Imaging and 2D Transforms

Transcription

1 Imaging and 2D Transforms Lecture 3 Comp 236 Spring 2005 Two-Dimensional Geometric Transforms Functions for mapping points from one place to another Geometric transforms can be applied to drawing primitives (lines, conics, triangles) pixel coordinates of an image 2/2/2005 Lecture 3 2 Translation Translations have the following form x= x + t x y= y + t y For every translation there exists an inverse function which undoes the translation. In our case the inverse looks like: x = x- t x y = y- t y There also exists a special translation, called the identity, that leaves every point unchanged. x= x + 0 y= y + 0 Groups and Composition For Translations: 1. There exists an inverse mapping for each function 2.There exists an identity mapping These properties might seem trivial at first glance, but they are actually very important, because when these conditions are shown for any class of functions it can be proven that such a class is closed under composition (i.e. any series of translations can be composed to a single translation). In mathematical parlance this the same as saying that translations form an algebraic group. 2/2/2005 Lecture 3 3 2/2/2005 Lecture 3 4

2 2D Rotations Another group of 2-transforms are the rotations about the origin. Euclidean Transforms The combination of translations and rotation functions defines the Euclidean Group Properties of Euclidean Transformations: They preserve distances They preserve angles How do you represent these functions? 2/2/2005 Lecture 3 5 2/2/2005 Lecture 3 6 Problems with this Form Must consider Translation and Rotation separately Computing the inverse transform involves multiple steps Order matters between the R and T parts These problem can be remedied by considering our 2 dimensional plane as a subspace within 3D. Choosing a Subspace We could use any planar subspace as long as it does not contain the origin WLOG assume the our 2D space of points lies on the 3D plane z = 1. Now we can express all Euclidean Transforms in matrix form: This gives a three parameter group of Transformations. 2/2/2005 Lecture 3 7 2/2/2005 Lecture 3 8

3 Playing with Euclidean Transforms In what order are the translation and rotation performed? Will this family of transforms always generate points on our chosen 3-D plane? Why? Similitude Transforms We can define a 4-parameter superset of Euclidean Transforms with additional capabilities Properties of Similtudes: Distances between points are changed by a fixed ratio Angles are preserved Maintains a Similar shape (similar triangles, circles map to circles, etc.) 2/2/2005 Lecture 3 9 2/2/2005 Lecture 3 10 Playing with Similitude Transforms Adds reflections Scales in x and y must be the same. Why? Order? Will this family of transforms always generate points on the chosen plane? Why? Affine Transformations A 6-parameter group of transforms 2/2/2005 Lecture /2/2005 Lecture 3 12

4 Affine Properties To the right is a simple illustration of how we can map our parameters into a six arbitrary numbers Properties of Affine Transforms: They preserve our selected plane (sometimes called the Affine plane) They preserve parallel lines Playing with Affine Transforms σ x scales the x-dimension σ y scales the y-dimension σ xy is often called the skew parameter 2/2/2005 Lecture /2/2005 Lecture 3 14 Determining Affine Transforms The coordinates of three corresponding points (non-colinear) uniquely determine an Affine Transform We set up 6 linear equations in terms of our 6 unknown values. In this case, we know the coordinates before and after the mapping, and we wish to solve for the entries in our Affine transform matrix. Solution Method If we know where we would like at least three points to map to, we can solve for an Affine transform that will give this mapping. This gives the following solution: 2/2/2005 Lecture /2/2005 Lecture 3 16

5 Projective Transformations The most general linear transformation that we can apply to 2-D points There is something different about this group of transformations. The result will not necessarily lie on our selected plane. Since our world (to this point) is 2D we need some way to deal with this. Projective Space The mapping of points from an N-D space to a M-D subspace (M < N) We need a rule for mapping points resulting from this transform back onto our plane z = 1. We will identify all points on lines through the origin of the 3-D space, and prefer the one that intersects our selected plane. Thus, Since the origin lies on all of these lines (and thus cannot be uniquely specified) we will disallow it. This is no big loss since it wasn t on our selected plane anyway (This is the real reason that we chose a plane not containing the origin). If we want to find the coordinate of any point in our selected plane we need only scale it such that its third coordinate, w, is 1. 2/2/2005 Lecture /2/2005 Lecture 3 18 Projective Transforms Since all of the resulting points are defined to within a non-zero scale factor. We can also scale the transformation by an arbitrary number and it will still give the same result. Degrees of Freedom A projective transform has 8 free-parameters Which can be expressed as the following linear rational equation: We might as well choose α so that one of the parameters of our matrix is 1 (i.e. p 33 = 1). Rearranging terms gives a linear expression in the coefficients: 2/2/2005 Lecture /2/2005 Lecture 3 20

6 2/2/2005 Lecture /2/2005 Lecture 3 24 Specifying a projective transform A projective transform can be uniquely defined by how it maps 4 points Projective Examples 2/2/2005 Lecture /2/2005 Lecture 3 22 Where are we going? A preview of assignment #1: A Recent Interest Geological Information Systems (GIS):

7 To get a head start Find GIS image data of some place you like in at least formats Topological Maps Watersheds Elevation Data Aerial Overheads Get them at a reasonably high resolution At least 1024 by 1024 Learn a tool for changing image formats 2/2/2005 Lecture 3 25 OpenGL Imaging gldrawpixels() Writes an array of pixels to the framebuffer glreadpixels() Reads an region of the framebuffer into an array of pixels in main memory glcopypixels() Copies a region from one part of the framebuffer to another glrasterpos*() Sets the current drawing position for gldrawpixels() and destination position of glcopypixels() 2/2/2005 Lecture 3 26 Loading an Image OpenGL Setup Code to read an image file into an array of unsigned bytes import Image class GLImage: def load(self, filename): global width, height self.name = filename im = Image.open(filename) im = im.transpose(image.flip_top_bottom) self.width, self.height = im.size self.data = im.tostring() self.x = (width - self.width)/2 self.y = (height - self.height)/2 self.channels = len(im.getbands()) return self Here we use gluortho2d() to make our world coordinates match the screen coordinates. This is typical when you do pixel imaging. def initialize(): glviewport(0, 0, width, height); glclearcolor(0, 0, 0, 1); glmatrixmode(gl_projection); glloadidentity(); gluortho2d(0, width, 0, height); glmatrixmode(gl_modelview); glloadidentity(); 2/2/2005 Lecture /2/2005 Lecture 3 28

8 Displaying an Image Here s code to display a list of images def display(): glclearcolor(0, 0, 0, 1) glclear(gl_color_buffer_bit) for im in imagelist: glrasterpos2i(im.x,im.y); if im.channels == 3: gldrawpixels(im.width, im.height, GL_RGB, GL_UNSIGNED_BYTE, im.data) else: gldrawpixels(im.width, im.height, GL_RGBA, GL_UNSIGNED_BYTE, im.data) glflush() 2/2/2005 Lecture 3 29 Next Time We ll make it do something interesting and touch on textures 2/2/2005 Lecture 3 30