Image formation & Geometrical Transforms Francisco Gómez J MMS U. Central y UJTL

Digital images Arrays of numbers Numbers can represent: – Intensity (gray level) – Range – X-ray absortion coefficient – Amount of light

Human eye Color sensors Intensity sensors 100,0000,000

Distribution of cones and rods

Sensor simplified Pupil Limit amount of light and incidence angle Lens Focus the light to a single image point Fotosensitive surface Film

Digital cameras – sensor type

Sensor type CCD type (Charge coupled device) Charge is accumulated during exposure CMOS type (complementary metal oxide on silicon) Light affects the conductivity (or gain) of each photodetector Pixel! To take into account: Number and size of sensor elements – Chip size – ADC resolution

Pinhole camera The pinhole camera and camera obscura principle illustrated in 1925, in The Boy Scientist.

Pinhole camera The «Camera obscura» was used by renaissance paienters to help to understand Perspective projection

Pinhole and lens model Pinhole Given a projection plane parallel to X-Y located at distance f where the point (p1,p2,p3) is going to be projected, i.e, which is the value for (u,v)? u=(p 1 /p 3 )f and v=(p 2 /p 3 )f

Pinhole and lens model This is the CCD sensor units in mm Image plane coordinate This is the real object in m

Example Camera – Focal length: 5mm You have an scene point located at (1m,2m,5m) – Where at the image plane coordinate this point is going to be located? – If the image plane is 10mm,10mm which is the FoV? – A building is 100m wide. How far away do we have to be in order that it fills the field of view? (0,0) Optical center (cx,cy) Optical center

Image buffer Optical center digital image Pixel Image Origin

Image Buffer

Example A camera observes a rectangle 2m away – The rectangle is known to be 50 cm x 30 cm – If the image in the rectangle measures 60 x 15 pixels Where is located the focal length in pixels?

Other kind of sensors (Color camera)

Other kind of sensors (Kinect)

Other kind of sensors

Other kind of sensors

Other kind of sensors (MRI)

Other kind of sensors (MRI)

Other kind of sensors (MRI)

Other kind of sensors

All is about tranform between frames

2D to 2d Transforms (Rigid) (1,1)(?,?) 2 2 xx (x’,y’) (?,?) yy  Preserves shape and size Number of degrees of freedom (3): t=(  x,  y) and 

2D to 2d Transforms (Euclidean - Rigid)  R is orthonormal Transpose is the inverse How to invert the transform?

Homogeneous coordinates - Homogeneous coordinates simply add an extra element 1 -If during operations the third element is different of 1 divide by this number -This representation is quite convenient to represent transformations to

Homogeneous coordinates

Example Transform the image point (10,40) using a rotation of 90 degrees and a translation of (15,-60)

2D to 2d Transforms (Similarity - Scaled) Preserve angles but not distances

2D to 2d Transforms (Afinne) Models rotation, translation, scaling, shearing, and reflection

Example I = imread('cameraman.tif'); tform = maketform('affine',[1 0 0; ; 0 0 1]); J = imtransform(I,tform); imshow(I), figure, imshow(J)

2D to 2d Transforms (Afinne) Models rotation, translation, scaling, shearing, and reflection

2D to 2d Transforms (Projective - Homography)

What preserves?

3d to 3d Transforms Coordinate frames – {A}, {B} How to describe points in {B} respect to {A} – We need a t – Rotation matrix R

3d to 3d Transforms (Rotation) 3 degrees of freedom! But we need a convention

Rotations around the axis

Rotation around the axis A rotation matrix can be expressed using a 3 x3 matrix Rotation in X Rotation in YRotation in Z

Rotational transformation R represents a rotational transformation of frame A to frame B From Point represented in frame B Point represented in frame A

Transforming a 3d point

In homogeneous coordinates Homogeneous What is the inverse?

Example Let p=(2,2,5) in frame A Frame B is located at (10,-5,6) and rotated 65 degrees around z-axis with respect to frame A Where is point p located respect to frame B?

Transformations are maps AB Transformation are answers to the question: If there are points in A how they can be written in B?

Small angle approximation sin(  sin(  Useful if you want to rotate objects in a video

Recovering angles

3d to 2d Transforms Projects 3d points into 2d

Perspective projection – Intrinsic matrix Optical center (cx,cy) How can we write the perspective projection as a transform? or How to project 3D points represented in the coordinate system attached to the camera, to the 2D image plane?

Extrinsic camera If 3D points are in world coordinates, we first need to transform them to camera coordinates We can write this as an extrinsic camera matrix, that does the rotation and translation, then a projection from 3D to 2D

World to camera

What about color?

Images in matlab

RGB color space

Hue Purity Luminosity