This folder contains the C++ source code of our projection tool. The folder Scripts contains a Python3 script used to test our streaming architecture
The projection tool can read a 360-degree video projected on any known 2D layout, can project this video onto any known 2D layout and can extract any Field of View (FoV) from the 360-degree video. For now the FoV extraction is static: i.e. the FoV center cannot move during the video.
You need a recent version of docker installed and running on your machine.
Inside the transformation repository, run the following command to build the docker image:
docker build -t trans360 .
You now have a docker image named trans360 that contains a debian buster installation with the trans executable located stored inside the /bin repository. The dockfile contains two stages: a build stage and stage that keeps only the minimum needed.
You have to run the build command each time you change a source file.
You have to mount the folder that contains the configuration file and the folder that contains the video inside the container. Paths inside the configuration file should be paths inside the container, not inside your machine. Output files should be writing to mounted folder if you want to get them on your host machine. The container working directory is /home/trans360.
For instance, to run the examples scripts, the configuration files and the input videos are inside the PATH_TO_TRANS_FOLDER/examples folder. The following command will mount the examples fodler into the working directory of the container, creates and starts the trans360 container, and finaly run the trans software on the configuration file named Config_1.ini:
docker run --rm -v PATH_TO_TRANS_FOLDER/examples:/home/trans360 trans360 trans -c Config_1.ini
Cf. How To Use Section for more information on the configuration file.
- A C++11 compiler (for instance gcc version 5 or more recent)
- CMake version 3 or more recent
- opencv at least version 3
- ffmpeg livav
- boost
OpenCV, libav ffmpeg (with the lastest API) and boost are supposed installed on you computer. If cmake cannot find those libs it will print an error message.
Create the build repository and move into it:
mkdir build cd build
Generate the Makefile:
cmake ..
Compile the software:
make
Now a software named trans should be in your build repository
- A C++11 compiler (for instance gcc version 5 or more recent)
- Python with the virtualenv package
- CMake version 3 or more recent
Note that during the first compilation step, * OpenCV, Boost and ffmpeg libav library will be download by conan (c.f. next section)
Create the build repository and move into it:
mkdir build cd build
Create a python virtualenv (tested with python3), use it and install the conan package inside:
python -m venv ConanEnv source ConanEnv/bin/activate pip install conan
Still inside the virtualenv, use conan to get the OpenCV, Boost and libav dependency and generate a part of the cmake configuration:
conan install .. -s compiler=gcc -s compiler.libcxx=libstdc++11 --build=missing deactivate
Generate the Makefile:
cmake -DUSE_CONAN=true ..
Compile the software:
make
Now a software named trans should be in your build repository
The trans software require one argument.
| -c | Path to the .ini configuration file. |
The ini file contains the configuration of the test scenario.
The Global section contains the global configuration of the test.
[Global]
;The number of frame per second
fps=24
;If true display the generated pictures and wait for a key to precessed before continuing
displayFinalPict=true
;Path to the output video file. If empty no output video is recorded.The flow id and the name of the last layout is used as an id for the generated output file
videoOutputName=
;codec ffmpeg should use (as known by ffmpeg). If empty or not available libx265 (HEVC) will be used. The container is selected from the extension set in the videoOutputName
videoOutputCodec=libx265
; bitrate in kbps of the output videos (0 if default crf)
videoOutputBitRate=0
;Path to the quality output file. If empty no quality is computed. The flow id and the name of the last layout is used as an id for the generated output file
qualityOutputName=
;Indicate which metric to use. "MS-SSIM", "SSIM", "PSNR" and "WS-PSNR" require the two final picture to have the same resolution.
;The "S-PSNR-NN" and "S-PSNR-I" are computed from a uniform sampling of 655362 points on the sphere. "S-PSNR-NN" uses the Nearest Neightboor interpolation and "S-PSNR-I" uses the Bicubic interpolation.
qualityToComputeList = ["MS-SSIM", "SSIM", "PSNR", "S-PSNR-NN", "S-PSNR-I", "WS-PSNR"]
;Index of the first frame of the input videos to process. If equal to n then the n first frames of the input videos will be skipped
startFrame=0
;Number of frame to process in the video
nbFrames= 5
;The layout flow indicate for each flow the input video, its layout and which transformation to perform. It is an array of array. The first string in an array is the path to the input video. The second string is the layout of the input video and the other string are section id of the layout onto which the video should be projected.
layoutFlow= [["../example.mp4", "Equirectangular", "EquirectangularTiled"], ["../example.mp4", "Equirectangular", "CubeMap", "FlatFixed"]]Each section id named in the layoutFlow attribute should be defined in the ini file. In the layout flow, the first string is the path to the input video, the second string the name of the section that describe the layout of the input video. The other strings are the name of the section the describe the layout onto which we want to project the video. There can be as many layout as we want and the video will be consecutively projected on each of those layout. It is not possible to do an other projection after a flat fixed view (a FoV extraction) projection.
equirectangular layout
[Equirectangular]
;type of the layout: equirectangular is the state-of-the-art equirectangular layout
type=equirectangular
;Width of the reference equirectangular video
refWidth=4096
;Width of the reference equirectangular video
refHeight=2048
;The rotation used to get the center of the front face: type = "euler" or "quaternion" or "angleAxis".
; if type is "euler" you have to give the yaw, pitch and roll angle in degree
; if type is "quaternion" you have to give the w, x, y, z value of the quaternion that represente the rotation. A normalization is performed by the software.
; if type is "angleAxis" you have to give an "angle" in degree and a direction vector x, y, z that generates the rotation axis (this vector should not be (0, 0, 0)).
rotation= {"type":"euler", "yaw":0.0, "pitch":0.0, "roll":0.0}
;rotation= {"type":"quaternion", "w":1.0, "x":0.0, "y":0.0, "z":0.0}
;rotation= {"type":"angleAxis", "angle":90, "x":0, "y":0, "z":1}
;if true, the resolution of the layout is compute based on the resolution of the previous layout (or on the resolution of the reference equirectangular video)
relativeResolution=true
;if relativeResolution=true ratio of the output layout width compare to the reference video; otherwise width size of the output video in pixel number
width=1
height=1
;bitrate ratio of the output video relative to the videoOutputBitRate (if it is the final layout and if the video is encoded).
bitrate=1flatFixed layout: FoV extraction
[FlatFixed]
;type of the layout: flatFixed means FoV extraction
type=flatFixed
;relativeResolution cannot be true for flatFixed layout
relativeResolution=false
;indicates if we want to use a static position set by the yaw, pitch and roll parameter or play a traces located at positionTrace
dynamicPositions=false
;The path to the position trace file. Used if dynamicPositions is set to true
positionTrace=pathToPositionTrace
;The rotation used to get the center of the front face: type = "euler" or "quaternion" or "angleAxis".
; if type is "euler" you have to give the yaw, pitch and roll angle in degree
; if type is "quaternion" you have to give the w, x, y, z value of the quaternion that represente the rotation. A normalization is performed by the software.
; if type is "angleAxis" you have to give an "angle" in degree and a direction vector x, y, z that generates the rotation axis (this vector should not be (0, 0, 0)).
rotation= {"type":"euler", "yaw":0.0, "pitch":0.0, "roll":0.0}
;rotation= {"type":"quaternion", "w":1.0, "x":0.0, "y":0.0, "z":0.0}
;rotation= {"type":"angleAxis", "angle":90, "x":0, "y":0, "z":1}
;width and height in pixels of the output image
width=1920
height=1080
;Horizontal angle of the display device (in degree)
horizontalAngleOfVision=110
;Vertical angle of the display device (in degree)
verticalAngleOfVision=90cube map layout
[CubeMapLower]
;type of the layout: cube map layout
type=cubeMap
;Size of the reference cube map video (ie if all faces have a size ratio of 1)
refWidth=4096
refHeight=2048
;use relative resolution for the face resolutions
relativeResolution=true
;The rotation used to get the center of the front face: type = "euler" or "quaternion" or "angleAxis".
; if type is "euler" you have to give the yaw, pitch and roll angle in degree
; if type is "quaternion" you have to give the w, x, y, z value of the quaternion that represente the rotation. A normalization is performed by the software.
; if type is "angleAxis" you have to give an "angle" in degree and a direction vector x, y, z that generates the rotation axis (this vector should not be (0, 0, 0)).
rotation= {"type":"euler", "yaw":0.0, "pitch":0.0, "roll":0.0}
;rotation= {"type":"quaternion", "w":1.0, "x":0.0, "y":0.0, "z":0.0}
;rotation= {"type":"angleAxis", "angle":90, "x":0, "y":0, "z":1}
;the edge of each face. If relativeResolution, ratio compared to the reference input video, otherwise size in pixel.
cubeEdgeLengthFront=1
cubeEdgeLengthLeft=1
cubeEdgeLengthRight=1
cubeEdgeLengthTop=1
cubeEdgeLengthBottom=1
cubeEdgeLengthBack=1
;if useTile is set to true, each face will be encoded into an independent video with a target bitrate set by the bitrateFront, bitrateLeft, etc. parameters
;if useTile is set to false, the whole projection will be encoded into a unique video (all faces on the same frame) and a global bitrate target set by the bitrate parameter
useTile=false
;Relative bitrate of each face (if final layout encoded). A ratio r means r*videoOutputBitRate/6
bitrateFront=1
bitrateLeft=1
bitrateRight=1
bitrateTop=1
bitrateBottom=1
bitrateBack=1
;bitrate is used only if useTile=false. A ratio r means a bitrate of r*videoOutputBitRate
bitrate=1
;indicate the position and the orientation of each face in the packed picture. position 1 is top left, 2 top right, 3 bottom left and 5 bottom right. Rotation can be 0, 90, -90 or 180.
facesPosition={"face1":"right", "face1Rotation":0, "face2":"back", "face2Rotation":0, "face3":"left", "face3Rotation":0, "face4":"top", "face4Rotation":-90, "face5":"front", "face5Rotation":-90, "face6":"bottom", "face6Rotation":-90}equal-area cubemap: EAC layout
[CubeMapEqualArea]
;type of the layout: cube map layout
type=EAC
;Size of the reference cube map video (ie if all faces have a size ratio of 1)
refWidth=4096
refHeight=2048
;use relative resolution for the face resolutions
relativeResolution=true
;The rotation used to get the center of the front face: type = "euler" or "quaternion" or "angleAxis".
; if type is "euler" you have to give the yaw, pitch and roll angle in degree
; if type is "quaternion" you have to give the w, x, y, z value of the quaternion that represente the rotation. A normalization is performed by the software.
; if type is "angleAxis" you have to give an "angle" in degree and a direction vector x, y, z that generates the rotation axis (this vector should not be (0, 0, 0)).
rotation= {"type":"euler", "yaw":0.0, "pitch":0.0, "roll":0.0}
;rotation= {"type":"quaternion", "w":1.0, "x":0.0, "y":0.0, "z":0.0}
;rotation= {"type":"angleAxis", "angle":90, "x":0, "y":0, "z":1}
;the edge of each face. If relativeResolution, ratio compared to the reference input video, otherwise size in pixel.
cubeEdgeLengthFront=1
cubeEdgeLengthLeft=1
cubeEdgeLengthRight=1
cubeEdgeLengthTop=1
cubeEdgeLengthBottom=1
cubeEdgeLengthBack=1
;if useTile is set to true, each face will be encoded into an independent video with a target bitrate set by the bitrateFront, bitrateLeft, etc. parameters
;if useTile is set to false, the whole projection will be encoded into a unique video (all faces on the same frame) and a global bitrate target set by the bitrate parameter
useTile=false
;Relative bitrate of each face (if final layout encoded). A ratio r means r*videoOutputBitRate/6
bitrateFront=1
bitrateLeft=1
bitrateRight=1
bitrateTop=1
bitrateBottom=1
bitrateBack=1
;bitrate is used only if useTile=false. A ratio r means a bitrate of r*videoOutputBitRate
bitrate=1
;indicate the position and the orientation of each face in the packed picture. position 1 is top left, 2 top right, 3 bottom left and 5 bottom right. Rotation can be 0, 90, -90 or 180.
facesPosition={"face1":"right", "face1Rotation":0, "face2":"back", "face2Rotation":0, "face3":"left", "face3Rotation":0, "face4":"top", "face4Rotation":-90, "face5":"front", "face5Rotation":-90, "face6":"bottom", "face6Rotation":-90}pyramid layout
[PyramidLower]
;type of the layout: pyramid map layout
type=pyramid2
;Size of the reference input equirectangular video
refWidth=4096
refHeight=2048
;use relative resolution for the face resolutions
relativeResolution=true
;Size of the Base face in the 3D space (the circumscribed sphere have a radius of 1). Should be strictly higher than 2.
pyramidBaseEdge=2.5
;The rotation used to get the center of the front face: type = "euler" or "quaternion" or "angleAxis".
; if type is "euler" you have to give the yaw, pitch and roll angle in degree
; if type is "quaternion" you have to give the w, x, y, z value of the quaternion that represente the rotation. A normalization is performed by the software.
; if type is "angleAxis" you have to give an "angle" in degree and a direction vector x, y, z that generates the rotation axis (this vector should not be (0, 0, 0)).
rotation= {"type":"euler", "yaw":0.0, "pitch":0.0, "roll":0.0}
;rotation= {"type":"quaternion", "w":1.0, "x":0.0, "y":0.0, "z":0.0}
;rotation= {"type":"angleAxis", "angle":90, "x":0, "y":0, "z":1}
;the edge of each face. If relativeResolution, ratio compared to the reference input video, otherwise size in pixel.
pyramidBaseEdgeLength=1
pyramidHeightLeft=1
pyramidHeightRight=1
pyramidHeightTop=1
pyramidHeightBottom=1
;if useTile is set to true, each face will be encoded into an independent video with a target bitrate set by the pyramidBaseBitrate, pyramidLeftBitrate, etc. parameters
;if useTile is set to false, the whole projection will be encoded into a unique video (all faces on the same frame) and a global bitrate target set by the bitrate parameter
useTile=true
;Relative bitrate of each face (if final layout encoded). A ratio r means r*videoOutputBitRate/5
pyramidBaseBitrate=1
pyramidLeftBitrate=0.25
pyramidRightBitrate=0.25
pyramidTopBitrate=0.25
pyramidBottomBitrate=0.25
;bitrate is used only if useTile=false. A ratio r means a bitrate of r*videoOutputBitRate
bitrate=1rhombicDodeca layout
[RhombicDodeca]
;type of the layout: rhombicDodeca
type=rhombicDodeca
;Size of the reference input equirectangular video
refWidth=4096
refHeight=2048
;use relative resolution for the face resolutions
relativeResolution=true
;The rotation used to get the center of the front face: type = "euler" or "quaternion" or "angleAxis".
; if type is "euler" you have to give the yaw, pitch and roll angle in degree
; if type is "quaternion" you have to give the w, x, y, z value of the quaternion that represente the rotation. A normalization is performed by the software.
; if type is "angleAxis" you have to give an "angle" in degree and a direction vector x, y, z that generates the rotation axis (this vector should not be (0, 0, 0)).
rotation= {"type":"euler", "yaw":199.4712, "pitch":-26.565, "roll":0.0}
;rotation= {"type":"quaternion", "w":0.16457888628862896, "x":0.2264437924852498, "y":0.03885174449037981, "z":0.9592327876917812}
;rotation= {"type":"angleAxis", "angle":161,05451359158116, "x":-0,22957428492645424, "y":-0,03938885390335991, "z":-0,9724937870694532}
;the length of each face. If relativeResolution, ratio compared to the reference input video, otherwise size in pixel.
rhombEdgeLengthFace1=1
rhombEdgeLengthFace2=1
rhombEdgeLengthFace3=1
rhombEdgeLengthFace4=1
rhombEdgeLengthFace5=1
rhombEdgeLengthFace6=1
rhombEdgeLengthFace7=1
rhombEdgeLengthFace8=1
rhombEdgeLengthFace9=1
rhombEdgeLengthFace10=1
rhombEdgeLengthFace11=1
rhombEdgeLengthFace12=1
;if useTile is set to true, each face will be encoded into an independent video with a target bitrate set by the rhombFace1Bitrate, rhombFace2Bitrate, etc. parameters
;if useTile is set to false, the whole projection will be encoded into a unique video (all faces on the same frame) and a global bitrate target set by the bitrate parameter
useTile=true
;Relative bitrate of each face (if final layout encoded). A ratio r means r*videoOutputBitRate/12
rhombFace1Bitrate=1
rhombFace5Bitrate=1
rhombFace2Bitrate=0.5
rhombFace3Bitrate=0.5
rhombFace4Bitrate=0.5
rhombFace6Bitrate=0.5
rhombFace7Bitrate=0.5
rhombFace8Bitrate=0.5
rhombFace10Bitrate=0.5
rhombFace11Bitrate=0.5
rhombFace9Bitrate=0.25
rhombFace12Bitrate=0.25
;bitrate is used only if useTile=false. A ratio r means a bitrate of r*videoOutputBitRate
bitrate=1equirectangularTiled layout
[EquirectangularTiledLower]
type=equirectangularTiled
;Size of the reference input equirectangular video
refWidth=4096
refHeight=2048
;use relative resolution for the face resolutions
relativeResolution=true
;The rotation used to get the center of the front face: type = "euler" or "quaternion" or "angleAxis".
; if type is "euler" you have to give the yaw, pitch and roll angle in degree
; if type is "quaternion" you have to give the w, x, y, z value of the quaternion that represente the rotation. A normalization is performed by the software.
; if type is "angleAxis" you have to give an "angle" in degree and a direction vector x, y, z that generates the rotation axis (this vector should not be (0, 0, 0)).
rotation= {"type":"euler", "yaw":0.0, "pitch":0.0, "roll":0.0}
;rotation= {"type":"quaternion", "w":1.0, "x":0.0, "y":0.0, "z":0.0}
;rotation= {"type":"angleAxis", "angle":90, "x":0, "y":0, "z":1}
;nbHTiles (resp. nbVTiles) indicate the number of horizontal (resp. vertical) tiles
;You have to compile the software with the RANGE_NB_H_TILES and RANGE_NB_V_TILES macro to be able to use this layout.
;For instance in this example the value 8 should be in the macro RANGE_NB_H_TILES and RANGE_NB_V_TILES otherwise a runtime error will be raised
nbHTiles=8
nbVTiles=8
;hTileRation_X and vTileRation_X indicate the relative horizontal and vertical ratio of the tile X. In this example each tile get 1/8 of the equirectangular picture
hTileRation_0=1
hTileRation_1=1
hTileRation_2=1
hTileRation_3=1
hTileRation_4=1
hTileRation_5=1
hTileRation_6=1
hTileRation_7=1
vTileRation_0=1
vTileRation_1=1
vTileRation_2=1
vTileRation_3=1
vTileRation_4=1
vTileRation_5=1
vTileRation_6=1
vTileRation_7=1
;if upscale is set to true, all the tiles will be upscaled to the original tile resolution
upscale=false
;if useTile is set to true, each face will be encoded into an independent video with a target bitrate set by the equirectangularTileBitrate_0_0, equirectangularTileBitrate_0_1, etc. parameters
;if useTile is set to false, the whole projection will be encoded into a unique video (all faces on the same frame) and a global bitrate target set by the bitrate parameter
useTile=true
;resolution of the tile (i,j)=(0,0)
equirectangularTile_0_0=1
; bitrate of the tile (i,j) = (0,0). Ratio of r means r*videoOutputBitRate/64
equirectangularTileBitrate_0_0=0.25
equirectangularTile_0_1=1
equirectangularTileBitrate_0_1=0.25
equirectangularTile_0_2=1
equirectangularTileBitrate_0_2=0.25
equirectangularTile_0_3=1
equirectangularTileBitrate_0_3=0.25
equirectangularTile_0_4=1
equirectangularTileBitrate_0_4=0.25
equirectangularTile_0_5=1
equirectangularTileBitrate_0_5=0.25
equirectangularTile_0_6=1
equirectangularTileBitrate_0_6=0.25
equirectangularTile_0_7=1
equirectangularTileBitrate_0_7=0.25
equirectangularTile_1_0=1
equirectangularTileBitrate_1_0=0.25
equirectangularTile_1_1=1
equirectangularTileBitrate_1_1=0.25
equirectangularTile_1_2=1
equirectangularTileBitrate_1_2=0.25
equirectangularTile_1_3=1
equirectangularTileBitrate_1_3=0.25
equirectangularTile_1_4=1
equirectangularTileBitrate_1_4=0.25
equirectangularTile_1_5=1
equirectangularTileBitrate_1_5=0.25
equirectangularTile_1_6=1
equirectangularTileBitrate_1_6=0.25
equirectangularTile_1_7=1
equirectangularTileBitrate_1_7=0.25
equirectangularTile_2_0=1
equirectangularTileBitrate_2_0=0.25
equirectangularTile_2_1=1
equirectangularTileBitrate_2_1=1
equirectangularTile_2_2=1
equirectangularTileBitrate_2_2=1
equirectangularTile_2_3=1
equirectangularTileBitrate_2_3=1
equirectangularTile_2_4=1
equirectangularTileBitrate_2_4=1
equirectangularTile_2_5=1
equirectangularTileBitrate_2_5=1
equirectangularTile_2_6=1
equirectangularTileBitrate_2_6=0.25
equirectangularTile_2_7=1
equirectangularTileBitrate_2_7=0.25
equirectangularTile_3_0=1
equirectangularTileBitrate_3_0=0.25
equirectangularTile_3_1=1
equirectangularTileBitrate_3_1=1
equirectangularTile_3_2=1
equirectangularTileBitrate_3_2=1
equirectangularTile_3_3=1
equirectangularTileBitrate_3_3=1
equirectangularTile_3_4=1
equirectangularTileBitrate_3_4=1
equirectangularTile_3_5=1
equirectangularTileBitrate_3_5=1
equirectangularTile_3_6=1
equirectangularTileBitrate_3_6=0.25
equirectangularTile_3_7=1
equirectangularTileBitrate_3_7=0.25
equirectangularTile_4_0=1
equirectangularTileBitrate_4_0=0.25
equirectangularTile_4_1=1
equirectangularTileBitrate_4_1=1
equirectangularTile_4_2=1
equirectangularTileBitrate_4_2=1
equirectangularTile_4_3=1
equirectangularTileBitrate_4_3=1
equirectangularTile_4_4=1
equirectangularTileBitrate_4_4=1
equirectangularTile_4_5=1
equirectangularTileBitrate_4_5=1
equirectangularTile_4_6=1
equirectangularTileBitrate_4_6=0.25
equirectangularTile_4_7=1
equirectangularTileBitrate_4_7=0.25
equirectangularTile_5_0=1
equirectangularTileBitrate_5_0=0.25
equirectangularTile_5_1=1
equirectangularTileBitrate_5_1=1
equirectangularTile_5_2=1
equirectangularTileBitrate_5_2=1
equirectangularTile_5_3=1
equirectangularTileBitrate_5_3=1
equirectangularTile_5_4=1
equirectangularTileBitrate_5_4=1
equirectangularTile_5_5=1
equirectangularTileBitrate_5_5=1
equirectangularTile_5_6=1
equirectangularTileBitrate_5_6=0.25
equirectangularTile_5_7=1
equirectangularTileBitrate_5_7=0.25
equirectangularTile_6_0=1
equirectangularTileBitrate_6_0=0.25
equirectangularTile_6_1=1
equirectangularTileBitrate_6_1=1
equirectangularTile_6_2=1
equirectangularTileBitrate_6_2=1
equirectangularTile_6_3=1
equirectangularTileBitrate_6_3=1
equirectangularTile_6_4=1
equirectangularTileBitrate_6_4=1
equirectangularTile_6_5=1
equirectangularTileBitrate_6_5=1
equirectangularTile_6_6=1
equirectangularTileBitrate_6_6=0.25
equirectangularTile_6_7=1
equirectangularTileBitrate_6_7=0.25
equirectangularTile_7_0=1
equirectangularTileBitrate_7_0=0.25
equirectangularTile_7_1=1
equirectangularTileBitrate_7_1=0.25
equirectangularTile_7_2=1
equirectangularTileBitrate_7_2=0.25
equirectangularTile_7_3=1
equirectangularTileBitrate_7_3=0.25
equirectangularTile_7_4=1
equirectangularTileBitrate_7_4=0.25
equirectangularTile_7_5=1
equirectangularTileBitrate_7_5=0.25
equirectangularTile_7_6=1
equirectangularTileBitrate_7_6=0.25
equirectangularTile_7_7=1
equirectangularTileBitrate_7_7=0.25
;bitrate is used only if useTile=false. A ratio r means a bitrate of r*videoOutputBitRate
bitrate=1Transformation on the Sphere
For all layout except for the FlatFixed layout, it is possible to plug a transformation the change the viewing direction corresponding to each pixel, and create a kind of warp on the projection. For instance, this is what Facebook Offset Cubemap projection do.
To plug a spherical transformation add on the layout configuration the field:
For the offset transformation:
For the horizontalOffset transformation (perform the offset only in the horizontal plan. Pixels corresponding to viewing direction with the same longitude before the projection will also have the same longitude after the projection [i.e. preserve "vertical" lines]):