WO2013023325A1 - 基于图像运动信息的2d转3d方法 - Google Patents

基于图像运动信息的2d转3d方法 Download PDF

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WO2013023325A1
WO2013023325A1 PCT/CN2011/001377 CN2011001377W WO2013023325A1 WO 2013023325 A1 WO2013023325 A1 WO 2013023325A1 CN 2011001377 W CN2011001377 W CN 2011001377W WO 2013023325 A1 WO2013023325 A1 WO 2013023325A1
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image
depth
value
pixel
input
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PCT/CN2011/001377
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English (en)
French (fr)
Inventor
冯涛
张彦丁
杨东
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北京世纪高蓝科技有限公司
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Priority to EP11870997.1A priority Critical patent/EP2629531A4/en
Priority to CN201180028889.9A priority patent/CN103053165B/zh
Priority to PCT/CN2011/001377 priority patent/WO2013023325A1/zh
Priority to US13/818,101 priority patent/US20130235155A1/en
Priority to JP2013540213A priority patent/JP2014504468A/ja
Publication of WO2013023325A1 publication Critical patent/WO2013023325A1/zh

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    • HELECTRICITY
    • H04ELECTRIC COMMUNICATION TECHNIQUE
    • H04NPICTORIAL COMMUNICATION, e.g. TELEVISION
    • H04N13/00Stereoscopic video systems; Multi-view video systems; Details thereof
    • H04N13/10Processing, recording or transmission of stereoscopic or multi-view image signals
    • H04N13/106Processing image signals
    • H04N13/128Adjusting depth or disparity
    • GPHYSICS
    • G06COMPUTING; CALCULATING OR COUNTING
    • G06TIMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T7/00Image analysis
    • G06T7/50Depth or shape recovery
    • G06T7/55Depth or shape recovery from multiple images
    • G06T7/579Depth or shape recovery from multiple images from motion
    • GPHYSICS
    • G06COMPUTING; CALCULATING OR COUNTING
    • G06TIMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T7/00Image analysis
    • G06T7/20Analysis of motion
    • G06T7/223Analysis of motion using block-matching
    • G06T7/238Analysis of motion using block-matching using non-full search, e.g. three-step search
    • HELECTRICITY
    • H04ELECTRIC COMMUNICATION TECHNIQUE
    • H04NPICTORIAL COMMUNICATION, e.g. TELEVISION
    • H04N13/00Stereoscopic video systems; Multi-view video systems; Details thereof
    • H04N13/20Image signal generators
    • H04N13/261Image signal generators with monoscopic-to-stereoscopic image conversion
    • H04N13/264Image signal generators with monoscopic-to-stereoscopic image conversion using the relative movement of objects in two video frames or fields
    • GPHYSICS
    • G06COMPUTING; CALCULATING OR COUNTING
    • G06TIMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T2207/00Indexing scheme for image analysis or image enhancement
    • G06T2207/20Special algorithmic details
    • G06T2207/20021Dividing image into blocks, subimages or windows
    • HELECTRICITY
    • H04ELECTRIC COMMUNICATION TECHNIQUE
    • H04NPICTORIAL COMMUNICATION, e.g. TELEVISION
    • H04N2213/00Details of stereoscopic systems
    • H04N2213/003Aspects relating to the "2D+depth" image format

Definitions

  • the present invention relates to the field of 2D to 3D technology, and in particular, to a 2D to 3D method based on image motion information.
  • 3D TV Three Dimensions
  • 3D applications are becoming more and more popular in people's lives, but despite the constant 3D filming, 3D sources are still unable to meet the current market needs.
  • Automatic conversion of 2D (Dimensions, 2D) sources to 3D becomes a new market need.
  • the conversion between 2D and 3D is to generate a second view video based on 2D view content.
  • the process includes two aspects of processing: one for depth estimation to obtain depth map (image); the other is based on depth map Depth Image Based Rendering (DIBR).
  • the depth map stores depth information in 8-bit grayscale values (0 grayscale represents the farthest value, and 255 grayscale represents the most recent value).
  • many algorithms have appeared in the field of 2D to 3D, and the more common ones are Based on the motion estimation based 2D to 3D algorithm, the method obtains the depth map of the input image by motion estimation.
  • the existing motion estimation based 2D to 3D algorithm is obtained.
  • the depth map is sparse, and the object can not be distinguished by different objects, which affects the image quality obtained by DIBR. Therefore, the popularization of the method is limited.
  • the technical problem to be solved by the present invention is: How to improve the quality of the image generated by the 2D to 3D method based on image motion information.
  • the present invention provides a 2D to 3D method based on motion estimation, the method comprising the steps of:
  • the motion estimation method is used to obtain the depth value of each pixel of the 2D image
  • step S4 The left eye image and the right eye image of step S4 are combined and output to obtain a 3D image.
  • step S1 further includes:
  • S1.2 calculates the depth value of each pixel according to the motion vector obtained in step S1.1.
  • the method of motion estimation is a diamond search algorithm.
  • step S2 further includes:
  • I ( x, y ) is the luminance value of the pixel at the (x, y ) position, and its value ranges from [0, 255]; SCALE is the scaling factor of the luminance value; width is the input 2D image The width value, height is the height value of the input 2D image; DEPTH SCALE is the depth value scaling factor,
  • step S2.1 further includes:
  • step S2.12 is performed
  • D(x,y)' min(D(xl,y)'+
  • SCALE 0.1.
  • DEPTH-SCALE 120.
  • step S3 further includes:
  • x1 and xr are the positions of the corresponding input 2D image xc positions in the left eye image and the right eye image
  • f is the focal length of the eye
  • tx is the distance between the eyes
  • Z is the distance of the pixel point from the human eye.
  • Dzero is the position of the zero plane, which has a value range of [0, 255];
  • Dzero 255.
  • the obtained depth map is continuous and dense, and the quality of the reconstructed image and the 3D visual effect are improved.
  • Figure 1/2 is a flow chart of a 2D to 3D method based on image motion information according to an embodiment of the present invention
  • Figure 2/2 is a schematic diagram of a visual model of a dual camera.
  • the 2D to 3D method based on image motion information includes the following steps: 51. Obtaining a depth value of each pixel of the input 2D image based on a motion estimation method;
  • step S4 The left eye image and the right eye image of step S4 are combined and output to obtain a 3D image.
  • step S1 further includes:
  • S1.1 Calculates the motion vector of each pixel based on the motion estimation method.
  • the motion estimation method adopts the diamond search algorithm, first searches for large diamonds, then searches for small diamonds, and finally moves to integer pixel precision.
  • Vector of course, other search algorithms are equally applicable here, not as a limitation on the method of the invention;
  • S1.2 calculates the depth value of each pixel according to the motion vector obtained in step S1.1.
  • y is the row of the pixel
  • X is the column of the pixel
  • D ( x, y ) is the depth value of the pixel at the unknown (x, y )
  • MV X and MV y are the horizontal and vertical movements of the pixel, respectively.
  • the input 2D image can be input before the motion search in step S1.1 is performed.
  • Performing a denoising process which is well known to those skilled in the art, does not perform motion vector discontinuity due to motion search. If the directly calculated depth map is sparse, the actual depth map should be dense. Therefore, the present invention accumulates the depth values calculated by the motion vectors based on the luminance information of each pixel.
  • step S2 further includes:
  • D(x,y)' min(D(xl,y) , +
  • D(x,y)' min(D(xl,y)'+
  • step S2.14 returns to step S2.ll if y ⁇ height, otherwise, outputs D(x, y) ' obtained in step S2.12 or S2.13.
  • height is the height value of the input 2D image;
  • the horizontal direction should keep the continuity of the depth value as much as possible, and avoid the excessive noise caused by the motion search. Therefore, the present invention does not use the horizontal gradient value for the scaling motion to obtain the depth value.
  • step S3 left The eye image is taken as an example, that is, the left eye image is reconstructed based on DIBR according to the depth map obtained in step S2 in step S3.
  • Equation (9) and (10) are the geometric relationships of the same pixel points in the corresponding Cl, Cr, and Cc in Fig. 2.
  • the xl or xr values of the corresponding input 2D image xc are calculated according to formulas (9) and (10). Then, the pixel value at (xc, y) is copied to the corresponding (xl, y) or (xr, y) (copy to (xl, y) in this embodiment).
  • step S3 further includes:
  • x1 and xr are the positions of the corresponding input 2D image xc positions in the left eye image and the right eye image
  • f is the focal length of the eye
  • tx is the distance between the eyes
  • Z is the distance of the pixel point from the human eye.
  • Dzero is the position of the zero plane, which has a value range of [0, 255];
  • the horizontal direction of the input 2D image is first scaled to improve the pixel precision during projection.
  • the image is stretched four times in the horizontal direction, and the X value of the 1/4 pixel accuracy corresponding to x1 per line is calculated based on the above human visual relationship. If the value of X corresponding to xl exceeds the image range, the pixel value of the xl position is obtained according to the interpolation; if multiple xl correspond to the same x, then D(x, y) is taken as the largest xl, and the other xl position values are passed. Interpolation is obtained; if X corresponding to xl is unique, the pixel value at position xl is the pixel value of the X position of the input 2D image.
  • the image reconstructed image obtained by the 2D-to-3D method based on the image motion information of the present invention has high quality and good 3D visual effect, and is important for promoting the automatic conversion of the 2D source to the 3D market.

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  • Engineering & Computer Science (AREA)
  • Multimedia (AREA)
  • Computer Vision & Pattern Recognition (AREA)
  • Physics & Mathematics (AREA)
  • General Physics & Mathematics (AREA)
  • Theoretical Computer Science (AREA)
  • Signal Processing (AREA)
  • Testing, Inspecting, Measuring Of Stereoscopic Televisions And Televisions (AREA)
  • Processing Or Creating Images (AREA)

Abstract

本发明公开了一种基于图像运动信息的2D转3D方法,涉及2D转3D技术领域。该方法包括步骤:S1.基于运动估计的方法得到输入的2D图像各像素的深度值;S2.根据各像素的亮度值,对各像素的深度值进行累加,得到输入的2D图像的深度图;S3.根据步骤S2得到的深度图,进行基于深度图的图像重构左眼和/或右眼图像;S4.将步骤S4的左眼图像和右眼图像合成并输出得到3D图像。在本发明的方法中,由于对运动估计得到的深度值进行了累加处理,得到的深度图连续且稠密,提高了重构图像的质量以及3D视觉效果。

Description

基于图像运动信息的 2D转 3D方法 技术领域
本发明涉及 2D转 3D技术领域, 尤其涉及一种基于图像运动信息的 2D转 3D方法。
背景技术
三维(Three Dimensions, 3D )电视席卷而来成为全球电视产业新的发展 方向, 各大电视生厂商都推出了自己的 3D电视。 3D应用在人们的生活中变得 越来越流行, 但是虽然不断有 3D电影拍摄, 3D片源仍不能满足目前的巿场需 要。 将二维(Two Dimensions, 2D )片源自动的转换为 3D成为新的巿场需要。
2D与 3D之间的转换即为生成基于 2D视图内容的第二视图视频,该过程包括两 个方面的处理: 一个为深度估计以得到深度图(depth map/image ); 另一个为 基于深度图的图像重构 (Depth Image Based Rendering, DIBR )。 深度图以 8 位灰度值存储了深度信息(0灰度表示最远值, 255灰度表示最近值), 在过去 的多年中, 2D转 3D这一领域出现了很多算法,较常用的为基于运动估计的 2D 转 3D算法, 该方法通过运动估计的方法得到输入图像的深度图, 但是, 由于 深度图需要相当的稠密度及精确度, 而现有的基于运动估计的 2D转 3D算法得 到的深度图较为稀疏, 在物体分解出不能区分出不同物体, 从而影响 DIBR得 到的图像质量, 因此, 限制了该方法的推广使用。
发明内容
(一) 要解决的技术问题
本发明要解决的技术问题是: 如何提高基于图像运动信息的 2D转 3D方法 生成图像的质量。
(二)技术方案
为解决上述问题, 本发明提供了一种基于运动估计的 2D转 3D方法, 该方 法包括步骤:
51. 基于运动估计的方法得到^入的 2D图像各像素的深度值;
52. 根据各像素的亮度值, 对所述各像素的深度值进行累加, 得到所述 输入的 2D图像的深度图;
53. 根据步骤 S2得到的深度图, 基于深度图的图像重构, 重构左眼和 /或 右眼图像;
54. 将步骤 S4的左眼图像和右眼图像合成并输出得到 3D图像。
优选地, 步骤 S1进一步包括:
S1.1 基于运动估计的方法计算各像素的运动矢量;
S1.2根据步驟 S1.1得到的运动矢量分别计算各像素的深度值。
为:
Figure imgf000004_0001
优选地, 所述运动估计的方法为菱形搜索算法。
优选地, 步骤 S2进一步包括:
S2.1 从所述输入的 2D图像的第一行开始, 对各像素的深度值进行累加 得到每一个像素的深度累加值 D(x,y)' :
S2.2 按照以下公式, 将所述深度累加值归一化到 [0, 255]之间, 得到归 一化的深度值 D(x,y)" :
D(x,y)" =
Figure imgf000004_0002
其中, I ( x,y )为(x,y )位置处的像素的亮度值, 其取值范围为 [0, 255]; SCALE为亮度值的缩放因子; width为所述输入的 2D图像的宽度值, height 为所述输入的 2D图像的高度值; DEPTH SCALE为深度值缩放因子,
sum
sum
sidth*height
sum= D(x,y)' |
x=o,y=o
优选地, 步骤 S2.1进一步包括:
S2.l l若 y为 0, 则 D(x,y)' = 0 , 否则, 执行步骤 S2.12;
S2.12若 y为奇数, 且 X为 0, 贝 lj D(X,y)' = D(x,y-l)'+D(X,y), 若 x不为 0,
D(x,y)' = min(D(x-l)y)'+|l(x+l,y)-l(x-l,y)|*SCALE,D(x)y-l),) + D(x,y)*(l+|l(x,y-l)-l(x,y+l)|*SCALE 否则执行步骤 S2.13; S2.13 若 x=width-l, 贝 lj D(x,y)' = D(x,y-l)'+D(x,y), 否则:
D(x,y)' = min(D(x-l,y)'+|l(x+l,y)-I(x-l,y)|*SCALE,D(x,y-l),) + D(x,y)*(l+|l(x,y-l)-l(x,y+l)|*SCALE
S2.14 若 < 1^§ , 则返回步骤 S2.l l, 否则, 输出步骤 S2.12或 S2.13 得到的 D(x,y)'。
优选地, SCALE=0.1。
优选地, DEPTH一 SCALE=120。
优选地, 步驟 S3进一步包括:
S3.1 按照下式重构左眼或右眼图像:
xl=xc+——
2 z
tx f
xr = xc
2 z
l/Z=Dz (x,y)"-Dzero
其中, xl、 xr分别为左眼图像及右眼图像中对应输入的 2D图像 xc位置 的位置, f为眼睛的焦距, tx为两眼之间的距离, Z为像素点离人眼的距离, Dzero为零平面的位置, 其取值范围为 [0,255];
S3.2 将(xc,y ) 处的像素值拷贝到对应的 (xl,y )或 (xr,y ) 处。
优选地, Dzero=255。
(三)有益效果
在本发明的方法中, 由于对运动估计得到的深度值进行了累加处理, 得 到的深度图连续且稠密, 提高了重构图像的质量以及 3D视觉效果。
附图说明
图 1/2是依照本发明一种实施方式的基于图像运动信息的 2D转 3D方法 流程图;
图 2/2是一种双摄像机的视觉模型示意图。
具体实施方式
本发明提出的基于图像运动信息的 2D转 3D方法,结合附图及实施例详 细说明如下。
如图 1/2所示, 依照本发明一种实施方式的基于图像运动信息的 2D转 3D方法包括步骤: 51. 基于运动估计的方法得到输入的 2D图像各像素的深度值;
52. 根据各像素的亮度值, 对各像素的深度值进行累加, 得到输入的 2D 图像的深度图;
53. 根据步骤 S2得到的深度图,进行基于深度图的图像重构左眼和 /或右 眼图像;
54. 将步骤 S4的左眼图像和右眼图像合成并输出得到 3D图像。
在本实施方式的方法中, 步骤 S1进一步包括:
S1.1 基于运动估计的方法计算各像素的运动矢量, 其中, 运动估计的方 法采用菱形搜索算法, 先进行大的菱形的搜索, 再进行小的菱形的搜索, 最 后的到整数像素精度的运动矢量, 当然, 其他的搜索算法在这里同样适用, 不作为对本发明方法的限制;
S1.2根据步骤 S1.1得到的运动矢量分别计算各像素的深度值。
其中, 深度值的计算公式为:
Figure imgf000006_0001
y为像素所在行, X为像素所在列, D ( x,y )为未知 (x,y )处的像素的 深度值, MVX以及 MVy分别为所述像素水平方向以及竖直方向的运动矢量, C为常量, 本实施方式中 C=l。
为了提高步骤 SI.1中搜索的精度, 减少噪声(特别是某些片源中添加的 椒盐噪声)对运动搜索精度的影响, 在进行步骤 S1.1的运动搜索之前, 可对 输入的 2D 图像进行去噪处理, 此为本领域的技术人员所公知的, 在此不做 由于运动搜索得到的运动矢量不连续, 如果直接计算得到的深度图很稀 疏, 而实际的深度图应该是稠密的, 所以本发明根据各像素的亮度信息对运 动矢量计算得到的深度值进行累加。
在本实施方式中, 步骤 S2进一步包括:
S2.1 从输入的 2D图像的第一行开始, 对各像素的深度值进行累加得到 每一个像素的深度累加值 D(x,y)', 该步骤进一步包括:
S2.l l若 y为 0, 则 D(x,y)' = 0, 否则, 执行步骤 S2.12; S2.12 若 y为奇数, 且 x为 0, j¾j D(x,y) ' = D(X,y-l)'+ D(x,y) , 若 x不为 0, 则:
D(x,y)' = min(D(x-l,y),+|l(x+l,y)-l(x-l,y)|*SCALE,D(x,y-l)1) + D(x,y)*(l+|l(x,y-l)-I(x,y+l)|*SCALE 否则执行步骤 S2.13;
S2.13 若 x=width-l, 则 D(x,y) ' = D(x,y-l)'+D(x,y), 否则:
D(x,y)' = min(D(x-l,y)'+|l(x+l,y)— I(x-l,y)|*SCALE,D(x,y-l)')+D(x,y)*(l+|l(x,y-l)-I(x,y+l)|*SCALE
S2.14若 y < height, 则返回步骤 S2.ll, 否则, 输出步骤 S2.12或 S2.13 得到的 D(x,y) '。
S2.2 按照以下公式, 将深度累加值归一化到 [0, 255]之间, 得到归一化 的深度值 D(x,y) ", 从而得到连续稠密的深度图:
D(x,y)" = ( 6 )
Figure imgf000007_0001
其中, I ( x,y )为(x,y )位置处的像素的亮度值, 其取值范围为 [0, 255]; SCALE为亮度值的缩放因子,本实施方式中 SCALE=0.1; width为输入的 2D 图像的宽度值, height为输入的 2D图像的高度值; DEPTH— SCALE为深度值 缩放因子, 本实施方式中, DEPTH一 SCALE=120;
, sum ( π
sum- \ Ί、 )
sidth*height
sum= D(x,y) ' ( 8 );
x=o, =o
S2.3 对步骤 S2.2得到的归一化深度值 D(x,y)"进行非对称高斯滤波,得到 最终的深度值 Dz (x,y)", 该非对称高斯滤波处理为本领域所述熟知的技术, 在 此不做赘述。
由于将在图像水平方向进行投影变换, 所以水平方向应尽量保持深度值 的连续性, 避免运动搜索带来的噪声过大影响, 所以本发明未将水平梯度值 用于缩放运动得到深度值。
由于人眼的视觉特性, 有 70%的人视觉感知依重于右眼, 20%的人依重 于左眼。 为了减少计算量, 本发明在使用 DIBR重构图像时仅重构用户不倚 重的那只眼, 不失一般性, 这里默认为左眼。 且这种情况下虽然重构帧的质 量较差, 但不影响 3D的视觉效杲。 因此, 本实施方式中在歩骤 S3中, 以左 眼图像为例, 即, 在步骤 S3中根据步骤 S2得到的深度图, 基于 DIBR重构 左眼图像。
如图 2/2所示, 其中, Cc为输入的 2D图像, C1为重构的左眼图像, Cr 为重构的右眼图像。 f为眼睛的焦距, tx为基线距离, 即两眼之间的距离, Z 为观测的像素点离人眼的距离, 按照公式(11 )进行计算。 Dzero为零平面的 位置, 取值 [0, 255], 在本实施方式中可取 255。 公式(9 )、 ( 10 )为图 2中对 应 Cl、 Cr、 Cc中同一像素点投影的几何关系, 根据公式 (9 )、 ( 10 )计算得 到对应输入的 2D图像 xc位置的 xl或 xr值, 然后将(xc,y )处的像素值拷贝 到对应的 (xl,y )或 (xr,y )处(本实施方式中拷贝到 (xl,y ) )。
即步骤 S3进一步包括:
S3.1 按照下式重构左眼或右眼图像:
( 9 )
2 z
xr = xc-—— ( 10 )
2 ζ
l/Z=Dz (x,y) "-Dzero ( 11 )
其中, xl、 xr分别为左眼图像及右眼图像中对应输入的 2D图像 xc位置 的位置, f为眼睛的焦距, tx为两眼之间的距离, Z为像素点离人眼的距离, Dzero为零平面的位置, 其取值范围为 [0,255];
S3.2 将(xc,y ) 处的像素值拷贝到对应的 (xl,y )或 (xr,y ) 处。
为了减少重构图像的锯齿效应, 先将输入的 2D 图像的水平方向进行缩 放, 以提高投影时的像素精度。 在本实施方式中, 在水平方向将图像拉伸到 原来的 4倍, 根据以上的人眼视觉关系计算得到每行 xl对应的 1/4像素精度 的 X值。若 xl对应的 X的值超出了图像范围, 则根据插值得到 xl位置的像素 值; 若多个 xl对应相同的 x, 则取 D(x,y) "最大的 xl, 其它的 xl位置值通过插 值得到; 若 xl对应的 X唯一, 则 xl位置的像素值为输入的 2D图像 X位置的 像素值。
以上实施方式仅用于说明本发明, 而并非对本发明的限制, 有关技术领 域的普通技术人员, 在不脱离本发明的精神和范围的情况下, 还可以做出各 种变化和变型, 因此所有等同的技术方案也属于本发明的范畴, 本发明的专 利保护范围应由权利要求限定。
工业实用性
使用本发明的基于图像运动信息的 2D转 3D方法得到的图像重构图像质 量高, 3D视觉效果好, 对推动 2D片源自动的转换为 3D的巿场发展具有重 要意义。

Claims

Figure imgf000010_0001
1、 一种基于图像运动信息的 2D转 3D方法, 其特征在于, 该方法包括 步骤:
51. 基于运动估计的方法得到输入的 2D图像各像素的深度值;
52. 根据各像素的亮度值, 对所述各像素的深度值进行累加, 得到所述 输入的 2D图像的深度图;
53. 根据步骤 S2 得到的深度图, 基于深度图的图像重构, 重构左眼和 / 或右眼图像;
54. 将步骤 S4的左眼图像和右眼图像合成并输出得到 3D图像。
2、 如权利要求 1所述的基于图像运动信息的 2D转 3D方法, 其特征在 于, 步骤 S1进一步包括:
S1.1 基于运动估计的方法计算各像素的运动矢量;
S1.2根据步骤 S1.1得到的运动矢量分别计算各像素的深度值。
3、 如权利要求 2所述的基于图像运动信息的 2D转 3D方法, 其特征在 于, 所述运动估计的方法为菱形搜索算法。
4、 如权利要求 3所述的基于图像运动信息的 2D转 3D方法, 其特征在 于, 步骤 S2进一步包括:
S2.1 从所述输入的 2D图像的第一行开始, 对各像素的深度值进行累加 得到每一个像素的深度累加值 D(x,y)' :
S2.2 按照以下公式, 将所述深度累加值归一化到 [0, 255]之间, 得到归 一化的深度值 D(X,y)" :
D(x,y)" = min 255, max 0, D(X,y) *DEPTH一 SCALE
V sum' 乂
其中, I ( x,y )为 ( x,y )位置处的像素的亮度值, 其取值范围为 [0, 255]; SCALE为亮度值的缩放因子; width为所述输入的 2D图像的宽度值, height 为所述输入的 2D图像的高度值; DEPTH SCALE为深度值缩放因子,
, sum
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Figure imgf000011_0001
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Figure imgf000011_0002
Z OX = JX
J xi z Z
+OX=IX
■W ^W^^M^f ^ res
°0n=3TV3S~HIdHa '
°IO=3TV3S Ί αε
Figure imgf000011_0003
、9 。擦# erz:s zvzs ' ^ 'nxs ^H^ 'V^ H>^ ^ PVZS
: ^ '(χ'χ)α+.(ι-χ'χ)α=,( χ)α「ιΜ 'i-ipp! =x errs TV3S*|(i+'x)i-(i 'x)i|+i)*( x)a+(,(i 'x)a'3TV3S*|( 'i-x)i-( 'i+x)i|+.( i-x)a)u!ui=,( x)a
'(χ'χ)α+,(ι-'χ)α=,(χ'χ)α
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