Multi-bands image analysis using local fractal dimension Aura Conci and Eldman O. Nunes IC - UFF

Introduction Use of fractals and image multiespectral bands to characterize texture. Considering inter-relation among bands the image FD є [ 0, number of bands + 2]. Improve the possibilies of usual false color segmentations (assigning satellite bands to RGB color). I t is not now limited to 3 band.

Eletromagnetic Spectrum : it is possible to measure waves in a strip that varies in frequency of 1 to 1024 Hz, or lengths with interval of values between  m and  m (micrometers).

areaspeculiar characteristics I.V. It includes radiation with wave lengths 0.75  m a 1.0 mm. I.V. radiation is absorbed easily by most of the substances (heating effect). Visivel Radiation capable to produce the vision sensation for human eye. Small variation of wave length (380 to 750 nm). Important for correlation with the visual human experience

The color sensations noticed by humans are combination of the intensities received by 3 types of cells cones. Combination of the 3 primary colors produces the others In the video: R=700 nm, G = 546,1 nm, B=435,1 nm.

Monocromatic : one color channel or one band. binary image: each pixel only 0 or 1 values. intensity level (grey level): each pixel one value from 0 to 255. Digital images

Multiband images: n band value for each pixel. examples: »color images »sattelite images »medical images

color images each pixel 3 values ( from 0 to 255 ) 3 bands: Red - Green -Blue.

Satellite Images

Band 1Band 2Band 3 Band 4Band 5Band 6Band 7 example : a LandSat-7 image is a collection of 7 images of same scene

sensor characteristics TMHRVAVHRR spacial resolution 30 m 120 m (Band 6) 20 m (Band 1 a 3) 10 m (Pan) 1.1 Km (nominal) spectral Bands (micro meters) Band Band Band Band Band Band Band Band Band Band Pan Band Band Band Band Band Radiometric resolution 8 bits 8 bits (1-3) 6 bits (Pan) 10 bits Temporal resolution 16 days26 days2 times a days

Landsat 7 - Sensor TM Channelspectral band (um)main applications Differentiation between soil and vegetation, conifers and deciduous trees healthy vegetation chlorophyll absortion, vegetation types biomass, water bodies penetrate smokes, snow surface temperature from -100 to 150 C hidrotermal map, buildings, soil trafficability

Band 4 (R), 5 (G), 3 (B) Band 4 (R), 3 (G), 2 (B) Multiespectral false color : l, m, n Bands to Red, Green and Blue.

Textures Texture is characterized by the repetition of a model on an area. Textons : size, format, color and orientation of the elements. Textons can be repeated in an exact way or with small variations on a same theme. Texture 1 Texture 2

Fractal Geometry self similar sets fractal dimensions and measures used to classify textures

FD for binary image Box Counting Theorem - 2D images. For a set A, N n (A) = number of boxes of side 1/2 n which interser the set A: DF = lim n  log N n (A) / log 2 n

n N n (A) 2n2n log N n (A) log 2 n 1 421,3860, ,4841, ,5832, ,6822, ,7803, ,8794,158

gray level images Box Counting Theorem extension for 3-dimensional object: third coordinate represents the intensity of the pixel. DF between 2 e 3.

Blanket Dimension - Blanket Covering Method The space is subdivided in cubes of sides SxSxS ’. Nn(A) denotes the number of cubes intercept a blanket covering the image: N n =  n n (i,j) On each grid (i,j), n n (i,j) = int ( ( max – min ) / s’ ) + 1

for multi-bands image a color R G B image is a subset of the pentadimensional space : N 5 ). Each pixel is defined by: (x, y, r, g, b) FD of this images: values from 2 to 5.

Generalizing: d-cube points (0D), segments (1D), squares (2D), cubes (3D) and for a n-dimensional : n-cube (nD) But what is d-cubos, and how many d-cubes appear in a divison of N d space?

Sweep representation : n-cube as translational swepps of (n-1) cube

Generalizing: d-Cube Counting - DCC: the experimental determination of the fractal dimension of images with multiple channels will imply in the recursive division of the N space in d-cubes of size r followed by the contagem of the numbers of d-cubes that intercept the image.

monochrome images: the space N 3 is divided by 3-cubos of size 1/2 n, and the number of 3- cubos that intercept the image it is counted. color images: the space N 5 is divided by 5- cubos of the same size 1/2 n, and the number of 5-cubos that intercept the image is counted. satellite images: the space N d is divided by d- cubes of size 1/2 n and the number of d-cubes that intercept the image is counted.

number of 1-cubes: N n 1-cubos = 2 1x n, where n is the number of divisions. number of 2-cubes: N n 2-cubos = 2 2x n, where n is the number of divisions. number of 3-cubos: N n 3-cubos = 2 3x n, where n is the number of divisions. Generalizing, the number of identical d-cube: N n d-cubes = 2 d x n, where d is the space dimension and n it is the number of divisions. Then FD of d-dimensional images can be obtained by: DF n = log (N n,d-cubo ) /log (2 n )

1-cube (segment) dimension divisions N n,1-cubos rule cube (square) dimension divisions N n,2-cubos rule cube (cube) dimension divisions N n,3-cubos rule

Imagedimension divisions N n,d-cubos Log (N n,d-cubos )Log 2 n FD max Binary2 14Log (4)Log (2)2 216Log (16)Log (4)2 364Log (64)Log (8)2 intensity level 3 18Log (8)Log (2)3 264Log (64)Log (4)3 3512Log (512)Log (8) Log (16)Log (2)4 2256Log (256)Log (4) Log (4096)Log (8)4 Color5 132Log (32)Log (2) Log (1024)Log (4) Log (32768)Log (8)5 Satelite 6 164Log (64)Log (2) Log (4096)Log (4) Log (262144)Log (8)6...

Results and Conclusions  binary images  gray scale  colored images  satellite images