Chapter 3 Binary Image Analysis Essay

Description

TitleEssays on model specification tests and on binary response models

NameShen, Xiangjin (author); Tsurumi, Hiroki (chair); Swanson, Norman Rasmus (internal member); Landon-Lane, John (internal member); Goldman, Elena (outside member); Rutgers University; Graduate School - New Brunswick

Date Created2013

Other Date2013-05 (degree)

SubjectEconomics, Bayesian statistical decision theory, Binary system (Mathematics)

Extentxi, 108 p. : ill.

DescriptionThis dissertation consists of three essays evaluating model selection criteria in both sampling theory and Bayesian analysis. In chapter one, I compare the Bayesian model selection criteria (DIC, PDIC and MSEF) and the conditional Kolmogorov test for the spot asset pricing models (Vasicek and CIR models); MCMC and block Bootstrap methods are applied. In chapter two, I compare parametric and semiparametric methods for the binary response models. The comparison is made by model specifications, ROC area, and marginal effects. Monte Carlo simulation, quasi-maximum likelihood and kernel density methods are applied. In chapter three, I compare two bandwidths of the kernel density: the standard bandwidth and computationally optimized bandwidth. The computationally optimized bandwidth is obtained by using the graphic processing unit (GPU) that shortens the computational time.

NotePh. D.

NoteIncludes bibliographical references

NoteIncludes vita

Noteby Xiangjin Shen

Genretheses

Persistent URLhttps://doi.org/doi:10.7282/T3C827WK

Languageeng

CollectionGraduate School - New Brunswick Electronic Theses and Dissertations

Organization NameRutgers, The State University of New Jersey

RightsThe author owns the copyright to this work.

Presentation on theme: "E.G.M. PetrakisBinary Image Processing1 Binary Image Analysis Segmentation produces homogenous regions –each region has uniform gray-level –each region."— Presentation transcript:

1 E.G.M. PetrakisBinary Image Processing1 Binary Image Analysis Segmentation produces homogenous regions –each region has uniform gray-level –each region is a binary image (0: background, 1: object or the reverse) –more intensity values for overlapping regions Binary images are easier to process and analyze than gray level images

2 E.G.M. PetrakisBinary Image Processing2 Binary Image Analysis Tasks Noise suppression Run-length encoding Component labeling Contour extraction Medial axis computation Thinning Filtering (morphological operations) Feature extraction (size, orientation etc.)

3 E.G.M. PetrakisBinary Image Processing3 Noise suppression Small regions are not useful information –apply “size filter” to remove such regions –all regions below T pixels in size are removed by changing the value of their pixels to 0 (background) –it is generally difficult to find a good value of T –if T is small, some noise will remain –if T is large, useful information will be lost

4 E.G.M. PetrakisBinary Image Processing4 original noisy image filtered Image T=10 original noisy image filtered image T=25 too high!

5 E.G.M. PetrakisBinary Image Processing5 Run-Length encoding Compact representation of a binary image Find the lengths of “runs” of 1 pixels sequences

6 E.G.M. PetrakisBinary Image Processing6 Component Labeling Assign different labels to pixels belonging to different regions (components) –connected components –not necessary for images with one region binary image labeled connected components

7 E.G.M. PetrakisBinary Image Processing7 Sequential Algorithm 1.Scan the image from left to right, top to bottom; if the pixel is 1 then a)if only one of the upper or left pixels has a label, copy this label to current pixel b)if both have the same label, copy this label c)if they have different labels, copy one label and mark these two labels as equivalent d)if there are no labeled neighbors, assign it a new label 2.Scan the labeled image and replace all equivalent labels with a common label 3.If there are no neighbors, go to 1

8 E.G.M. PetrakisBinary Image Processing8 Contour Extraction Find all 8-connected pixels of a region that are adjacent to the background –select a starting pixel and track the boundary until it comes back with the starting pixel binary region boundary

9 E.G.M. PetrakisBinary Image Processing9 Boundary Following Scan the image from left to right and from top to bottom until an 1 pixel is found 1)stop if this is the initial pixel 2)if it is 1, add it to the boundary 3)go to a 0 4-neighbor on its left 4)check the 8-neighbors of the current pixel and go to the first 1 pixel found in clockwise order 5)go to step 2

10 E.G.M. PetrakisBinary Image Processing10 Area – Center Binary (or gray) region B[i,j] –B[i,j] = 1 if (i,j) in the region, 0 otherwise –Area: –Center of gravity:

11 E.G.M. PetrakisBinary Image Processing11 Orientation Angle with the horizontal direction Find angle θ minimizing θ r ij ρ = x cos θ + y sin θ x y

12 E.G.M. PetrakisBinary Image Processing12 Computing Orientation E = a sin 2 θ – b sin θ cos θ+c cos 2 θ, where

13 E.G.M. PetrakisBinary Image Processing13 Computation Orientation (cont.) From which we get Differentiating with respect to θ and setting the result to zero –tan2θ = b/(a-c) unless b = 0 and a = c Consequently The solution with the + minimizes E The solution with the – maximizes E

14 E.G.M. PetrakisBinary Image Processing14 Computation Orientation (cont.) Compute E min, E max minimum and maximum of the least second moment E The ratio e = E max /E min represents roundness –e  0 for lines –e  1 for circles

15 E.G.M. PetrakisBinary Image Processing15 Distance Transform Compute the distance of each pixel (i,j) from the background S  at iteration n compute F n [i,j]:  F 0 [i,j] = f[i,j] (initial values)  F n [i,j] = F 0 [i,j] + min(F n-1 [u,v])  (u,v) are the 4-neighbor pixels of (i,j) that is pixels with D([i,j],[u,v]) = 1  repeat until no distances changes

16 E.G.M. PetrakisBinary Image Processing16 Example of Distance Transform Distance transform of an image after the first and second iterations –on the first iteration, all pixels that are not adjacent to S are changed to 2 –on succeeding iterations only pixels further away from S change

17 E.G.M. PetrakisBinary Image Processing17 Skeleton (Medial Axis) Set of pixels with locally maximum distance from the background S Take the distance transform and keep (i,j) Keep a point (i,j) if it is the max in its 4-neighborhood: D([i,j],S): locally maximum that is D([i,j],S) >= D([u,v],S) where (u,v) are the 4-neighbors of (i,j) The region can be reconstructed from its skeleton –take all pixels within distance D(i,j) from each pixel (i,j) of the skeleton

18 E.G.M. PetrakisBinary Image Processing18 Examples of Medial Axis Transform the medial axis transform is very sensitive to noise

19 E.G.M. PetrakisBinary Image Processing19 Thinning Binary regions are reduced to their center lines –also called skeletons or core lines –suitable for elongated shapes and OCR –each region (e.g., character) is transformed to a line representation –further analysis and recognition is facilitated Iteratively check the 8-neighbors of each pixel –delete pixels connected with S unless the 8-neighbor relationship with the remaining pixels is destroyed and except pixels at the end of a line –until no pixels change

20 E.G.M. PetrakisBinary Image Processing20 no pixels change after 5 iterations

21 E.G.M. PetrakisBinary Image Processing21 Filtering Operations Expansion: background pixels adjacent to the region are changed from 0 to 1 –the region is expanded –fills gaps, the region is smoothed Shrinking: pixels are changed from 1 to 0 –the region is shrinked –removes noise, thinning A combination of Expansion with Shrinking may achieve better smoothing

22 E.G.M. PetrakisBinary Image Processing22 noisy image expanding followed by shrinking filled holes but did not eliminate noise shrinking followed by expanding eliminated noise but did not fill the holes

23 E.G.M. PetrakisBinary Image Processing23 Morphological Filtering Image filtering using 4 filtering operations –two basic: Dilation, Erosion plus –two derived: Opening, Closing –and a Structuring Element (SE) whose size and shape may vary SE: rectangle with R=1,2,3 SE: circle with R=1,2,3

24 E.G.M. PetrakisBinary Image Processing24 Erosion - Dilation Erosion: apply the SE on every pixel (i,j) of the image f –(i,j) is the center of the SE –if the whole SE is included in the region then, f[i,j] = 1 –otherwise, f[i,j] = 0 –erosion shrinks the object Dilation: if at least one pixel of the SE is inside the region f[i,j] = 1 –dilation expands the region

25 E.G.M. PetrakisBinary Image Processing25 original image Structuring Element (SE) Erosion with the SE at various positions of the original image

26 E.G.M. PetrakisBinary Image Processing26 the erosion of the original image with the SE the bold line shows the border of the original image the dilation of the original image with the SE the bold line shows the border of the original image

27 E.G.M. PetrakisBinary Image Processing27 Opening - Closing Opening: erosion followed by dilation with the same SE –filters out “positive” detail, shirks the region Closing: dilation followed by erosion with the same SE –smoothes by expansion, fills gaps and holes smaller than the SE

28 E.G.M. PetrakisBinary Image Processing28 initial erosion initial dilation succeeding dilation succeeding erosion opening closing

29 E.G.M. PetrakisBinary Image Processing29 Pattern Spectrum Succeeding Openings with the same SE Succeeding Closings with the same SE i=0 i=-1i=-2 i=-3i=-4 i=0i=1i=2i=3i=4

30 E.G.M. PetrakisBinary Image Processing30 Example of Pattern Spectrum Size vector p=(a -n,a -n+1,a -n+2,….a, a 1, a 2, a 3, …, a n ) Distance between shapes A,B: opening original closing -4-3-201234

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