Structured Multidimensional Information · N×M Array of values (attribute), (typically two spatial dimensions): f (x,y) ~ “bidimensional signals ”. N×M×L Array of values, (typically

(C) Copyright - Jorge Marquez 2010 CCADET UNAM

Representation Primitives, Location and Attribute(s)

Structured Multidimensional Information

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(Medical) Imaging

Energy,Informationor Stimuli

Interaction

Image Formation (Physics)

Sensors Acquisition

Image Production (Technology)

Multidimensional InformationProcessing, Analysis

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Medical ImagingInput stimuli

Light (coherent)EM-field

Radiation:

νx-rayα,β,γe− e+

UltrasoundLabelled probesContrast agentsBiochemical signalsElectric signalsMechanical signalsCognitive activity

Endoscopy, keratopography, thermographyMoiré-pattern imaging, polarographyMicroscopies (confocal, multi-photon, ...Radiography, Fluoroscopy, CT, axial, spiralContrast angiography, dynam. MR-angio.NMRI, nuclear medicine, functional NMRScintigraphy, gamma-camera,...TEM, SEM, synchrotron EM, PET, SPECTFlow, tensor, parametric imagingUS mode A,b, Doppler, Time-of-flightDiffusion / perfusion imagingImpedance tomography, cardiographyMagnetoelectroencephalography map ... .... . .

Multidimensional InformationN-dimensional reconstruction

Processing, AnalysisQuantification, Visualization

Interpretation Diagnosis

DataBase of Atlases and PhantomsMapping & registrationInter-comparisonGround truth & referenceDBs: anatomical, functional, Simulation 3

Unstructured and Structured Information


Samples

Signals

Images

Volumes & Videos

Hyper-vols

Set of attribute values (no order, no spatial or temporalstructure; may be point samples in N dimensions)

Ordered set of values (attribute), in one dimension(typically time or one spatial dimension): f (t)

N×M Array of values (attribute), (typically two spatialdimensions): f (x,y) ~ “bidimensional signals ”.

N×M×L Array of values, (typically three spatialdimensions): f(x,y,z) and sequences of frames f(x,y,t)

N×M×L×K array of values; e,g, volume sequence f (x,y,z,t)

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Structured Information II


Boundaries&

Manifolds

N-dimensional local-structured sets in M dimensions andboundaries of other manifolds (topology differences):

• Lifting of 2D images to 3D volumes (or ND to (N+1)D)• Boundaries of objects in M dimensions:

o Endpoints, sets of endpoints, hot points.o Contours (planar and embedded in M ).o Surface boundaries

• Parametric curves: (x(s), y(s)), (x(s), y(s), z(s))• Time evolving sets (shape and topology changes):

o Snakes: (x(s), y(s), z(s))(t)o Lightings (branching snakes)o Balloons and branching balloons

• Networks, branching curves, attributed graphs in M

• Parametric surfaces, interfaces, waves, patches andmembranes: (x(r, s), y(r, s), z(r, s))

• Hypersurfaces, branching manifolds and branes• Mixed manifolds 5

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩


Location: x, y, z, t,σ,...(spatial) configuration

Associated Attribute(s) & Features :

Representation primitive of discrete samples

Scalar value (intensity, depth, density)Vector value: chanels (color, optical flux)…

Structured Multidimensional Information

Orthogonal discrete grid

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Where is the “Pixel”?

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*Euclidean Representation Primitives

Facets (triangles, voxel-faces, polygons)

Points and nodes, pixels, voxels, hyper-voxels

Edges, lines, arcs, arrows, flux-lines, planes

Tetrahedra, spheres, polytopes, ellipsoids

Polygons, disks

Non Euclidean:Geodesic pixels, spels, surfels and “patches”

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*Other Representation Elements(may have location and attributes)

Contours (iso-…), discrete boundaries, meshes, finite/volume element, solid boundaries

Medial Axis (regional-reps) = skeletons Graphs

∂iV , iV

Skel i V , Γi V

Smoothed ParticlesPDFs (Point Spread Functions, LDFs, EDFs)

Texels (textural primitives), symbol sets, icons, images

Cellular complexes (vertices+edges +facets+voxels)

Polynomials, Splines, Basis Functions, Wavelets

Neighborhoods, cliques, structuring elements, kernels

A 変

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Location: x, y, z, t,σ,... Geometry, (spatial) configuration, domain

Orthogonal discrete gridHoneycomb or hexagonal grid

Set of points, (cartesian) array, grid, lattice or tiling, irregular mesh/graph

Offset-square gridInterlaced tiling

Spheroidal polar coordinatesStatistically regular tri-mesh

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vertex-and-edge grid(= graph)


Scalar image (e.g. intensity, density, opacity, transparency)Binary attribute (boolean value, orientation, “up/down”)Parametric image (e.g. range images = depth) Vector image (n-channels), feature vector (color, optical flux, α-blend,…)Transform domain (Fourier, Cosine, Hermite, EDT, MAT,…)Output result of detectors or operatorsLabels (classification, contours,…)Local Information (fractal dimens, texture, coherence, directionality,…)Field image (distance, tensor = transformation, displacement, warp)An image itself (per point or pixel)Neighborhood relations, regional, hierarchical, connectivity, ...Meta-information (data structures) & pointers to information

Associated Attributes & Features:

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Scalar and Vector attributes can be visualized through:• Gray levels, colors and Look-Up-Tables (indexed color)• Patterns, textures, symbols, icons and widgets• Material properties: texture, transparence, shinniness, etc.• Animation, blinking, stretching, morphing• Lifting from 2D to 3D, projections, shades, depth cues• Image fusion techniques, color-channel (de)composition• Optical techniques: Moïré, polarization, filters, phase-

interference, holography

Attribute Visualization:

(Visualization and Computer Graphics domain)

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OpenGL Primitives

GL_QUAD_STRIP

GL_POLYGON

GL_TRIANGLE_STRIPGL_TRIANGLE_FAN

GL_POINTS

GL_LINES

GL_LINE_LOOP

GL_LINE_STRIP

GL_TRIANGLES

13(C) Copyright - Jorge Marquez 2010 CCADET UNAM


A 2-bit indexed color image. The color of eachpixel is represented by a number; each number(the index) corresponds to a color in the colortable (the palette).

Visualizing 2D information

Scalar attribute grayscale attribute

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In classical halftone images, the primitives are dots or squares of varying size and fixed intensity (Black or Color). An intensity is visualized by using a size proportional to gray-level (or color) intensities.

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In modern halftones, the primitive is a B/W dot, with a

quasi-random position; the attribute is gray-level and

visualization is doneby varying spatial-location

statistics: dots agglomerate in function of intensity

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In “ASCII” art, the primitives are a set of B/W characters,

with fixed location. The gray-level attribute is visualized by

using sorted characters in function of their average gray-level by area unity.

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Floyd-Steinberg dithering algorithm, based on error diffusion


A vector field may be a displacement field to model an image deformation: each pixel has a displacement

vector associated (Δx(x,y), Δy(x,y))

Δx(x,y)

Δy(x,y)

I(x,y) I(x+Δx,y+Δy)

The displacement Δx, Δy changes at each point

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T(I(x,y)) + ΔI(x,y)

T(x,y) (I(x,y)) ⋅M(x,y) + ΔI(x, y)

Transfer function uout = T(uin )

Position-dependent transfer function

Scalar-field “deformations” (intensity-only)

Offset field(background)

Modulation field

Pixel-by-pixel addition (+) and product (⋅)


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Vector Images (Format .svg)

Filled vectorized contours(polygons)


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L: → 2 Discrete Polyline of N vertices:L = {(xs,ys)}s=0,…, N−1

Edge vectors: vs = (xs,ys)−(xs−1 ,ys−1)

L is a (closed) contour of N−1 verticesif (x0,y0) = (xN−1, yN−1)

Region A (polygon) = solid /geometric interior of ∂A. We call ∂A the boundary of A.

L may be a vectorized curve C. Set of pixels {(x, y)} fitting C = rasterized curve:

C: → 2 parametrical curve, C=(x(s), y(s)), s may be the arc of lenght

A (very) simple manifold


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{ },

3

, 1

2

0, [0,1]s s s s s N

t t tt= … −

∈+ + +a b c d

cubic-spline interpolant


Spatial Discretization Spacing / sample geometry

Atributte

x, y, t, etc.

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Atributte(s) Quantization Scale of L values (integers)

Atributte

x, y, t, etc.

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Digitization = Discretization AND Quantization

Atributte

x, y, t, etc.

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Continuous Appearance – Discrete Model

Atributte

(x, y)

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“2.5D” Intensity Surface= (binary 3D object) 2D Image (gray levels)

29(C) Copyright - Jorge Marquez 2010 CCADET UNAM

3D Surface = (binary 3D object)

2D Image (gray levels)Iso-intensity sets representation

30(C) Copyright - Jorge Marquez2010 CCADET UNAM

Processing, Analisis & Visualization:• On representation primitives

o Format conversions, coding, compression, cryptingo Re-sampling, texturing, meshing, finite element

• On attributes and parameterso Transfer functions, segmentation & labelingo Convolutions (average = special case)o Integer transforms (e.g., FFT)o Functions and arithmetic operationso Logic operations, predicate segmentationo Fusiono Re-sampling, Interpolation

• On localization/coordinateso Geometric transformations – Distortions/warpso Registration, image fusiono Re-sampling, zoom

• On visualization attributes• On attributed manifolds


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1. Scalar image (e.g. Intensity)2. Parametric image (e.g. range images) 3. Features vector (color, optical flux,…)4. Transform domain (Fourier, EDT, MAT,…)5. Output of detectors or operators6. Labels (classification, contours,…)7. Local Information (fractal dimension,

texture, coherence, directionality,…)8. Field image (distance, tensor)9. Neighborhood relations, regional,

hierarchical, connectivity, ...10. Meta-information (data structures)11. Pointers to local information12. Etcétera…

Location & Geometry: x,y,z,t,σ,...

Associated Attributes & Features:

Representation:

Facets, meshes

DiscreteBoundaries

Medial Axis& Graphs

∂iV , iV

Skel i V , Γi V

Smooth Particles32


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HERE: manifold examples• Parametric curves, polylines, polygons (vectorized contours)• Discrete oriented boundaries: Facet, voxel, triangle mesh,

hamiltonian paths = (directed closed) graphs• Interfaces, thin-plate splines, bilinear Coons patches• Möebius strips, Klein bottle, Penrose triangle• Branching structures: Medial Axis Transform and SKIZ,

coraline manifold, mixed sets• Topology properties

Documents

Structured Multidimensional Information · N×M Array of values (attribute), (typically two spatial dimensions): f (x,y) ~ “bidimensional signals ”. N×M×L Array of values, (typically