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Images as Functions 2 A Binary image is a function B(x,y) Є [0,1] Gray-tone image is a function: g(x,y) Є [0.,1,….L-1] A color image is represented by three functions f(x,y) =(f1(x,y), f2(x,y), f3(x,y)) or a vector-valued function: f(x,y) Multi-spectral Image: f(x,y) =(f1(x,y), f2(x,y),…, fn(x,y))

Gray-tone Image as Function 3

Image vs Matrix 4 There are many different file formats.

5 Digital Image Terminology: pixel (with value 94) its 3x3 neighborhood binary image gray-scale (or gray-tone) image color image multi-spectral image labeled image region of medium intensity resolution (7x7)

Image File Formats  PGM Portable Gray Map, older form  GIF was early commercial version Graphics Interchange Format  JPEG (JPG) is modern version Joint Photographic Experts Group  MPEG for motion Motion Picture Experts Group 6

How do you create an İmage file format?  Given image, 1. Find a compact mathematical representation 2. Code the mathematical representation 7

 Many formats exist: header plus data  Do they handle color?  Do they provide for compression?  Are there good packages that use them or at least convert between them? 8

PGM image with ASCII info.  P2 means ASCII gray  Comments  W=16; H=8  192 is max intensity  Can be made with editor  Large images are usually not stored as ASCII 9

PBM/PGM/PPM Codes 10 P1: ascii binary (PBM) P2: ascii grayscale (PGM) P3: ascii color (PPM) P4: byte binary (PBM) P5: byte grayscale (PGM) P6: byte color (PPM)

JPG current popular form  Public standard  Allows for image compression; often 10:1 or 30:1 are easily possible  8x8 intensity regions are fit with basis of cosines  Error in cosine fit coded as well  Parameters then compressed with Huffman coding  Common for most digital cameras 11

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From 3D Scenes to 2D Images 13 Object World Camera Real Image Pixel Image

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Binary Image B(r,c) 15 0 represents the background 1 represents the foreground

Binary Image Analysis  is used in a number of practical applications, e.g. 16 Part inspection Shape analysis Enhancement Document processing

What kinds of operations? 17 Separate objects from background and from one another Aggregate pixels for each object Compute features for each object

Example: red blood cell image  Many blood cells are separate objects  Many touch – bad!  Salt and pepper noise from thresholding  How useable is this data? 18

Results of analysis  63 separate objects detected  Single cells have area about 50  Noise spots  Gobs of cells 19

Binary Image Operations 20 1.Thresholding a gray-tone image 2.Convolution 3.Morphology 4.Feature extractions (area, centroid) 5.Connected components analysis

1. Thresholding 21 Convert gray level or color image into binary image Use histogram Definition: The Histogram of a gray-level image I is defined as H(m) = { (r,c) : I(r,c) =m) } Where m spans the gray values

Histogram-Directed Thresholding 22 How can we use a histogram to separate an image into 2 (or several) different regions? Is there a single clear threshold? 2? 3?

Histogram  Background is black  Healthy cherry is bright  Bruise is medium dark  Histogram shows two cherry regions (black background has been removed) 23 gray-tone values pixel counts 0 256

Automatic Thresholding: Otsu’s Method 24 Assumption: the histogram is bimodal t Method: find the threshold t that minimizes the weighted sum of within-group variances for the two groups that result from separating the gray tones at value t. Grp 1 Grp 2

Thresholding Example 25 original gray tone imagebinary thresholded image

2. Convolution  Given a gray level image I(r,c) and a mask m(r,c) convolution is I(r,c)*m(r,c)= ΣΣ I(k,l). m(r-k,m-l) 26 Masks

SMOOTHING MASKS

Convolution of an İmage with a Mask 28

Image Enhancement WITH SMOOTING MASKS

Averaging blurrs the image

Image Enhancement WITH AVERAGING AND THRESHOLDING Image Enhancement WITH AVERAGING AND THRESHOLDING

Restricted Averaging  Apply averaging to only pixels with brightness value outside a predefined interval. Mask m(i,j) = 1For g(k+i,l+j)€ [min, max] 0 otherwise

Median Filtering  Find a median value of a given neighborhood.  Removes sand like noise

EDEGE DETECTİON BY CONVOLUTION EDGE PROFILES EDEGE DETECTİON BY CONVOLUTION EDGE PROFILES Edges are the pixels where the brightness changes abrubtly. It is a vector variable with magnitude and direction

EDGES, GRADIENT AND LAPLACIAN

SMOOT EDGES, NOISY EDGES

Continuous world  Gradient  Δg(x,y) = ∂g/ ∂x + ∂g/ ∂y  Magnitude: |Δg(x,y) | = √ (∂g/ ∂x) 2 + (∂g/ ∂y) 2  Phase : Ψ = arg (∂g/ ∂x, ∂g/ ∂y) radians

Discrete world  Use difference in various directions  Δi g(i,j) = g(i,j) - g(i+1,j)  or  Δj g(i,j) = g(i,j) - g(i,j+1)  or  Δij g(i,j) = g(i,j)- g(i+1,j+1)  or  |Δ g(i,j) | = |g(i,j)- g(i+1,j+1) | + |g(i,j+1)- g(i+1,j) |

GRADIENT EDGE MASKS Approximation in discrete grid GRADIENT EDGE MASKS Approximation in discrete grid

GRADIENT EDGE MASKS

Horizontal Horizontal vertical diagonal edges

Vertical, Horisontal and diagonal edge detection

EdgesEdges

LAPLACIAN MASKS

LAPLACIAN of GAUSSIAN EDGE MASKS

EDGE DETECTION

3. Mathematical Morphology  Morphology: Study of forms of animals and plants  Mathematical Morphology: Study of shapes  Similar to convolution  Arithmetic operations Set Operations 50

Need to define Image as a Set  Given a binary image I (r,c), assume 1 correspond to object 0 correspond to backround. Define a set with elements to the coordinates of the object  X = { (r1,c1), (r2,c2),….} 51

 X= 52

Set Operations

Set Operations on Images AND, OR Set Operations on Images AND, OR

TRANSLATION REFLECTION

Set Operations

Morphologic Operations 57 Binary mathematical morphology consists of two basic operations dilation and erosion and several composite relations closing and opening

Dilation: A+B ={ Z: (Bz)∩ A≠ Φ} 58 Dilation expands the connected sets of 1s of a binary image. It can be used for 1. growing features 2. filling holes and gaps

Structuring Elements 59 A structuring element is a shape mask used in the basic morphological operations. They can be any shape and size that is digitally representable, and each has an origin. box hexagon disk something box(length,width) disk(diameter)

Dilation with Structuring Elements 60 The arguments to dilation and erosion are 1.a binary image B 2.a structuring element S dilate(B,S) takes binary image B, places the origin of structuring element S over each 1-pixel, and ORs the structuring element S into the output image at the corresponding position origin B S dilate B  S

DILATION

Erosion 63 Erosion shrinks the connected sets of 1s of a binary image. It can be used for 1. shrinking features 2. Removing bridges, branches and small protrusions

Erosion with Structuring Elements 64 erode(B,S) takes a binary image B, places the origin of structuring element S over every pixel position, and ORs a binary 1 into that position of the output image only if every position of S (with a 1) covers a 1 in B B S origin erode B S

EROSİON: A-B = {z: (Bz) A

IMAGE ENHANCEMENT WİTH MORPHOLOGY

Example to Try erode dilate with same structuring element S B

Opening and Closing 68 Closing is the compound operation of dilation followed by erosion (with the same structuring element) Opening is the compound operation of erosion followed by dilation (with the same structuring element)

Use of Opening 69 Original Opening Corners 1.What kind of structuring element was used in the opening? 2.How did we get the corners?

Morphological Image Processing

OBJECT DETECTION

BOUNDARY EXTRACTİON

HOW DO YOU REMOVE THE HOLES

Morphological Image Processing

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Gear Tooth Inspection 85 original binary image detected defects How did they do it?

Some Details 86

Region Properties 87 Properties of the regions can be used to recognize objects. geometric properties (Ch 3) gray-tone properties color properties texture properties shape properties (a few in Ch 3) motion properties relationship properties (1 in Ch 3)

Geometric and Shape Properties 88 area centroid perimeter perimeter length circularity elongation mean and standard deviation of radial distance bounding box extremal axis length from bounding box second order moments (row, column, mixed) lengths and orientations of axes of best-fit ellipse Which are statistical? Which are structural?

2. Connected Components Labeling 89 Once you have a binary image, you can identify and then analyze each connected set of pixels. The connected components operation takes in a binary image and produces a labeled image in which each pixel has the integer label of either the background (0) or a component. binary image after morphology connected components

Methods for CC Analysis Recursive Tracking (almost never used) 2. Parallel Growing (needs parallel hardware) 3. Row-by-Row (most common) Classical Algorithm (see text) Efficient Run-Length Algorithm (developed for speed in real industrial applications)

Equivalent Labels Original Binary Image

Equivalent Labels The Labeling Process 1  2 1  3

Run-Length Data Structure U N U S E D row scol ecol label Rstart Rend Runs Row Index Binary Image

Run-Length Algorithm 94 Procedure run_length_classical { initialize Run-Length and Union-Find data structures count <- 0 /* Pass 1 (by rows) */ for each current row and its previous row { move pointer P along the runs of current row move pointer Q along the runs of previous row

Case 1: No Overlap 95 |/////| |/////| |////| |///| |///| |/////| Q P Q P /* new label */ count <- count + 1 label(P) <- count P <- P + 1 /* check Q’s next run */ Q <- Q + 1

Case 2: Overlap 96 Subcase 1: P’s run has no label yet |///////| |/////| |/////////////| Subcase 2: P’s run has a label that is different from Q’s run |///////| |/////| |/////////////| Q Q P P label(P) <- label(Q) move pointer(s) union(label(P),label(Q)) move pointer(s) }

Pass 2 (by runs) 97 /* Relabel each run with the name of the equivalence class of its label */ For each run M { label(M) <- find(label(M)) } where union and find refer to the operations of the Union-Find data structure, which keeps track of sets of equivalent labels.

Labeling shown as Pseudo-Color 98 connected components of 1’s from thresholded image connected components of cluster labels

Region Adjacency Graph 99 A region adjacency graph (RAG) is a graph in which each node represents a region of the image and an edge connects two nodes if the regions are adjacent

Morphological Image Processing