8.1. Principles of Photogrammetry

Photogrammetry is governed by principles of projective geometry. It relies on the formation of images through the perspective projection of a three-dimensional object onto a two-dimensional image plane.

Photogrammetry is built on projective geometry, a mathematical framework that helps us understand how images are formed. The main concept here is that when we take a photograph of a three-dimensional object, this object is reduced to a two-dimensional representation on a photo. This process involves perspective projection, where light rays travel from points of the object to the camera lens, creating an image. Think of it as looking at a sculpture in a museum; while you see it in full three dimensions, a photograph will only capture its height and width, without depth.

Imagine holding a cube in front of you and taking a picture of it. The picture will show the front face of the cube, making it look flat. You lose the depth of the cube (the third dimension) in the photo, similar to how we see a 2D drawing of a 3D object. This is exactly how photogrammetry captures the essence of an object: by compressing its three-dimensional form into a flat image.

• The fundamental principle involves capturing an image through a single exposure station where light rays from object points converge to a point (lens or pinhole). • The image formed is a perspective view, leading to geometric distortions that must be corrected for accurate measurements.

Central Projection is a key idea in photogrammetry. When we capture an image from a single point, such as through a camera lens, all light rays that come from the different points on the object focus on a single point inside the camera (often called the pinhole). As a result, the photo taken shows a perspective view of the scene. However, this method can introduce distortion; if we do not correct these distortions, the measurements we take from the image may not be accurate. For instance, objects that are far from the camera may appear smaller than those that are closer, which is a geometric distortion that needs addressing.

Think about looking at a building from the sidewalk directly in front of it. The top of the building appears smaller than the base because of your angle of view. If you were to photograph it from ground level, it wouldn't truly reflect how towering it is compared to your height; it would look shorter. Photogrammetry must correct this by understanding these relationships of perspective—just like how artists add depth to a painting.

• Relationship between object space and image space is defined through collinearity equations. • These equations relate coordinates of object points with image points, camera focal length, and orientation parameters.

Perspective Geometry in photogrammetry refers to the mathematical relationships that help translate coordinates from the real world (object space) into the flat image (image space). The collinearity equations are crucial here because they establish how a point in 3D space corresponds to a point in the 2D image captured by the camera. This relationship involves several factors: the focal length of the camera, meaning how zoomed in or out the image will appear, and the orientation of the camera at the time of capturing the image, which affects the angle at which the image is taken.

Imagine trying to create a map of a park by measuring distances between trees and landmarks. You take pictures from different angles, and you want to ensure your map reflects the actual layout. The relationships defined by perspective geometry are like using a compass and ruler to ensure everything is correctly marked in your drawing. Without this geometric understanding, your map might not accurately represent the real-world distances and positions.

--