The Art, Science and Algorithms Rays of Scene to pixels of - PDF document

• General topic for next couple of weeks • Cameras : Pinhole Camera and Optics The Art, Science and Algorithms – Rays of Scene to pixels of Photography – Camera without optics – Lens in the camera system Lenses & Depth of Field (DOF) – The Lens Equation Optics I CSCI 4900/6900 Maria Hybinette Cameras captures Rays of Light à Get it ta a sensor { Film or Digital} and that creates a photograph 1 2 Camera & Rays of Light History-Science-Art of the Camera • Mathematicians, scientists (or artists • Recall: Context of Computational deeply scientifically motivated) have been Photography the key pushers of the advancement of • How we capture a 3D the camera scene into 2D array of • Idea has been around about the 4 th , 5 th pixels • Rays are fundamental and 6 th Century B.C. (‘aperture’) primitives • Illumination (Light Rays) – Chines philosopher Mo-Ti “the collecting follows a path from the plate”, “locked treasure room”; source to the scene – Aristotle & Euclid’s ‘Optica’ talked about a – Geometry to extract pinhole camera, or the camera obscura ) information of the scene • Computation can – Byzantine mathematician Anthemius of control the parameters Tralles used a type of camera obscura in his of the optics, sensor and experiments. 6th • Rays to Pixels illumination Camera = Vaulted room h@p://en.wikipedia.org/wiki/History_of_photography Obscura = Dark h@p://www.moGobscura.com/moG_about.html Camera Obscura = Darkroom 3 Slide: Irfan Essa 4 Maria Hybinette Camera Obscura (pinhole camera) • Abu Ali al-Haytham, mathematician (around 1000 AD) v Lights streams of particles travel in straight lines • Johannes Kepler (“ Astronomiae Pars Optica” (1604)) v Corrected theory on how camera obscura operated v Inverse square laws of light v Astronomical laws v Human optics h@p://en.wikipedia.org/wiki/Johannes_Kepler h@p://arts.jrank.org/pages/9526/Camera-obscura.html v Later “ Dioptrice “1611- telescope 5 Maria Hybinette 6 Maria Hybinette

Other (Drawing) Mechanisms Camera as Art • More recently: Tim’s Vermeer • https://en.wikipedia.org/wiki/Tim %27s_Vermeer Law of 1 pt AlberG’s AlberG’s Dürer's Silhoue@e Camera Lucida PerspecGve “ArGst’s “Grid” or Perspectograph Machine, William Hyde GilberG, Glass” “Veil” 1525-1538 1780 Wollaston Brunelleschi 1450 1450 1807 1413-1425 (Kepler Inspired) h@p://en.wikipedia.org/wiki/PerspecGve_(graphical) Camera = Vaulted room Lucida= Lighted, Lit, Shining h@p://www.acmi.net.au/AIC/DRAWING_MACHINES.html Camera Lucida= Lighted room 7 Maria Hybinette 8 Evolution of the Camera Single-Lens Reflex Camera • Mirrors that direct light from the lens to the viewfinder : – view through the lens and see exactly what will be captured “before taking • Behind the Camera • Today picture • Single Lens vs Non Single Evolution BC 1839 1907 1948 Lens – View finder with its own lens. • Reflex : Mirror Reflect the a portion of the light to the viewfinder Formal to Casual Figure: h@p://electronics.howstuffworks.com/ camera5.htm Slide Adapted from Irfan Essa 9 10 Capturing Rays: Objective and Tools Try to capture When you take a picture 3D Scene Illumination Optics Sensor Geometry Light Processing (Perspective) Scattering Display Using: User • Objective: Light source – we want to capture reflection from the scene that is illuminated by the source (sun) • Scenes must be illuminated must have a lightsource Sensor / Color Slide: Irfan Essa Optics / Lens Filter 11 12

Add a Sensor • Add a barrier • Add a Sensor Film or Sensor • Do we get a reasonable image. 13 14 Camera Obscura Sensor • Rays of light come straight from each • Image point of an object Camera = Vaulted room 15 16 Why not use Sensors without Optics ? • 1 Pinhole: Rays of • Theoretically , light come straight • No distortion: from each point of an object Straight Lines – no distortion. remain straight • Infinite depth of – straight lines are still straight field: Everything – infinite DOF Byelorussky Station: commons.wikimedia.org in focus (but • Larger Pinhole there is optical – Fuzzy image, due to lurring) DOF geometric blur h@p://www.pracGcalphysics.org/go/Guidance_93.html 17 18

Larger Pinhole Smaller Pinhole • Diffraction limit, smaller apertures means more diffraction • Small hole does not create a bright dot but a diffused circular disk, called an Airy’s disc, surrounded by concentric circular rings • Geometric Blur : Aperture h@p://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/cirapp2.html 20 h@p://www.huecandela.com/hue-x/pin-pdf/Prober-%20Wellman.pdf 19 Pinhole Size Summary • Large pinhole gives • Large Pinhole = geometric blur (2mm) • Optimal pinhole gives Geometric Blur • Small Pinhole = little light (0.35mm) – Maximum sharpness, Diffraction Blur aperture is proportional to its • Best Pinhole = distance from the Very Little Light image plane. • Small pinhole gives diffraction blur (0.07mm) Hecht & Ganesan “OpGcs 4th ed.” p205 21 22 Replacing pinhole with a lens • For d (pinhole diameter), � r 1 f (distance from d = 2 2 f π pinhole to sensor), and • π (wavelength of light): π f 23 24

Replacing pinhole with a lens Geometrical Optics • 1 Pinhole: Rays of light come • Parallel rays converge to a point located at focal straight from each point of an length f from the lens, � object its principal focal point – no distortion, – straight lines are still straight – infinite DOF • ‘ Center’ rays going though center of lens are • Larger Pinhole fuzziness not deviated – so points are shown at the same • Lens : Need to collect rays emanating from a ‘near’ point in the scene through ‘ different pinholes ’, so that they so converge at a point at perspective the sensor. flange f in camera 25 26 Gauss’s ray tracing construction Same Lens : Changing Focus Distance f f • Focus distance (from Principal Focal Point camera to the object in scene). – To focus on objects at different focus distances, move the sensor relative to the lens. Scene Inside Camera • Focused at infinite à Parallel rays -- • Rays coming from points on a plane parallel to Converges at the principal focus the lens are focused on another plane parallel point • As subject gets closer: to the lens • distances decreases to object, • distance increases to image 28 Marc Levoy 2010 27 • Further away from the prin. fp.* Deriving the Lens Formula Deriving Lens Formula • Rays from points on a plane parallel to the lens, focuses on a plane parallel to the lens on the other side (and upside down). • Flip d 0 to other side so we can relate it to • Lens Equation: Relates distances: to object, to the other parameters à blue triangle image and focal length • Relate lines d 0 , d 1 to sides A, B by looking • Step 1: Flip d 0 to other side so we can relate it to at resulting green triangles derived the other parameters à create blue triangle… from blue triangles (verify they are similar). 29 30

Deriving Lens Formula Deriving Lens Formula • Flip d 0 to other side so we can relate it to the other • Flip d 0 to other side so we can relate it to the other parameters à blue triangle parameters à blue triangle • Relate lines d 0 , d 1 to sides A, B by looking at • Relate lines d 0 , d 1 to sides A, B by looking at resulting green triangles derived from blue resulting green triangles derived from blue triangles (verify they are similar). triangles (verify they are similar). • NOW Relate principal focal length f to A/B by • Relate principal focal length f to A/B by finding finding new similar triangles à red/pink triangles new similar triangles à red/pink triangles 31 32 Deriving Lens Formula Deriving Lens Formula: f, d 0 , d 1 • Flip d 0 to other side so we can relate it to the other • Flip d 0 to other side so we can relate it to the other parameters à parameters à blue triangle blue triangle • Relate lines d 0 , d 1 to sides A, B by looking at • Relate lines d 0 , d 1 to sides A, B by looking at resulting green resulting green triangles derived from blue triangles derived from blue triangles (verify they are similar). • Relate principal focal length f to A/B by finding new similar triangles (verify they are similar). • Relate principal focal length f to A/B by finding triangles à red/pink triangles : Algebra to get to yellow and new similar triangles à red/pink triangles determine scaling. 33 34 Deriving Lens Formula: Heights Points of Interest: 2F Focal Distance • Use same pink triangles and A/B to relate the heights of object and • When the object is at 2F the image is of the the image (scaling factor) same size as object size. 35 36