An Introduction to Images Chapter 01/02 G&WModule 1: GoalsWhat Is An Digital Image?Digital Image Acquisition ProcessImage As A Mapping (Function)Image As A Mapping: IssuesExamples of “Images” as FunctionsImages As 2D Functions3D Images – VolumesDigital Image: Continuous to DiscreteMultivalued ImagesImage As Grid of ValuesSampling Effect of spatial resolutionQuantization: Effect of intensity levels Where Do Digital Images Come From?CCD CamerasX-RaysX-Ray ImagesComputed TomographyCT (CAT)Magnetic Resonance ImagingMRINuclear Medicine PET, SPECT, …Serial SectioningSerial Section Transmission Electron MicroscopyExamplesFingerprint imagesFingerprint matchingArray vs. Matrix OperationsArithmetic Operations on ImagesArithmetic operations: f + gArithmetic operations: f - gImage Subtraction: Motion DetectionArithmetic operations: f x gArithmetic operations: f / gOperations on Cartesian Image GridsImage Coordinates and ResolutionIndex SetsNeighborhoodAdjacencyPathsDistances in ImagesDistances in ImagesManhattan Distance / City Block DistanceManhattan Distance / City Block DistanceManhattan Distance / City Block DistancePixels with D4 distance from centerPixels with D8 distance from centerConnected ComponentConnected ComponentA Simple Algorithm: Flood FillA Simple Algorithm: Flood FillConnected Component AnalysisCC – PurposeCC Output/Extensions1An Introduction to ImagesChapter 01/02 G&WCS6640/BIOENG6640Guido Gerig, modified from Ross WhitakerSCI Institute, School of ComputingUniversity of Utah2Module 1: Goals• Understand images as mappings– Understand the difference – continuous vs discrete– Be able to identify domain and range of an image in a precise way• Know several examples of images– How they are used– How they are formed• Understand domain topology, physical dimensions, and resolution of images• Understand and be able to use (e.g. reason about and implement)– Arithmetic operations, neighborhoods, adjacency, connected components3What Is An Digital Image?• A file you download from the web (e.g. image.jpg)• What you see on the screen• An array (regular grid) of data values•A mapping from one domain to another– A discrete sampling (approximation) of a functionDigital Image Acquisition Process45Image As A Mapping (Function)rangedomain6Image As A Mapping: Issues• Dimensionality of domain (n = ?)• Dimensionality of range (m = ?)• Typically use shorthand of Rnor Rm• Discrete or continuous– Discrete reasoning/math– Continuous math (calculus) –> discrete approximation– Issues for both domain and range8Examples of “Images” as Functions“Graph”“x” or “t”9Images As 2D Functions“Graph”xyyProject brightness on a screen with values at coordinates (x,y) proportional to value of f(x,y)103D Images – VolumesxzImpossible to graph:SlicingVolume renderingScientific visualizationySliceDigital Image: Continuous to Discrete11Sampling (space) and Quantization (intensity)12Multivalued Images• Color images: mappings to some subset of R3– Color spaces: RBG, HSV, etc.• Spectral imagery– Measure energy at different bands within the electromagnetic spectrum• E.g. Satellite images13Image As Grid of Values• Two views– Domain is a discrete set of samples– Samples are points from an underlying continuous function• How is the grid organized?– Unstructured• Points specified by position and value– Structured grids• Position inferred from structure/index• 1D, 2D, 3D, ….•Sizes w, w x h, w x h x df1 x1 y1f2 x2 y2f3 x3 y3… … …1234Unstructured gridStructured gridsSampling Effect of spatial resolution14Quantization: Effect of intensity levels 1516Where Do Digital Images Come From?• Digitizing film or paper– Rasterize, sample reflectance/transmission on grid17CCD CamerasOptics Image plane18X-RaysProjection Image plane19X-Ray Images20Computed TomographySeries of projections VolumeReconstruction21CT (CAT)22Magnetic Resonance ImagingRandom orientation (water molecules)Magnetic field –align spinsSpins become random (generate field)Coil CoilGenerate fieldDetect/measure field23MRI24Nuclear MedicinePET, SPECT, …DetectorsEmissionInjection&detection Volume/ImageReconstruction25Serial SectioningProjection (Film)Electron Beam/LightThin SectionSpecimen26Serial Section Transmission Electron Microscopy27Examples• Quality control of surface-mount packaging• Retinal architecture from serial section TEM• Image-based phenotypingImagesMutantImagesWild Types28Tolga Tasdizen - ECE 6962 Lecture Notes, U. of UtahFingerprint images• Ink technique– spread ink– press on paper– capture with CCD camera or scanner• Latent fingerprints • Live-scan– Optical sensors– Capacitive sensor– Thermal sensor– Pizoelectric (pressure)29Tolga Tasdizen - ECE 6962 Lecture Notes, U. of UtahFingerprint matching• Fingerprint patterns are unique to the individual• Matching– using the ridges directly is hard– often singularity points are used• Local: Minutiae • Global: Loop, delta, ...DeltaLoopMinutiaeRidges37Array vs. Matrix OperationsMatrix multiply( MATLAB A*X )Array multiply( MATLAB A.*X )Images can be represented as matrices, but the operations refer to array operations unless otherwise specified38Arithmetic Operations on Images• Arithmetic operations on pixel values– Multiple images with the same domain– Image become arguments• Implied that the operation is applied pointwise across the domain– Addition, subtraction, multiply, divide, boolean39Arithmetic operations: f + g© 1992–2008 R. C. Gonzalez & R. E. WoodsAveraging (adding) multiple images can reduce noise40Arithmetic operations: f - g© 1992–2008 R. C. Gonzalez & R. E. WoodsDigital Subtractive Angiography(DSA)41Image Subtraction: Motion Detection© 1992–2008 R. C. Gonzalez & R. E. Woods42Arithmetic operations: f x g© 1992–2008 R. C. Gonzalez & R. E. Woods43Arithmetic operations: f / gCaptured imageIllumination Corrected image44Operations on Cartesian Image Grids• Grid resolution• Neighborhoods• Adjacency and connectivity• Paths• Connected components• Flood fill45Image Coordinates and Resolution• A single point on an image grid is a “pixel”– Sometimes this is just the location, sometimes also the value• References to pixels– Single index (implied ordering) “i” or “f(i)”– Multiple index (gives position on logical grid) “i,j” or “f(i,j)”• Physical coordinates – Logical coordinates place