method acquire a planar image, the processing and positioning, the correction process will encounter many practical difficulties, and difficult to handle real big picture and for controlled stereoscopic depth.

另外,申请号为200510087641.6的中国发明专利,公开了一种用于创建全景图像的数字成像设备及其方法,其中,数字成像设备包括用于连续捕获多个图像的捕获部分;用于分别检测从捕获部分输出的多个图像的图像信息检测部分;用于图像转换的全景图像生成部分。 Moreover, Application No. 200510087641.6 Chinese invention patent, discloses a digital imaging apparatus and a method for creating a panoramic image, wherein the digital imaging apparatus comprises means for continuously capturing capturing section plurality of images; means for detecting, respectively, from captured image information detecting part of the plurality of images of the portion of the output; a panoramic image generating section image conversion. 通过选择从图像信息检测部分输出的多个图像信息中的一组,通过对所选择的图像信息进行合并转换,从而创建全景图像。 From the plurality of image information of the image detecting part of the output information in a group, by the image information of the selected merger conversion, thereby creating a panoramic image by selecting. 但该方法实际处理速度较慢,且只能用于专用的全景数字成像设备。 However, the method of the actual processing speed is slow, and only a dedicated panoramic digital imaging device.

现有的全景图像生成方法存在的缺点是:1、若采用直接生成全景图像的方法,则对硬件设备要求较高,价格昂贵,难以普及应用;2、现有的通过图像拼合技术生成全景图像的方法,要么需要专用鱼眼镜头设备,要么实现困难、智能化程度低、对图像要求高、定位和校正困难、普通相机照片无法使用,处理速度较慢等,甚至最终合成的全景图效果也较差。 Existing panoramic image generation method has the disadvantage that: 1, the use of direct generating a panoramic image method, then the hardware demanding, expensive, difficult to spread the application; 2, existing through image mosaic technique to generate a panoramic image the method requires either a dedicated fisheye lens device, or difficult to realize, low intelligence, high image requirements, locate and correct difficult, ordinary camera photos can not be used, processing speed is slow, and even the final synthesis panorama effect poor.

发明内容 SUMMARY

为了克服已有的全景图像生成方法存在的缺点和不足,本发明提供了一种对设备要求低(仅需普通数码相机或普通平面照片)、实用廉价、实现简便、智能化程度高、处理速度快、全景图清晰、真实感强、适用性广、易于普及的基于序列静态图像的360°环视全景生成方法。 In order to overcome the problems of the conventional panoramic image generating method shortcomings and deficiencies, the present invention provides an equipment that require low (only ordinary digital camera or an ordinary photograph of a plane), cheap and practical, to achieve easy, high intelligence, processing speed fast panorama clear, strong sense of reality, wide applicability, easy to spread based on 360 ° a sequence of still images panoramic generation method.

本发明解决其技术问题所采用的技术方案是:一种基于序列静态图像的360°环视全景生成方法,该方法包括如下步骤:(1)用相机拍摄所需的序列图像;(2)对序列图像进行预处理:采用中值滤波进行去噪,直方图均衡化处理;(3)序列图像的拼接:通过将相邻前后图像的拼接形成360°环视全景图像,包括以下步骤:(3.1)把拍摄的图片通过变换投影到360°水平柱面; The present invention solves the technical problem using the technical solution is: based on 360 ° a sequence of still images Panoramic generation, the method comprising the steps of: (1) capturing a desired sequence of images by the camera; (2) sequence image preprocessing: median filtering denoising, histogram equalization; stitching (3) a sequence of images: forming 360 ° panoramic image by stitching the front and rear of the adjacent image, comprising the steps of: (3.1) the pictures taken by the transform projected to 360 ° horizontal cylinder;

(3.2)在读入的两幅相邻图像的前一幅图像中,选定一块大小和位置均合适的图像作为模板,确定在后一幅图像中的搜索范围,得到相邻两幅图像的最佳匹配位置li,依次得到序列图像中每相邻两幅图像的最佳匹配位置;(3.3)在两幅相邻图像的重叠区域S和T,将其对应的各象素按一定权值合成到新的图像。 (3.2) in the previous image two adjacent images read, the selected block size, and location of the right image as a template search range is determined after an image to give adjacent two images best matching position li, sequentially obtained image sequences best matches the position of each adjacent two of the image; (3.3) in the overlapping area S two adjacent images and T, each of the pixels by a certain weight corresponding synthesis of the new image. 各象素在各图像上的权值计算公式如下:WValue=cos(π2*|x-x02x02|)*cos(π2*|y-y02y02|)---(1)]]>式中WValue表示权值,(x0,y0)为重合部分的中心位置,(x,y)为象素坐标;将相邻图像的重叠区域S和T对应的各象素值按一定的权值合成新的图像;重合部分的象素值可表示为:IN′=IN×WValue1+IN+1×WValue2(2)其中,IN和IN+1分别表示两个相邻图像的某个相应重叠位置的象素在各自原图像中的象素值,WValue1和WValue2是根据公式(1)计算出的该象素在各自图像上的权值,其取值范围为(0,1),且和为1。 Each pixel weight is calculated in each image as follows: WValue = cos (& pi; 2 * | x-x02x02 |) * cos (& pi; 2 * | y-y02y02 |) --- (1)]]> wherein WValue is the weight, (x0, y0) as the overlap center position portion, (x, y) is the pixel coordinates; neighboring each pixel value of the overlap region of image S and T corresponding to a certain weight the new image synthesis; the pixel values ​​of the overlapping portions can be expressed as: iN '= iN × WValue1 + iN + 1 × WValue2 (2) wherein, iN and iN + 1 each represent a respective overlapping position of two adjacent images pixel in the respective pixel values ​​of the original image, WValue1 and WValue2 are according to the formula (1) to calculate the pixel weights on the respective images, which is in the range (0,1), and and for the 1.

作为优选的一种方案:在所述的(2)中,采用直方图均衡化处理变换函数:sk=T(rk)=Σi=0knin,k=0,1,2···,L-1---(3)]]>其中,n是图像中象素的总合,nk是灰度级为rk的象素个数,L为图像中可能的灰度级数,一般取值为256;通过公式(3)将输入图像中灰度级为rk的象素映射为输出图像灰度级为sk的对应象素。 Preferred examples of a solution: in (2) above, using the histogram equalization transfer function: sk = T (rk) = & Sigma; i = 0knin, k = 0,1,2 & CenterDot; & CenterDot; & CenterDot; , L-1 --- (3)]]> wherein, n being the pixels in the image aggregate, nk is the gray level number of pixels rk, L is the image of the possible gray levels, in general value is 256; (3) the input image gradation by formula pixmap rk as an output image gray levels sk corresponding pixel.

作为优选的另一种方案:在所述的(3.1)中,360°水平柱面投影变换公式为: As a preferred another solution: in the (3.1) in, 360 ° horizontal cylindrical projection conversion formula is:

其中, among them, k=r2+(W2-x)2,]]>(x,y)为输入图像上的任意一点,(x1,y1)为该点经过360°水平柱面投影变换后的坐标值,θ为投影角度,W为图像的宽度,H为图像的高度。 k = r2 + (W2-x) 2,]]> (x, y) as input any point on the image one o'clock, (x1, y1) for the point after 360 ° horizontal coordinate value of the cylindrical projection conversion, θ projection angle, W is the width of the image, H is the height of the image.

作为优选的再一种方案:在所述的(3.2)中,定义图像的绝对误差函数为:ϵ(i,j,mk,nk)=|si,j(mk,nk)-s^(i,j)-T(m,n)+T^|---(5)]]>其中,s^(i,j)=1M2Σm=1MΣn=1Msi,j(m,n),]]>T^=1M2Σm=1MΣn=1MT(m,n),]]>T为模板,S为被搜索图,si,j为子图,即模板覆盖下的那块搜索图,i,j为子图左上角象素点在S中的坐标,M为模板的宽和高。 Preferable still another embodiment: in the (3.2), the absolute error function defined image is: & epsiv; (i, j, mk, nk) = | si, j (mk, nk) -s ^ ( i, j) -T (m, n) + T ^ | --- (5)]]> wherein, s ^ ​​(i, j) = 1M2 & Sigma; m = 1M & Sigma; n = 1Msi, j (m, n) ,]]> T ^ = 1M2 & Sigma; m = 1M & Sigma; n = 1MT (m, n),]]> T as a template, S is searched FIG, si, j for the sub-picture, i.e., the piece covered by the template search FIG, i, j is the coordinate of the child upper left corner pixel dots in S, M being the template width and height.

将模板中心与所在图像的垂直中心线的水平距离,记作x1之后,遍历模板图像和第一个位置的子图像中所有的象素点,计算对应象素点的ε(i,j,mk,nk)并累加后将其作为阈值的初始值T0。 ε (i, j, mk the horizontal and vertical center line of the template center location distances of the images, after referred to as x1, all the pixel points subpicture traverse the template image and the first position, computing the corresponding pixel points , nk) after and accumulate its initial value T0 as a threshold value.

再计算模板图像和下一个位置子图像中对应象素点的ε(i,j,mk,nk)并累加,记作T,在计算并累加的过程中比较T与T0的大小,若在完全遍历模板图像和子图像的象素点之前,得到T≥T0,则停止计算,并将子图像移动到下一个位置,重新开始新一轮的计算;若完全遍历模板图像和子图像的象素点后,得到T<T0,则更新阈值T0,并将此时子图像中心象素点的坐标位置(i,j)记录下来,得到i值与被搜索图像垂直中心线的水平距离,记作x2,取x1与x2的平均值作为相邻两幅图像的最佳匹配位置li,即li=(x1+x2)/2。 Then calculate ε (i, j, mk, nk) corresponding to the pixel points template image and the next position of the sub image and accumulating, denoted by T, compare T size T0 in the computing and accumulating in the process, if the full before traversing the pixel dots template image and a sub image, obtained T≥T0, then stop counting, and the sub-picture moves to the next position, to start a new round of calculation; post if fully traversed pixel dots template image and a sub image to give T <T0, the updated threshold value T0, and at this time the coordinate positions of the sub image center pixel point (i, J) is recorded, to obtain the horizontal distance i value is the search image vertical center line, denoted x2, take x1 and the average value of x2 as a best matching position li adjacent two images, i.e., li = (x1 + x2) / 2.

进一步,在每累加相应位置的一行或者一列的ε(i,j,mk,nk)之后,将T与T0进行大小比较。 Further, after the row corresponding to the position of each of the accumulation or ε (i, j, mk, nk) one of the T and T0 size comparison.

再进一步,在所述的(3.3)中,对于拼接后的图像,并不是直接取IN′的值,而是引入一个阈值K,首先计算该点在平滑前的灰度值和加权平均值的差值,若此值小于阈值,则取IN′为此点的灰度值,反之,则取平滑前的灰度值为该点的灰度值。 Still further, in the (3.3), the image after the stitching, not directly take IN 'value, but the introduction of a threshold value K, first calculates the gray values ​​and the weighted average of the prior smooth the the difference, if this value is less than the threshold value, then take iN 'gray value for this point, on the contrary, it takes the gradation value of the gradation before smoothing value of the point.

更进一步,在所述的步骤(1)中,序列图像的张数不少于12张,相邻两幅图像必须具有重叠部分,且重叠部分在30%到50%之间。 Still further, in said step (1), the number of sequences of images not less than 12, the adjacent two images must have an overlapping portion, and 30% to 50% between the overlapping portions.

在所述的步骤(1)中,对拍摄一个完整360度环视实景全景或者仅部分视角的环视实景场景,在一个固定位置,按顺时针或者逆时针方向等角度水平旋转拍摄。 In said step (1), the captured a full 360 ° view real panoramic or only partially perspective looking around real scene, in a fixed position, an angle horizontal clockwise or counterclockwise direction like the rotational imaging.

在所述的步骤(1)中,对于实物实体360°立体造型展示的虚拟拍摄,将实物实体放在赤道仪或者带刻度转盘上,逆时针或顺时针方向等角度地旋转载物仪盘进行拍摄。 In said step (1), with respect to the virtual physical entity 360 ° three-dimensional shape display of the shooting, the physical entity on the mount or belt scale dial clockwise or counterclockwise direction equiangularly screwed reproduced was meter disk shooting.

本发明的技术构思为:本发明提出一种利用照相机得到一组部分重叠的图像序列,通过图像的预处理、图像的拼接和融合算法来生成全景图像。 Technical concept of the present invention are: to provide a use of a camera to obtain an image sequence of a set of partial overlap, generating a panoramic image by stitching and image fusion algorithm pre-processing, the image of the present invention. 使用普通相机就可以完成360°环视全景图虚拟场景构造所需图像的采集工作,方便实用,易于普及。 Using an ordinary camera can be completed 360 ° Panoramic FIG virtual scene structure acquisition working the desired image, convenient and practical, easy to spread. 鉴于360°环视全景图实现快速、简便,又不影响全景图像浏览的真实性;并且对于一些场景只需要360°环视,因此360°环视全景图就有很广泛的适用性。 In view of the 360 ​​° Panoramic view of fast, easy, and does not affect the authenticity of the panoramic image viewing; and for some scenarios need only 360 ° look around, so 360 ° Panoramic map we have very broad applicability.

本发明的有益效果主要表现在:对设备要求低、普通相机照片即可、成本低廉、实现简便、智能化程度高、处理速度快、全景图清晰、无需完备空间、真实感强、适用性广,易于普及及具有广泛的适用性。 The advantages of the main problems: low equipment requirements, ordinary camera photos can be low cost, easy to implement, high intelligence, processing speed, panorama clear, no complete space, realistic, wide applicability , easy to spread and has broad applicability.

附图说明 BRIEF DESCRIPTION

图1是本发明所述方法生成的360°环视全景图样本图像示例。 Figure 1 is according to the present invention, a method of generating the 360 ​​° Panoramic FIG sample image example.

图2是本发明中由2D图像到360°水平柱面投影变换示意图。 FIG 2 is a 2D image to a 360 ° horizontal cylindrical projection transformation schematic diagram of the present invention.

图3是图像拼接算法的流程图。 FIG 3 is a flowchart of the image stitching algorithm.

图4是360°环视全景图的生成过程示意图。 FIG 4 is a generation process 360 ° Panoramic FIG schematic.

具体实施方式 Detailed ways

下面结合附图对本发明作进一步描述。 DRAWINGS The invention will be further described below in conjunction.

参照图1~图4,一种基于序列静态图像的360°环视全景生成方法,包括以下步骤:(1)用相机拍摄所需的序列图像;(2)序列图像的预处理;(3)序列图像的拼接。 Referring to FIGS. 1 to 4, based on 360 ° a sequence of still images Panoramic generating method, comprising the steps of: (1) capturing a desired sequence of images by the camera; Pretreatment (2) a sequence of images; (3) sequence mosaic image.

拍摄所需的序列图像是指采用普通相机镜头,对不同的场景采用不同的拍摄方法所拍摄的一组序列图像,每组图像不少于12张且相邻两幅图像必须具有一定的重叠部分,且重叠部分在30%到50%之间。 Sequences required for the image capturing means using an ordinary camera lens, using a set of sequences of images having different imaging method for different scenarios, each image is less than 12 and adjacent two image must have a certain overlap portion and an overlapping portion between 30% to 50%. 对拍摄一个完整360度环视实景全景或者仅部分视角的环视实景场景,在一个固定位置,按顺时针或者逆时针方向等角度水平旋转拍摄。 Photographing a complete 360 ​​° view real panoramic or only partially perspective looking around real scene, in a fixed position, an angle horizontal clockwise or counterclockwise direction like the rotational imaging. 对于实物实体360°立体造型展示的虚拟拍摄,将实物实体放在赤道仪或者带刻度转盘上,逆时针或顺时针方向等角度地旋转载物仪盘进行拍摄。 For virtual physical entity 360 ° three-dimensional shape display of the shooting, the physical entity on the mount or belt scale dial clockwise or counterclockwise direction equiangularly screwed reproduced was meter disk imaging.

为了得到效果较好的全景图像,本实施例提出了在图像拼接之前,对序列图像采用中值滤波进行去噪、直方图均衡化处理,以平衡光照条件不同带来的影响。 In order to obtain a better panoramic image, the present embodiment proposes a prior image stitching, sequence of images the median filtering is denoised, histogram equalization, to balance the light of different impact conditions. 我们采用如下直方图均衡化处理变换函数:sk=T(rk)=&Sigma;i=0knin,k=0,1,2&CenterDot;&CenterDot;&CenterDot;,L-1---(3)]]>其中,n是图像中象素的总数,nk是灰度级为rk的象素个数,L为图像中可能的灰度级数,一般取值为256。 We use the following histogram equalization transfer function: sk = T (rk) = & Sigma; i = 0knin, k = 0,1,2 & CenterDot; & CenterDot; & CenterDot;, L-1 --- (3)]]> wherein , n being the total number of pixels in the image, nk is the gray level number of pixels rk, L is the image of the possible gray levels, the general value of 256. 通过公式(3)将输入图像中灰度级为rk的象素映射为输出图像灰度级为sk的对应象素。 By the equation (3) the input image gray level pixel map of rk output image gray levels sk corresponding pixel.

在完成图像的预处理后,我们通过三个步骤来完成对12幅图像的拼接从而完成一幅360°环视全景图的生成,这三个步骤为:①图像的变换;②图像的匹配;③图像的平滑处理。 After completion of the pre-image, we accomplished by three steps of 12 mosaic image to complete a 360 ° surveying generated panorama, these three steps: conversion ① image; matching ② image; ③ smoothing the image. 本发明所述的方法首先要对序列图像进行360°水平柱面投影变换,将各投影平面的重叠图像映射到一个标准投影即360°水平柱面投影上得到360°水平柱面投影图像。 The method of the present invention is the first to be 360 ​​° horizontal cylindrical projection transformation sequence of images, maps superimposed image for each projection plane into a standard projection obtained 360 ° horizontal cylindrical projection image horizontal cylindrical projection i.e. 360 °. 在平面透视投影中,对于一个固定的观点,任意两个2D平面的透视变换可以由矩阵相乘完成:x&prime;y&prime;w&prime;=m0m1m2m3m4m5m6m7m8xyw]]>其中,(x,y)代表第一幅图像的象素坐标,(x′,y′)是(x,y)在第二幅图像上的对应坐标,首先把拍摄的图片通过变换投影到标准360°水平柱面,相应的投影变换公式为: In a plane perspective projection, with respect to a fixed point of view, any perspective transformation two 2D planes may be multiplied completed by the matrix: x & prime; y & prime; w & prime; = m0m1m2m3m4m5m6m7m8xyw]]> where, (x, y) representative of the first image the pixel coordinates, (x ', y') are (x, y) corresponding to the coordinates on the second image, first the pictures taken by the conversion is projected onto a standard 360 ° horizontal cylinder, corresponding projection conversion formula is : 其中, among them, k=r2+(W2-x)2,]]>(x,y)为输入图像上的任意一点,(x1,y1)为该点经过柱面投影变换后的坐标值,θ为投影角度,W为图像的宽度,H为图像的高度。 k = r2 + (W2-x) 2,]]> (x, y) as input any point on the image one o'clock, (x1, y1) for the point after the coordinate values ​​of cylindrical projection conversion, θ is the projection angle, W for the width of the image, H is the height of the image.

参见图2,360°水平柱面投影变换的核心是投影变换公式,首先要建立坐标系,如图2(a)所示假定所有相机运动都发生在XZ平面,以原始输入图对应的观察方向为Z轴,以原始输入图像所在的视平面为XY平面,坐标圆点就是光轴与图像平面的交点,并将原始输入图像标定为I,柱面投影图像标定为J,投影柱面标定为K,观察点标定为O(投影中心)。 Referring to Figure 2,360 ° horizontal central cylindrical projection transformation is projective transformation formula, first create a coordinate system, as shown in FIG 2 (a) shown assume that all camera motion occurs in the XZ plane, the viewing direction of the original input view corresponding to Z-axis, the original input image in the viewing plane where the XY plane, the coordinates of the dots is the intersection of the optical axis of the image plane, and the original input image calibration is I, cylindrical projection image calibration is J, their nominally cylindrical projection K, the observation point calibration is O (projection center). 现在是要在O点观察图像I在柱面K上的投影像J。 Now to observe the image I projected image at the point O in the cylinder K, J.

图2(b)和图2(c)分别是横向观察方向(XZ平面)和纵向观察方向(YZ平面)的柱面投影示意图,图中标定了原始输入图像和柱面投影图像上的两个对应点M和N的相互位置关系,以及投影柱面半径、横向观察角α和纵向观察角β的情况。 FIG 2 (b) and 2 (c) are respectively a lateral observation direction (XZ plane) and longitudinal viewing direction (YZ plane) of the cylindrical projection schematic drawing calibrated two on the original input image and the cylindrical projection image the mutual positional relationship between corresponding points M and N, and a projection cylinder radius, transverse to observe the angle α and the vertical viewing angle β of.

本实施例提出的一种自适应阈值序列序贯相似性检测算法(SSDA,Sequential Similarity Detection Algorithms),该算法中包括确定阈值的选取、模板的大小、模板位置和搜索范围。 An adaptive threshold sequence similarity detection algorithm sequential embodiment proposed (SSDA, Sequential Similarity Detection Algorithms) according to the present embodiment, the algorithm includes determining the size of the selected template threshold value, the template position and the search range. 首先,定义图像的绝对误差函数为:&epsiv;(i,j,mk,nk)=|si,j(mk,nk)-s^(i,j)-T(m,n)+T^|---(5)]]>其中,s^(i,j)=1M2&Sigma;m=1M&Sigma;n=1Msi,j(m,n),]]>T^=1M2&Sigma;m=1M&Sigma;n=1MT(m,n),]]>T为模板,S为被搜索图,si,j为子图,即模板覆盖下的那块搜索图,i,j为子图左上角象素点在S中的坐标,M为模板的宽和高。 First, the absolute error function defined image is: & epsiv; (i, j, mk, nk) = | si, j (mk, nk) -s ^ (i, j) -T (m, n) + T ^ | --- (5)]]> wherein, s ^ ​​(i, j) = 1M2 & Sigma; m = 1M & Sigma; n = 1Msi, j (m, n),]]> T ^ = 1M2 & Sigma; m = 1M & Sigma; n = 1MT (m, n),]]> T as a template, S is searched FIG, si, j for the sub-picture, a piece of the search of FIG lower, i.e. covered by the template, i, j is the sub-upper left corner pixel points in S coordinates, M being the width and height of the template.

参照图3,在读入的两幅相邻图像的前一幅图像中,选定一块大小和位置均合适的图像作为模板,将模板中心与所在图像的垂直中心线的水平距离记作x1之后,确定在后一幅图像中的搜索范围(即子图的遍历范围)。 After 3, before an image of two adjacent images is read, the selected block size, and location of the right image as a template, the horizontal and vertical center line of the template center location distances of the images referred to as x1 determining a search range after an image (i.e., traverse range of the sub-graph). 遍历模板图像和第一个位置的子图像中所有的象素点,计算对应象素点的ε(i,j,mk,nk)并累加后将其作为阈值的初始值T0。 Its initial value T0 as the threshold after all pixel points subpicture traverse the template image and the first position, the computing ε corresponding pixel point (i, j, mk, nk) and accumulated. 再计算模板图像和下一个位置子图像中对应象素点的ε(i,j,mk,nk)并累加,记作T,在计算并累加的过程中比较T与T0的大小。 Then calculate ε (i, j, mk, nk) corresponding to the pixel points template image and the next position of the sub image and accumulating, denoted by T, compare T size T0 in the computing and accumulating process. 为了加快匹配拼接速度,在每累加相应位置的一行或者一列的ε(i,j,mk,nk)之后,将T与T0进行大小比较。 In order to speed up the matching piecing speed, after the row corresponding to the position of each of the accumulation or ε (i, j, mk, nk) one of the T is compared in magnitude with T0. 对于某一位置,若在完全遍历模板图像和子图像的象素点之前,得到T≥T0,则停止计算,并将子图像移动到下一个位置,重新开始新一轮的计算。 For a certain position, if prior to complete traverse pixel dots template image and a sub image, obtained T≥T0, then stop counting, and the sub-picture moves to the next position, to start a new round of calculation. 若完全遍历模板图像和子图像的象素点后,得到T<T0,则更新阈值T0,并将此时子图像中心象素点的坐标位置(i,j)记录下来,得到i值与被搜索图像垂直中心线的水平距离,记作x2。 After if fully traversed pixel dots template image and a sub image, to obtain T <T0, the updated threshold value T0, and at this time the coordinate positions of the sub image center pixel point (i, J) is recorded, to obtain the value of i to be searched horizontal image vertical center line distance, denoted by x2. 阈值T0按下式取值:T0=TT&le;T0T0T>T0,]]>这样在搜索范围内遍历子图像,即可找到x2。 If the threshold T0 following formula values: T0 = TT & le; T0T0T> T0,]]> so traverse the sub-picture can be found within the search range x2. 取x1与x2的平均值作为相邻两幅图像的最佳匹配位置li,即li=(x1+x2)/2。 Take x1 and the average value of x2 as a best matching position li adjacent two images, i.e., li = (x1 + x2) / 2. 同理可求得序列图像中每相邻两幅图像的最佳匹配位置。 The same sequence of images can be obtained best matching position of each adjacent two images.

本实施例为了使得拼接后的图像效果更好,更具有整体感,提出一种改进的带权缝合的方法进行拼接后的平滑过渡,在两幅相邻图像的重叠区S和T,将其对应的各象素按照一定权值合成到新的图像。 The present embodiment to make the image effect after splicing better, more overall sense, proposes an improved method of weighted suture is a smooth transition after the splice, the overlap region S two adjacent images and T, which corresponding to the new image each pixel composited according to a certain weight. 各象素在各图像上的权值计算公式如下:WValue=cos(&pi;2*|x-x02x02|)*cos(&pi;2*|y-y02y02|)---(1)]]>式中WValue表示权值,(x0,y0)为重合部分的中心位置,(x,y)为象素坐标。 Each pixel weight is calculated in each image as follows: WValue = cos (& pi; 2 * | x-x02x02 |) * cos (& pi; 2 * | y-y02y02 |) --- (1)]]> wherein WValue is the weight, (x0, y0) as the overlap center position portion, (x, y) is the pixel coordinates. 将相邻图像的重叠区域S和T对应的各象素值按一定的权值合成新的图像。 Each pixel value of the overlap region S and T corresponding to adjacent image synthesizing new image by a certain weight.

重合部分的象素值可表示为:IN′=IN×WValue1+IN+1×WValue2(2)其中,IN和IN+1分别表示两个相邻图像的某个相应重叠的象素在各自原图像中的象素值,WValue1和WValue2是根据公式(1)计算出的该象素在各自图像上的权值,其取值范围为(0,1),且和为1。 Overlapped portion of the pixel value can be expressed as: IN '= IN × WValue1 + IN + 1 × WValue2 (2) wherein, IN and IN + 1 each represent a respective two adjacent pixels overlap in the respective original images the pixel values ​​in the image, WValue1 and WValue2 are according to the formula (1) to calculate the pixel weights on the respective images, which is in the range (0,1), and up to one.

对于拼接后的图像,并不是直接取IN′的值,而是引入一个阈值K,首先计算该点在平滑前的灰度值和加权平均值的差值,若此值小于阈值,则取IN′为此点的灰度值,反之,则取平滑前的灰度值为该点的灰度值。 For the stitched image, not directly take IN 'value, but the introduction of a threshold value K, first calculates the difference between the gradation value and the weighted average before smoothing this point, if this value is less than the threshold value, then take IN "gradation value for this point, on the contrary, it takes the gradation value of the gradation before smoothing value of the point.

如图4所示,是整个360°环视全景图像的生成过程。 4, the entire 360 ​​° Panoramic image generation process. 首先根据所拍摄场景的不同选择不同的拍摄方式,根据401拍摄序列图像。 First, depending on the scene being photographed different options photographing mode, photographing image sequence in accordance with 401. 对拍摄一个完整360度环视实景全景或者仅部分视角的环视实景场景,在一个固定位置,按顺时针或者逆时针方向等角度水平旋转拍摄;对于实物实体360°立体造型展示的虚拟拍摄,将实物实体放在赤道仪或者带刻度转盘上,逆时针或顺时针方向等角度地旋转载物仪盘进行拍摄。 Photographing a panoramic real full 360 ° view angle of view or only partially look around real scene, in a fixed position, horizontal angle clockwise or counterclockwise direction by rotating the like captured; for virtual modeling of physical entities 360 ° stereoscopic imaging display, the kind entity on the mount or belt scale dial clockwise or counterclockwise direction equiangularly screwed reproduced was meter disk imaging. 拍摄时相邻两幅图像要有一定的重叠,且对同一场景图像的张数不少于12张。 There should be some overlapping of adjacent two images when shooting, and for the number of copies of the same image of the scene less than 12. 402对序列图像进行中值滤波去噪、直方图均衡化处理等图像预处理,以平衡光照条件不同带来的影响,采用直方图均衡化处理变换函数:sk=T(rk)=&Sigma;i=0knin,k=0,1,2&CenterDot;&CenterDot;&CenterDot;,L-1---(3)]]>其中,n是图像中象素的总数,nk是灰度级为rk的象素个数,L为图像中可能的灰度级数,一般取值为256。 402 pairs of image sequences denoising median filtering, histogram equalization image pre-processing and the like, in order to balance the impact of different lighting conditions, using the histogram equalization transfer function: sk = T (rk) = & Sigma; i = 0knin, k = 0,1,2 & CenterDot; & CenterDot; & CenterDot;, L-1 --- (3)]]> wherein, n being the total number of image pixels, nk is the gray level pixel number of rk number, L is the number of gray levels in the image may be, the general value of 256. 通过公式(3)将输入图像中灰度级为rk的象素映射为输出图像灰度级为sk的对应象素。 By the equation (3) the input image gray level pixel map of rk output image gray levels sk corresponding pixel.

403通过360°水平柱面投影变换公式将各投影平面的重叠图像映射到一个标准投影即360°水平柱面投影上得到360°水平柱面图像,相应的变换公式为: 403 cylindrical projection conversion formula superimposed images of the projection plane by mapping 360 ° horizontal to a standard obtained by projecting 360 ° horizontal cylindrical image on i.e. 360 ° horizontal cylindrical projection, the corresponding transformation formula is:

其中, among them, k=r2+(W2-x)2,]]>(x,y)为输入图像上的任意一点,(x1,y1)为该点经过360°水平柱面投影变换后的坐标值,θ为投影角度,W为图像的宽度,H为图像的高度。 k = r2 + (W2-x) 2,]]> (x, y) as input any point on the image one o'clock, (x1, y1) for the point after 360 ° horizontal coordinate value of the cylindrical projection conversion, θ projection angle, W is the width of the image, H is the height of the image.

404采用一种自适应阈值序列序贯相似性检测算法(SSDA),先完成相邻两幅图像的拼合,最终通过同样的方法完成一组序列图像的拼合。 404 using an adaptive threshold sequence sequential similarity detection algorithm (as SSDA), to complete the split adjacent two images, the final completion of a set of split sequence of images by the same method.

405为消除痕迹采用改进的带权缝合的方法进行拼接后的平滑过渡,在该步骤中并不是直接采用按照一定权值合成到新的图像,而是在之前引入一个阈值,首先计算该点在平滑前的灰度值和加权平均值的差值,若此值小于阈值,则将该点对应的象素按一定权值合成到新的图像,反之则取平滑前的灰度值为该点的灰度值。 405 to eliminate smooth transition after the trace method weighted stapling improved stitching, in this step, not directly with the new images are combined according to a certain weight, but the introduction of a threshold value before the first calculating the point the difference between the gradation value and the weighted average of the previous smoothed, if this value is less than the threshold value, then the point corresponding to the pixel according to a certain weight synthesizes a new image, and vice versa before taking smooth gradation value of the point gradation value. 最终完成序列图像的全景生成。 Panoramic finalized sequence generated images.

Claims (10)

1.一种基于序列静态图像的360°环视全景生成方法,其特征在于:该方法包括如下步骤:(1)用相机拍摄所需的序列图像;(2)对序列图像进行预处理:采用中值滤波进行去噪,直方图均衡化处理;(3)序列图像的拼接:通过将相邻前后图像的拼接形成360°环视全景图像,包括以下步骤:(3.1)把拍摄的图片通过变换投影到360°水平柱面;(3.2)在读入的两幅相邻图像的前一幅图像中,选定一块大小和位置均合适的图像作为模板,确定在后一幅图像中的搜索范围,得到相邻两幅图像的最佳匹配位置li,依次得到序列图像中每相邻两幅图像的最佳匹配位置;(3.3)在两幅相邻图像的重叠区域S和T,将其对应的各象素按一定权值合成到新的图像。 1. Based on 360 ° a sequence of still images Panoramic generating method, characterized in that: the method comprising the steps of: (1) capturing a desired sequence of images by the camera; (2) the sequence of images pretreatment: use of median filtering denoising, histogram equalization; stitching (3) a sequence of images: forming 360 ° panoramic image by stitching the front and rear of the adjacent image, comprising the steps of: (3.1) the pictures taken by the conversion is projected onto 360 ° horizontal cylinder; (3.2) in the previous image two adjacent images read, the selected block size, and location of the right image as a template search range is determined after an image to give best matching position li adjacent two images successively obtained image sequences best matches the position of each adjacent two of the image; (3.3) in the overlapping area S two adjacent images and T, which correspond to the respective pixel according to a certain weight synthesize new image. 各象素在各图像上的权值计算公式如下:WValue=cos(&pi;2*|x-x02x02|)*cos(&pi;2*|y-y02y02|)---(1)]]>式中WValue表示权值,(x0,y0)为重合部分的中心位置,(x,y)为象素坐标;将相邻图像的重叠区域S和T对应的各象素值按一定的权值合成为新的图像,重合部分的象素值可表示为:IN′=IN×WValue1+IN+1×WValue2(2)其中,IN和IN+1分别表示两个相邻图像的某个相应重叠位置的象素在各自原图像中的象素值,WValue1和WValue2是根据公式(1)计算出的该象素在各自图像上的权值,其取值范围为(0,1),且和为1。 Each pixel weight is calculated in each image as follows: WValue = cos (& pi; 2 * | x-x02x02 |) * cos (& pi; 2 * | y-y02y02 |) --- (1)]]> wherein WValue is the weight, (x0, y0) as the overlap center position portion, (x, y) is the pixel coordinates; neighboring each pixel value of the overlap region of image S and T corresponding to a certain weight synthesis of a new image, the pixel values ​​of the overlapping portions can be expressed as: iN '= iN × WValue1 + iN + 1 × WValue2 (2) wherein, iN and iN + 1 each represent a respective overlap two adjacent images pixel values ​​of pixel positions in the respective original image, WValue1 and WValue2 are according to the formula (1) to calculate the pixel weights on the respective images, which is in the range (0,1), and, and 1.
2.如权利要求1所述的一种基于序列静态图像的360°环视全景生成方法,其特征在于:在所述的(2)中,采用直方图均衡化处理变换函数:sk=T(rk)=&Sigma;i=0knin,k=0,1,2&CenterDot;&CenterDot;&CenterDot;,L-1---(3)]]>其中,n是图像中象素的总数,nk是灰度级为rk的象素个数,L为图像中可能的灰度级数,一般取值为256;通过公式(3)将输入图像中灰度级为rk的象素映射为输出图像灰度级为sk的对应象素。 As claimed in claim 1 which is based on the sequence of still images 360 ° Panoramic generation method, comprising: in the (2), the use of histogram equalization transfer function: sk = T (rk ) = & Sigma; i = 0knin, k = 0,1,2 & CenterDot; & CenterDot; & CenterDot;, L-1 --- (3)]]> wherein, n being the total number of image pixels, nk is the gray level pixel number rk, L is the image of the possible gray levels, the general value of 256; by the equation (3) the input image gray level pixel map of rk output image gray level sk the corresponding pixel.
3.如权利要求1所述的一种基于序列静态图像的360°环视全景生成方法,其特征在于:在所述的(3.1)中,360°水平柱面投影变换公式为: 3. one of the claim 1 Panoramic generating method based on 360 ° sequence still image, wherein: in the (3.1), 360 ° horizontal cylindrical projection conversion formula is: 其中, among them, k=r2+(W2-x)2,]]>(x,y)为输入图像上的任意一点,(x1,y1)为该点经过柱面投影变换后的坐标值,θ为投影角度,W为图像的宽度,H为图像的高度。 k = r2 + (W2-x) 2,]]> (x, y) as input any point on the image point, (x1, y1) for the point after the coordinate values ​​of cylindrical projection conversion, θ is the projection angle, W for the width of the image, H is the height of the image.
4.如权利要求1-3之一所述的一种基于序列静态图像的360°环视全景生成方法,其特征在于:在所述的(3.2)中,定义图像的绝对误差函数为:&epsiv;(i,j,mk,nk)=|si,j(mk,nk)-s^(i,j)-T(m,n)+T^|---(5)]]>其中,s^(i,j)=1M2&Sigma;m=1M&Sigma;n=1Msi,j(m,n),T^=1M2&Sigma;m=1M&Sigma;n=1MT(m,n),]]>T为模板,S为被搜索图,si,j为子图,即模板覆盖下的那块搜索图,i,j为子图左上角象素点在S中的坐标,M为模板的宽和高。 4. The one of claims 1-3 panorama generating method of surveying based on the 360 ​​° sequence still image, wherein: in the (3.2), the absolute error function defined image is: & epsiv; (i, j, mk, nk) = | si, j (mk, nk) -s ^ (i, j) -T (m, n) + T ^ | --- (5)]]> wherein, s ^ (i, j) = 1M2 & Sigma; m = 1M & Sigma; n = 1Msi, j (m, n), T ^ = 1M2 & Sigma; m = 1M & Sigma; n = 1MT (m, n),]]> T as a template, S is searched FIG, si, j for the sub-picture, a piece of the search of FIG lower, i.e. covered by the template, i, j is the coordinate of top left corner of FIG pixel dots in S, M being the width and height of the template. 将模板中心与所在图像的垂直中心线的水平距离记作xi之后,遍历模板图像和第一个位置的子图像中所有的象素点,计算对应象素点的ε(i,j,mk,nk)并累加后将其作为阈值的初始值T0。 After the horizontal and vertical center line of the template center location distances of the images referred to as xi, all the pixel points subpicture traverse the template image and the first position, the computing ε corresponding pixel point (i, j, mk, nk) after and accumulate its initial value T0 as a threshold value. 再计算模板图像和下一个位置子图像中对应象素点的ε(i,j,mk,nk)并累加,记作T,在计算并累加的过程中比较T与T0的大小,若在完全遍历模板图像和子图像的象素点之前,得到T≥T0,则停止计算,并将子图像移动到下一个位置,重新开始新一轮的计算;若完全遍历模板图像和子图像的象素点后,得到T<T0,则更新阈值T0,并将此时子图像中心象素点的坐标位置(i,j)记录下来,得到i值与被搜索图像垂直中心线的水平距离,记作x2,取x1与x2的平均值作为相邻两幅图像的最佳匹配位置li,即li=(x1+x2)/2。 Then calculate ε (i, j, mk, nk) corresponding to the pixel points template image and the next position of the sub image and accumulating, denoted by T, compare T size T0 in the computing and accumulating in the process, if the full before traversing the pixel dots template image and a sub image, obtained T≥T0, then stop counting, and the sub-picture moves to the next position, to start a new round of calculation; post if fully traversed pixel dots template image and a sub image to give T <T0, the updated threshold value T0, and at this time the coordinate positions of the sub image center pixel point (i, J) is recorded, to obtain the horizontal distance i value is the search image vertical center line, denoted x2, take x1 and the average value of x2 as a best matching position li adjacent two images, i.e., li = (x1 + x2) / 2.
5.如权利要求4所述的一种基于序列静态图像的360°环视全景生成方法,其特征在于:在每累加相应位置的一行或者一列的ε(i,j,mk,nk)之后,将T与T0进行大小比较。 5. An claimed in claim Panoramic generating method based on 360 ° a sequence of still images, comprising: a row corresponding to the position of each of the accumulation or ε (i, j, mk, nk) one after the T and T0 size comparison.
6.如权利要求5所述的一种基于序列静态图像的360°环视全景生成方法,其特征在于:在所述的(3.3)中,对于拼接后的图像,并不是直接取IN′的值,而是引入一个阈值K,首先计算该点在平滑前的灰度值和加权平均值的差值,若此值小于阈值,则取IN′为此点的灰度值,反之,则取平滑前的灰度值为该点的灰度值。 The value of (3.3), with respect to the stitched image, not directly take IN ': 6. An claimed in claim Panoramic generating method based on 360 ° a sequence of still images, wherein , but the introduction of a threshold value K, first calculates the difference between the gradation value and the weighted average before smoothing this point, if this value is less than the threshold value, then take the 'gray value for this point iN, and vice versa, then take smooth grayscale value before the gradation value point.
7.如权利要求1-3之一所述的一种基于序列静态图像的360°环视全景生成方法,其特征在于:在所述的(1)中,序列图像的张数不少于12张,相邻两幅图像必须具有重叠部分,且重叠部分在30%到50%之间。 7. The one of claims 1-3 Panoramic generating method based on 360 ° sequence still image, wherein: in said (1), the number of a sequence of images not less than 12 , the adjacent two images must overlap portion at 30% to 50% between the overlapping portions and.
8.如权利要求4所述的一种基于序列静态图像的360°环视全景生成方法,其特征在于:在所述的(1)中,一组序列图像的张数不少于12张,相邻两幅图像必须具有重叠部分,且重叠部分在30%到50%之间。 8. An claimed in claim Panoramic generating method based on 360 ° sequence still image, wherein: in said (1), the number of a set sequence of images not less than 12, with o two images must overlap portion at 30% to 50% between the overlapping portions and.
9.如权利要求8所述的一种基于序列静态图像的360°环视全景生成方法,其特征在于:在所述的(1)中,对拍摄一个完整360度环视实景全景或者仅部分视角的环视实景场景,在一个固定位置,按顺时针或者逆时针方向等角度水平旋转拍摄。 9. An according to claim 8 panorama generating method of surveying based on the 360 ​​° sequence still image, wherein: in said (1), the captured a full 360 ° view real panoramic or only partially Perspective Looking around real scene, in a fixed position, an angle horizontal clockwise or counterclockwise direction like the rotational imaging.
10.如权利要求8所述的一种基于序列静态图像的360°环视全景生成方法,其特征在于:在所述的(1)中,对于实物实体360°立体造型展示的虚拟拍摄,将实物实体放在赤道仪或者带刻度转盘上,逆时针或顺时针方向等角度地旋转载物仪盘进行拍摄。 10. An according to claim 8 Panoramic generating method based on 360 ° sequence still image, wherein: in said (1), with respect to the virtual physical entity 360 ° three-dimensional shape display of the shooting, the physical entity on the mount or belt scale dial clockwise or counterclockwise direction equiangularly screwed reproduced was meter disk imaging.
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