COMP30019 Graphics and Interaction Perspective Geometry

Transcription

1 COMP30019 Graphics and Interaction Perspective Geometry Department of Computing and Information Systems The

2 Lecture outline Introduction to perspective geometry Perspective Geometry Virtual camera Centre of projection Human perspective

3 Perspective geometry How are three-dimensional objects projected onto two-dimensional images? Aim: understand point-of-view, projective geometry. Reading: Foley Sections 6.1 to 6.4 (excluding example 6.1, we ll cover matrices later). Akenine-Moller Section 2.3.

4 Viewing

5 Viewport

6 Geometry of image formation Mapping from 3D space to 2D image surface, more specifically, a mapping from 3D directions (rays to/ from the observer). You can think perspective as a transformation as a way of moving from a higher dimensional image to a lower dimensional form. The X, Y, Z points in the three dimensional world, sometimes called voxels, are transformed in to x, y pixels in a two-dimensional image. Simplest device that does this is the pin-hole camera that gives perspective projection. Practical cameras with lenses ideally give the same projection, aside from greater light gathering, and issues like focus.

7 Pinhole Camera projection screen for image (maybe translucent waxed paper) light ray from object image of object (upside down) pinhole in box light-tight box object in 3D scene

8 Perspective geometry (X,Y,Z) f X x O Z

9 Perspective geometry Basically an abstraction of pin-hole camera. Look at XOZ plane (same thing happens in YOZ plane). Actual point in 3D space is (X, Y, Z ) 0 is origin (focal point) or centre of projection. Z is distance from actual point to origin. f is focal distance (focal length). x is the image (upside down) with respect to real world.

10 Virtual camera

11 Virtual camera geometry (X,Y,Z) f X x O Z

12 Virtual camera geometry Image projection surface imagined to be in front of projection centre. Geometrically equivalent Often more convenient to think about projection

13 Perspective Formulas Point P = (X, Y, Z ) in 3D space has projection (x, y) in the image where or x f y f = X Z = Y Z x = Xf Z y = Yf Z f being the focal distance (sometimes f is called d). Look at similar triangles in the previous diagram.

14 Perspective Formulas Look at perspective projection diagram to convince yourself of this triangles xof and XOZ have the same proportions. Rearranging gives equations shown below. These formulas apply only for this special coordinate system, sometimes called camera-centred coordinates, for which perspective projection has a particularly simple form. For other coordinate systems, some 3D transformation will be necessary (see later).

15 Camera transformation

16 Centre of projection A A Projectors Center of projection A' Figure 6.03 (a) B' B Projection plane Center of projection at infinity Projectors (b) A' B' B Projection plane Foley,

17 One point perspective projection (Foley, Figure 6.04) z-axis vanishing point y y z-axis vanishing point z x z x

18 One-point perspective projection (Foley, Figure 6.05) Projection plane y Center of projection z x Projection plane normal

19 Two-point perspective

20 Three-point perspective

21 Vanishing points In 3D, parallel lines meet only at infinity, so the vanishing point can be thought of as the projection of a point at infinity. If the set of lines is parallel to one of the three principal axes, the vanishing point is called an axis vanishing point. So called one-point, two-point, and three-point perspectives are just special cases of perspective projection, depending on how image plane lines up with significant planes in scene. Talking about these cases specifically is mainly an artifact of artists or architects dealing with horizontals and verticals in built environments. In fact, there are an infinity of vanishing points, one for each of the infinity of directions in which a line can be oriented.

22 House example (Foley Section 6.4) y (0, 10, 54) (8, 16, 30) (16, 10, 30) (16, 0, 30) x z (16, 0, 54)

23 One-point, centred perspective projection example y x v Foley Figures 6.21 and 6.22 VRP z n VUP VPN DOP PRP = (8, 6, 30) CW u Window on view plane

24 Exercise Which of the below is the centre of project in Foley Figure 6.22? VRP (view reference point) PRP (projection reference point) VPN (view plane normal) DOP (direction of projection) VUP (view-up vector) Is the view plane inbetween the centre of projection and the house or behind the centre of projection?

25 Two-point perspective projection example In a two-point projection of a house, left, the viewplane (defined by the view plane normal, VPN), right, cuts the z and x axes (Foley Figures 6.17 and 6.25). y View plane v x u z VPN

26 Parallel projection Parallel projection introduces no perspective distortion - centre of projection plane (focal point) is at infinity Along with its variants it is useful in engineering drawings, where measurements must be taken. oblique projection if view plane is not perpendicular to projection.

27 Geometric project classes Subclasses of planar geometric projections (Foley Figure 6.10). Planar geometric projections Parallel Perspective Orthographic Oblique One-point Top (plan) Cabinet Front elevation Side Axonometric Cavalier elevation Other Two-point Three-point Isometric Other

28 Perspective of the human eye Human eye effectively uses a kind of spherical projection: Retina is curved, though projection centre (in lens) isn t at centre of the eyeball (therefore not planar geometric projection). Doesn t exactly match perspective projection. Only a problem for very wide fields of view. Perspective is basically the right projection for putting a 3D scene onto a flat surface for human viewing. Other projections are possible for special effects, e.g. fish-eye lens.

29 Summary Perspective geometry is based loosely on the pin-hole camera model that maps 3D points onto a 2D image plane The image plane may thought of either behind a focal point or in between a vanishing point and the object. Computer graphics largely concerns planar geometric projections, generally perspective projection and sometimes parallel projection for specific applications. One-point, two-point and three-point projection variants arise according to how many times the viewplane cuts the axis planes.