Binary Images

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Binary Images

Each pixel is 0 or 1, background or foreground

Image processing to Enhance separation of objects of interest

Separate and count multiple objects

Understand the shapes of multiple objects

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From image to binary image

Classification: dividing pixels into "foreground" and "background" Thresholding

If a pixel has a value in range (min, max) it is foreground Often min is 0 or max is maximum pixel value Choice of range can be manual or automatic (E.g. look for peaks / valleys in histogram)

More complex operations Use information from neighboring pixels Use properties besides pixel value (e.g. location) …

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Image Neighborhoods Neighborhoods can be defined for each pixel

The two most common neighborhoods 4-neighborhood

8-neighborhood

NW E

S

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Applying a Mask

Mask is a set of relative pixel positions. One is designated the origin (0,0) - usually at center

Each mask element is weighted

To apply the mask, put the origin pixel over the image pixel and multiply weights by the pixels under them, then add up all the values.

Usually this is repeated for every image in the pixel. Assumptions must be made for pixels near the edge of the image.

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Mask application example

Mask = 1 1 1, origin at center

Apply to every pixel in image:

0 0 0 0 1

0 0 0 1 1

0 0 1 1 1

0 1 1 1 1

1 1 1 1 1

Result is

0 0 0 1 1

0 0 1 2 2

0 1 2 3 2

1 2 3 3 2

2 3 3 3 2

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Shapes in a binary image

0 0 0 0 0 0 0 0 0 0 0

0 1 1 1 1 0 0 0 0 0 0

0 1 1 1 1 0 0 0 0 1 0

0 1 1 1 1 0 0 0 1 1 0

0 0 0 0 0 0 0 1 1 1 0

0 0 0 0 0 0 1 1 0 1 0

0 0 0 0 0 1 1 1 1 1 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

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Masks for object counting

External corners (origin = top left)

0 0 0 0 1 0 0 1

0 1 1 0 0 0 0 0

Internal corners (origin = top left)

1 1 1 1 1 0 0 1

1 0 0 1 1 1 1 1

Apply using exact match (result is 1 if every pixel in mask matches the image)

If any of the 4 external corner masks matches, the corner is external (& same for internal)

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Counting 4-Connected Objects

The number of 4-connected objects in an image is

(E - I) / 4

where E is the number of external corners and I is the number of internal corners.

Assumptions: Objects are 4-connected (2 diagonal pixels are 2 objects)

Objects do not contain "holes" within them

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Connected Components

"Blobs" are connected components 4-connected (diagonal neighbors don't count)

8-connected (diagonal neighbors are connected)

The following diagram has 3 4-connected components (red, blue, black) or 2 8-connected components (non-black, black)

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Recursive Connected Components

Copy the image

While at least one 1-pixel exists in the existing image Create a new label

REC_LABEL(label, pixel)

REC_LABEL(label, pixel) Label and remove the pixel

For each non-zero neighbor

REC_LABEL(label, neighbor)

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Recursive Connected Components Example

First pixel

After 4 recursive calls, no 4-neighbors neighbors to color

Start again with a new color on an unmarked pixel

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Two-Pass Connected Components

Pass 1 For every pixel in the image (left-right, top-bottom)

If the pixel is non-zero

If the pixel has no labeled neighbors (above/left)

Create a new label & label it

Else if all labeled neighbors are the same

Give the pixel the same label

Else if neighbors have 2 different labels

Give the pixel the largest label

Mark the smaller label as a parent of the larger one (parent [larger] = smaller)

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Two-Pass Connected Components

Pass 2 Renumber all pixels using only one value for each

"equivalence class"

This value is the root of the tree

While (parent[x] != 0) x = parent[x];

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Two-pass Connected Component Example

2 lines scanned

3 lines scanned

?

4th line -- Conflict - set black = blue

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Binary Image Morphology

Morphology = "Study of Shape"

Set of operations that are useful for processing connected components ("blobs") based on shape

Examples Remove small holes or outcroppings

Remove objects below a given size

Smooth corners

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Structuring Element

Mask used for binary morphology

Like convolution masks, they slide over the image and operate on the pixels under them

Common elements: Box (square or rectangle)

Disk (digital filled circle)

Bar (horizontal or vertical)

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Morphology Operations

Dilation Result is mask OR original

Erosion Result pixel is 1 if origin pixel is 1 and all pixels covered

by the mask are also 1

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Morphology Operations

Closing = dilation followed by erosion

Opening = erosion followed by dilation

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Applications of Morphology

General Closing closes holes (up to size of element)

Opening opens spaces (up to size of element)

Specialized Choose elements of size/shape based on your object

Eliminate objects that are too small / large

Isolate interesting features

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Grassfire Transform

Each pixel is distance to closest “1” in the original image.

2 1 1 1 1 2 3 4 3 2 3

1 0 0 0 0 1 2 3 2 1 2

1 0 0 0 0 1 2 2 1 0 1

1 0 0 0 0 1 2 1 0 0 1

2 1 1 1 1 2 1 0 0 0 1

3 2 2 2 2 1 0 0 1 0 1

4 3 3 2 1 0 0 0 0 0 1

5 4 4 3 2 1 1 1 1 1 2

6 5 5 4 3 2 2 2 2 2 3

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Another Example

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

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Two-Pass Algorithm for Grassfire

Set all boundary pixels to Max

First pass: top left to bottom right If original pixel was 1, pixel is 0

Else pixel = min(above + 1, left + 1);

Second pass: bottom right to top left Pixel = min(pixel, below+1,right+1);

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Distance Transforms and Medial Axis

Grassfire as described here measures 4-connected distance to the region of interest

Variations Use a different distance transform, e.g. Euclidean

Use 0 instead of a large value at the boundaries

Medial axis is set of points furthest from a boundary Set of pixels with maximal values in grassfire

Very sensitive to changes in boundary

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Examples

Figure 3.9 (f = dilated, g=grassfire, h=components)

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Other Useful Binary Image Operations

Pixelwise AND, OR

Pixelwise Subtraction 0 if first image is 0, or both images are 1

Translation Every pixel is copied (dx, dy) away from its original

position

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Shape Representation

Shape = list of features Boundary points, segments Region features (center, convex hull, etc.) “Interesting” points, e.g. corners Shape invariants

Shape = mathematical description Function type & parameters (e.g. circle, radius 5, centered

at (0,0)) Mathematical function approximation of boundary, e.g. B-

spline Shape Classes

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Shape Features

Region features Simple properties (area, Euler’s #, projections, bounding

box, eccentricity, elongatedness, direction, compactness) Statistical moments Convex Hull Region decompositions (hierarchical representation) Skeleton

Boundary features Boundary points (e.g. chain code) Geometric properties (perimeter, curvature, chords, etc.)

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Simple Properties

Area Number of pixels in the region

Perimeter Number of boundary pixels

Compactness = (Perimeter & Perimeter) / Area Maximal for circle, minimal for thin, long rectangle

Centroid Average x value, average y value

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Statistical Moments

In a binary image, a moment is the sum of relevant x’s and y’s.

(0,0) moment = area

(1,0) moment / area = average x coordinate

(0,1) moment / area = average y coordinate

x0y0

(x , y) in shape∑

x(x , y) in shape

∑

m p, q = xpyq

x, y∈shape∑

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Variations on Statistical Moments

Central moment First translate the shape so that its center is (0,0), i.e.

subtract the (average x, average y) values from all pixel locations

Scaled central moment Divide the central moment by a power of the scale factor

and the area

Unscaled central moment Divide the central moment by a power of the area only

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2nd Order Moments and Ellipses

Central (2,0) moment relates to the horizontal axis of an ellipse approximating the shape

Central (0, 2) moment relates to the vertical axis of an ellipse approximating the shape

Central (1,1) moment relates to the rotation of an ellipse approximating the shape

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More Region Properties

Bounding box Min, max x-value, min-max y-value

Extremal points (on the bounding box) Topmost left, topmost right, leftmost top, leftmost bottom,

etc.

Lengths of extremal axes (e.g. top left -> bottom right)

These approximate the Convex Hull of the object Convex hull is the shape you get by pounding nails into

the black pixels and then wrapping them with a rubber band.

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Boundary Pixels and Perimeter

4-connected object (black)

Boundary pixels have at least one white 8-neighbor

8-connected object (black)

Boundary pixels have at least one white 4-neighbor

If the object is 4-connected, the background is 8-connected and vice versa

Perimeter is the number of boundary pixels

Chain code: start at uppermost, leftmost boundary pixel - list DIRECTION to next pixel until the first one is reached again

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Where is the center?

Centroid, center of mass

(average x over all pixels, average y over all pixels)

Center of contour

(average x, y over boundary pixels only)

Example (note sensitivity to contour!)

CM = CC CM CC

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Region Skeletons

Basis: thinning (many algorithms!) Maximum values of grassfire transform (medial axis)

Last pixels to disappear with repeated erosion

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Hierarchical or Graph Shape Description

Define a set of primitive shapes

Define a set of operations (concatenation, union, intersection, etc.)

Define a shape as a network of primitive shapes (parts) connected by operations

Recognize a shape by recognizing its parts and the relationships between them.

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Region Adjacency Graph

Primitive shape = "connected set of pixels"

Operations = "adjacent to" Element of region 1 is in the neighborhood of element of

region 2

In binary images, all regions with no holes are adjacent to the single background region

All holes are adjacent to the objects that contain them