如何从 Projection 和 ModelView 矩阵中获取 CATransform3D？

Question

所有，

我有一个 iphone 项目，它使用 OpenGL-ES 为给定的模型视图矩阵和给定的投影矩阵绘制 3D 模型。我需要用 CALayer 替换 3D 模型，因此我将模型视图矩阵的值放入 CATransform3D 结构中并将其分配给layer.transform。它运行良好，图层可见并按预期在屏幕上移动，但一段时间后我意识到我的图层行为不够精确，我应该考虑投影矩阵。然后出现了一个问题：当我简单地连接两个矩阵时，我的层看起来很奇怪（它非常小，大约 2 个像素，而它应该是 300 左右，因为它很远）或者根本不可见。我该如何解决？

这是一段代码：

- (void)adjustImageObjectWithUserInfo:(NSDictionary *)userInfo
{
    NSNumber *objectID = [userInfo objectForKey:kObjectIDKey];
    CALayer *layer = [self.imageLayers objectForKey:objectID];
    if (!layer) { return; }

    CATransform3D transform = CATransform3DIdentity;
    NSArray *modelViewMatrix = [userInfo objectForKey:kModelViewMatrixKey];

      // Get raw model view matrix;
    CGFloat *p = (CGFloat *)&transform;
    for (int i = 0; i < 16; ++i)
    {
        *p = [[modelViewMatrix objectAtIndex:i] floatValue];
        ++p;
    }

      // Rotate around +z for Pi/2 
    transform = CATransform3DConcat(transform, CATransform3DMakeRotation(M_PI_2, 0, 0, 1));

      // Project with projection matrix
    transform = CATransform3DConcat(transform, _projectionMatrix);

    layer.transform = transform; 
}

任何帮助将不胜感激。

Answer 1

我最近遇到了这个确切的问题（我也在使用 ARToolKit），看到您没有找到答案，我感到非常失望。我想你现在已经继续前进了，但我想通了，我将它发布给任何其他可能遇到同样问题的迷失灵魂。

对我来说最令人困惑的是，每个人都在谈论通过将 m34 变量设置为一个小的负数来进行 CALayer 透视变换。尽管这确实有效，但它的信息量并不大。我最终意识到，变换的工作原理与其他变换完全一样，它是齐次坐标的列主变换矩阵。唯一的特殊情况是它必须结合模型视图和投影矩阵，然后在一个矩阵中缩放到 openGL 视口的大小。我首先尝试使用 m34 是一个小的负数的样式的矩阵，如更详细的解释here，但最终切换到打开的 GL 样式透视变换，如here 解释的那样。它们实际上是彼此之间的equivalent，它们只是代表了对转换的不同思考方式。

在我们的例子中，我们试图让 CALayer 变换完全复制一个开放的 GL 变换。所需要的只是将模型视图、投影和缩放矩阵相乘并翻转 y 轴以说明设备屏幕原点位于左上角而 open GL 位于左下角的事实。只要图层锚点在 (.5,.5) 并且它的位置正好在屏幕的中心，结果将与 open GL 的变换相同

void attach_CALayer_to_marker(CATransform3D* transform, Matrix4 modelView, Matrix4 openGL_projection, Vector2 GLViewportSize)
{
//Important: This function assumes that the CA layer has its origin in the 
//exact center of the screen.

Matrix4 flipY = {   1, 0,0,0,
                    0,-1,0,0,
                    0, 0,1,0,
                    0, 0,0,1}; 

//instead of -1 to 1 we want our result to go from -width/2 to width/2, same 
//for height
CGFloat ScreenScale = [[UIScreen mainScreen] scale];
float xscl = GLViewportSize.x/ScreenScale/2;
float yscl = GLViewportSize.y/ScreenScale/2;

//The open GL perspective matrix projects onto a 2x2x2 cube.  To get it onto the
    //device screen it needs to be scaled to the correct size but
//maintaining the aspect ratio specified by the open GL window.
    Matrix4 scalingMatrix = {xscl,0   ,0,0,
                               0,   yscl,0,0,
                               0,   0   ,1,0,
                               0,   0   ,0,1};

//Open GL measures y from the bottom and CALayers measure from the top so at the
//end the entire projection must be flipped over the xz plane.
//When that happens the contents of the CALayer will get flipped upside down.
//To correct for that they are flipped updside down at the very beginning,
//they will then be flipped right side up at the end.

Matrix flipped = MatrixMakeFromProduct(modelView, flipY);
Matrix unscaled = MatrixMakeFromProduct(openGL_projection, flipped);
Matrix scaled = MatrixMakeFromProduct(scalingMatrix, unscaled);

//flip over xz plane to move origin to bottom instead of top
Matrix Final = SiMatrix4MakeFromProduct(flipY, scaled);
*transform = convert_your_matrix_object_to_CATransform3D(Final);
}

此函数采用打开的 GLprojection 和 openGL 视图大小，并使用它们为 CALayer 生成正确的变换。 CALayer 大小应以 open GL 场景的单位指定。 OpenGL 视口实际上包含 4 个变量，[xoffset,yoffset,x,y] 但前两个无关紧要，因为 CALayer 的 Origin 位于屏幕中心以对应 OpenGL 3d Origin。

只需将 Matrix 替换为您可以访问的任何通用 4x4 列主要矩阵类。只要确保你以正确的顺序乘以你的矩阵，任何事情都会奏效。所有这一切本质上都是在复制 OpenGL 管道（减去裁剪）。

Answer 2

我刚刚摆脱了投影矩阵，这是我得到的最好的变体：

- (void)adjustTransformationOfLayerWithMarkerId:(NSNumber *)markerId forModelViewMatrix:(NSArray *)modelViewMatrix
{   
    CALayer *layer = [self.imageLayers objectForKey:markerId];

            ...

    CATransform3D transform = CATransform3DIdentity;        
    CGFloat *p = (CGFloat *)&transform;
    for (int i = 0; i < 16; ++i) {
        *p = [[modelViewMatrix objectAtIndex:i] floatValue];
        ++p;
    }

    transform.m44 = (transform.m43 > 0) ? transform.m43/kZDistanceWithoutDistortion : 1;

    CGFloat angle = -M_PI_2;
    if (self.delegate.interfaceOrientation == UIInterfaceOrientationLandscapeLeft)      { angle = M_PI; }
    if (self.delegate.interfaceOrientation == UIInterfaceOrientationLandscapeRight)     { angle = 0; }
    if (self.delegate.interfaceOrientation == UIInterfaceOrientationPortraitUpsideDown) { angle = M_PI_2; }
    transform = CATransform3DConcat(transform, CATransform3DMakeRotation(angle, 0, 0, -1));

    transform = CATransform3DConcat(CATransform3DMakeScale(-1, 1, 1), transform);

        // Normalize transformation
    CGFloat scaleFactor = 1.0f / transform.m44;
    transform.m41 = transform.m41 * scaleFactor;
    transform.m42 = transform.m42 * scaleFactor;
    transform.m43 = transform.m43 * scaleFactor;
    transform = CATransform3DScale(transform, scaleFactor, scaleFactor, scaleFactor);
    transform.m44 = 1;

    BOOL disableAction = YES;

            ...

    [CATransaction begin];
    [CATransaction setDisableActions:disableAction]; // Disable animation for layer to move faster
    layer.transform = transform;                
    [CATransaction commit];
}

它不是绝对精确，但对于我的目的来说已经足够精确了。当 x 或 y 位移大约为屏幕尺寸时，偏转变得明显。

Answer 3

两个可能的问题：

1) 连接顺序。经典矩阵数学是从右到左。所以试试

CATransform3DConcat(_projectionMatrix, transform)

，和

2) 投影系数的值是错误的。您使用的值是什么？

希望这会有所帮助。

Answer 4

您是否考虑到图层渲染到已经应用了正射投影矩阵的 GL 上下文中？

在 Mac 上查看介绍性评论；这个类在 iPhone 上是私有的，但是原理是一样的。

此外，与 CATransform3D 相比，OpenGL 矩阵在内存中进行了转置。也要考虑到这一点；虽然大多数结果看起来都一样，但有些则不然。

Answer 5

作为问题的直接答案：投影矩阵旨在输出 -1 ... 1 范围内的内容，而 CoreAnimation 通常适用于像素。烦恼由此而来。所以要解决这个问题，你应该在将模型视图乘以投影矩阵之前，减少你的模型以很好地适应 -1 ... 1 范围（取决于你的世界是你可以将它划分为 bounds.size）和投影矩阵乘法之后你可以返回像素。

所以下面是一个Swift中的sn-p代码（我个人很讨厌Swift，但是我的客户喜欢它）相信可以看懂。它是为 CoreAnimation 和 ARKit 编写的，并测试了它的工作原理。我希望 ARKit 矩阵可以被认为是 opengl 的

func initCATransform3D(_ t_ : float4x4) -> CATransform3D
{
    let t = t_.transpose //surprise m43 means 4th column 3rd row, thanks to Apple for amazing documenation for CATransform3D and total disregarding standard math with Mij notation
    //surprise: at Apple didn't care about CA & ARKit compatibility so no any easier way to do this
    return CATransform3D(
        m11: CGFloat(t[0][0]), m12: CGFloat(t[1][0]), m13: CGFloat(t[2][0]), m14: CGFloat(t[3][0]),
        m21: CGFloat(t[0][1]), m22: CGFloat(t[1][1]), m23: CGFloat(t[2][1]), m24: CGFloat(t[3][1]),
        m31: CGFloat(t[0][2]), m32: CGFloat(t[1][2]), m33: CGFloat(t[2][2]), m34: CGFloat(t[3][2]),
        m41: CGFloat(t[0][3]), m42: CGFloat(t[1][3]), m43: CGFloat(t[2][3]), m44: CGFloat(t[3][3])
    )
}

override func updateAnchors(frame: ARFrame) {
    for animView in stickerViews {
        guard let anchor = animView.anchor else { continue }

        //100 here to make object smaller... on input it's in pixels, but we want it to be more real.
        //we work in meters at this place
        //surprise: nevertheless the matrices are column-major they are inited in "transposed" form because arrays/columns are written in horizontal direction
        let mx = float4x4(
            [1/Float(100), 0, 0, 0],
            [0, -1/Float(100), 0, 0],    //flip Y axis; it's directed up in 3d world while for CA on iOS it's directed down
            [0, 0, 1, 0],
            [0, 0, 0, 1]
        )

        let simd_atr = anchor.transform * mx //* matrix_scale(1/Float(bounds.size.height), matrix_identity_float4x4)
        var atr = initCATransform3D(simd_atr)   //atr = anchor transform

        let camera = frame.camera
        let view = initCATransform3D(camera.viewMatrix(for: .landscapeRight))
        let proj = initCATransform3D(camera.projectionMatrix(for: .landscapeRight, viewportSize: camera.imageResolution, zNear: 0.01, zFar: 1000))

        //surprise: CATransform3DConcat(a,b) is equal to mathematical b*a, documentation in apple is wrong
        //for fun it's distinct from glsl or opengl multiplication order
        atr = CATransform3DConcat(CATransform3DConcat(atr, view), proj)
        let norm = CATransform3DMakeScale(0.5, -0.5, 1)     //on iOS Y axis is directed down, but we flipped it formerly, so return back!
        let shift = CATransform3DMakeTranslation(1, -1, 0)  //we should shift it to another end of projection matrix output before flipping
        let screen_scale = CATransform3DMakeScale(bounds.size.width, bounds.size.height, 1)
        atr = CATransform3DConcat(CATransform3DConcat(atr, shift), norm)
        atr = CATransform3DConcat(atr, CATransform3DMakeAffineTransform(frame.displayTransform(for: self.orientation(), viewportSize: bounds.size)));
        atr = CATransform3DConcat(atr, screen_scale)    //scale back to pixels

        //printCATransform(atr)
        //we assume center is in 0,0 i.e. formerly there was animView.layer.center = CGPoint(x:0, y:0)
        animView.layer.transform = atr
    }
}

附：我认为对于使用左右坐标系、Y 和 Z 轴方向、列主要矩阵、浮点和 CGFloat 不兼容以及缺乏 CA 和 ARKit 集成创建所有可能混合的开发人员来说，地狱中的一个地方将特别热门。 .