The previous article reviewed the concepts of space, ing, and transformation. Next, we will introduce the affine, bilinear, and perspective commonly used in image processing. As discussed below
Two-dimensional.
Affine
(Affine)An affine transformation is a linear transformation. After the transformation, the image will be straight (straight line or straight line) and parallel (parallel line after transformation ). Affine transformations are implemented through a series of atomic transformations: translation, scaling, flip, rotation, and sheer ). In fact, affination is performed in the affined space, rather than the vector space. Points in the affined space are represented as a datum point and a vector's sum. Here we do not introduce the affined space, see the http://courseware.ecnudec.com/zsb/zsx/Zsx08/ZSX089/ZSX08901/zsx089010.htm for the one-dimensional situation, there is an affined transform f (x) = ax + B two-dimensional general form: V = MW + B V, W are two-dimensional spatial coordinates, M is the transformation matrix [A0 A1] [B0 B1], so there is a general form of [x'] = [a0x + a1y + A2] [y'] [b0x + b1y + B2] Translation: M = [1 0] B is [dx] [0 1] [dy] rotation: M = [cos θ-sin θ] [sin θ cos θ] Twist M = [1 UX] [Uy 1] scaling/scaling (scaling when S1 = S2) M = [S1 0] [0 s2] for image processing The mark is not necessarily an integer, so grayscale interpolation is required.
Orthogonal
(Orthogonal)The inner product of two vectors in a linear space (point Multiplication can be understood as ry) is 0, which is called orthogonal. For example, in ry, the two vertical vectors are orthogonal, and we can see that the orthogonal vector group has linear independence. The transformation applied by orthogonal matrix is called orthogonal transformation. Fourier transformation, cosine transformation, and the power-Hadamard transformation are all orthogonal transformations. For more information, see Space Transformation
Bilinear
(Bilinear)If F (k1a1 + k2a2, B) = k1 F (A1, B) + k2f (A2, B) and F (, k1b1 + k2b2) = k1f (A, B1) + k2f (A, B2) is called F as a bilinear function on V. Common bilinear function form: F (x, y) = A0 + a1x + a2y + a3xy. For example, this form is used in bilinear interpolation of images.
Perspective
(Perspective)Perspective transformation is a geometric transformation. For example, in Mobile Robot Vision Research, the camera and the ground have an angle, rather than vertical projection (positive projection), you need to use perspective transformation to correct it into a positive projection. Has the following forms:
