cse 554 lecture 1: binary picturestaoju/cse554/lectures/lect01_binarypictu… · cse 554 lecture 1:...
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CSE554 Binary Pictures Slide 1
CSE 554
Lecture 1: Binary Pictures
Fall 2016
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CSE554 Binary Pictures Slide 2
2xy ][][ ySinxSinz
Triangle surfaces
Polyline
Curves Surfaces
Geometric Forms
• Continuous forms
– Defined by mathematical functions
– E.g.: parabolas, splines, subdivision
surfaces
• Discrete forms
– Disjoint elements with connectivity
relations
– E.g.: polylines, triangle surfaces, pixels
and voxels
Pixels Voxels
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CSE554 Binary Pictures Slide 3
Digital Pictures
• Made up of discrete points associated with colors
– Image: 2D array of pixels
2D Image
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CSE554 Binary Pictures Slide 4
Digital Pictures
• Color representations
– Grayscale: 1 value representing grays from
black (lowest value) to white (highest value)
• 8-bit (0-255), 16-bit, etc.
– RGB: 3 values each representing colors from
black (lowest value) to pure red, green, or
blue (highest value).
• 24-bit (0-255 in each color)
– XYZ, HSL/HSV, CMYK, etc.
Grayscale
Low High
RGB
Low R High R
Low G High G
Low B High B
R G
B
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CSE554 Binary Pictures Slide 5
Digital Pictures
• Made up of discrete points associated with colors
– Volume: 3D array of voxels
3D Volume
Voxel
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CSE554 Binary Pictures Slide 6
Binary Pictures
• A grayscale picture with 2 colors: black (0) and white (1)
– The set of 1 or 0 pixels (voxels) is called object or background
– A “blocky” geometry
Background
ObjectAnalogy: Lego, Minecraft
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CSE554 Binary Pictures Slide 7
Binary Pictures
Creation Processing
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CSE554 Binary Pictures Slide 8
Segmentation
• Separating object from background in a grayscale picture
– A simple method: thresholding by pixel (voxel) color
• All pixels (voxels) with color above a threshold is set to 1
Grayscale picture Thresholded binary picture
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CSE554 Binary Pictures Slide 9
Segmentation
• Separating object from background in a grayscale picture
– A simple method: thresholding by pixel (voxel) color
– Other methods:
• K-means clustering
• Watershed
• Region growing
• Snakes and Level set
• Graph cut
• …
– More details covered in Computer Vision course
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CSE554 Binary Pictures Slide 10
Rasterization
• Filling the interior of a shape by pixels or voxels
– Known as “scan-conversion”, or “pixelization / voxelization”
– More details covered in Computer Graphics course
2D Polygon Binary Picture
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CSE554 Binary Pictures Slide 11
Binary Pictures
Creation Processing
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CSE554 Binary Pictures Slide 12
Binary Pictures
Removing
islands and
holes
Smoothing
jagged
boundaries
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CSE554 Binary Pictures Slide 13
Removing Islands & Holes
Take the largest 2 connected
components of the object
Invert the largest connected
component of the background
• Observations:
• Islands (holes) of the object are holes (islands) of the background
• Islands (holes) are not as well-connected as the object (background).
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CSE554 Binary Pictures Slide 14
Connected Components
• Definition
– A maximum set of pixels (voxels) in the object or background, such that
any two pixels (voxels) in the set are connected by a path of connected
pixels (voxels)
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CSE554 Binary Pictures Slide 15
Connected Components
How many connected components are there in
the object? What about background?
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CSE554 Binary Pictures Slide 16
Connectivity (2D)
• Two pixels are connected if
their squares share:
– A common edge
• 4-connectivity
– A common vertex
• 8-connectivity
Two connected
pixels
x
x
All pixels
connected to x
4-connectivity
8-connectivity
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CSE554 Binary Pictures Slide 17
Connectivity (2D)
Object: 8-connectivity (1 comp) Object: 4-connectivity (4 comp)
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CSE554 Binary Pictures Slide 18
Connectivity (2D)
• What connectivity should be used for the background?
Object: 8-connectivity (1 comp)
Paradox: a closed curve does not disconnect the background,
while an open curve does.
Background: 8-connectivity (1 comp)
Object: 4-connectivity (4 comp)
Background: 4-connectivity (2 comp)
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CSE554 Binary Pictures Slide 19
Connectivity (2D)
• Different connectivity for object (O) and background (B)
– 2D pixels: 4- and 8-connectivity respectively for O and B (or B and O)
Object: 8-connectivity (1 comp)
Background: 4-connectivity (2 comp)
Object: 4-connectivity (4 comp)
Background: 8-connectivity (1 comp)
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CSE554 Binary Pictures Slide 20
Connectivity (3D)
• Two voxels are connected if
their cubes share:
– A common face
• 6-connectivity
– A common edge
• 18-connectivity
– A common vertex
• 26-connectivity
• Use 6- and 26-connectivity
respectively for O and B (or B and O)
6-connectivity
18-connectivity
26-connectivity
Two connected
voxels
All voxels connected
to the center voxel
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CSE554 Binary Pictures Slide 21
Finding Connected Components
• The “flooding” algorithm
– Start from a seed pixel/voxel, expand the connected component
– Either do depth-first or breadth-first search (a LIFO stack or FIFO queue)
// Finding the connected component containing an object pixel p
1. Initialize
1. Create a result set S that contains only p
2. Create a Visited flag at each pixel, and set it to be
False except for p
3. Initialize a queue (or stack) Q that contains only p.
2. Repeat until Q is empty:
1. Pop a pixel x from Q.
2. For each unvisited object pixel y connected to x, add y
to S, set its flag to be visited, and push y to Q.
3. Output S
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CSE554 Binary Pictures Slide 22
Finding Connected Components
• Why using a “visited” flag?
– Otherwise, the program will not terminate
• Why not checking to see if y is in S?
– Checking the visited flag is much faster ( O(1) vs. O(log n) )
1. …
2. Repeat until Q is empty:
1. Pop a pixel x from Q.
2. For each unvisited object pixel y connected to x, add y
to S, set its flag to be visited, and push y to Q.
3. Output S
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CSE554 Binary Pictures Slide 23
Connectivity (2D)
• Connected components containing the blue pixel:
8-connectivity 4-connectivity
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CSE554 Binary Pictures Slide 24
Finding Connected Components
• Labeling all components in an image:
– Loop through each pixel (voxel). If it is not labeled, use it as a seed to find
a connected component, then label all pixels (voxels) in the component.
8-connected object 4-connected object
One
component
Two
components
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CSE554 Binary Pictures Slide 25
Using Connected Components
• Pruning isolated islands from the main object
• Filling interior holes of the object
Take the largest components of
the object
Invert the largest components of
the background
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CSE554 Binary Pictures Slide 26
Morphological Operators
• Smoothing out object boundary
Opening
Closing
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CSE554 Binary Pictures Slide 27
Morphological Operators
• Operations to change shapes
– Erosion
– Dilation
– Opening: first erode, then dilate.
– Closing: first dilate, then erode.
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CSE554 Binary Pictures Slide 28
Mathematical Morphology
Object (A) Structure element (Bx)
x
DilationErosion
𝑨⊖𝑩 = {𝒙 ∈ 𝑨|𝑩𝒙 ⊆ 𝑨} 𝑨⊕𝑩 =
𝒙∈𝑨
𝑩𝒙
Input:
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CSE554 Binary Pictures Slide 29
Mathematical Morphology
• Structure element B is symmetric if:
• Examples:
𝒙 ∈ 𝑩𝒚 ⟺ 𝒚 ∈ 𝑩𝒙
𝒙𝒚
𝒙
𝒚
Circle Square
𝒙𝒚
Triangle
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CSE554 Binary Pictures Slide 30
Mathematical Morphology
• Duality (for symmetric structuring elements)
– Erosion (dilation) is equivalent to dilation (erosion) of the background
DilationErosion
𝑨⊖𝑩 = 𝑨⊕𝑩 𝑨⊕𝑩 = 𝑨⊖𝑩
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CSE554 Binary Pictures Slide 31
Mathematical Morphology
• Opening (erode, then dilate)
– Union of all B that can fit inside A
• Shaves off convex corners and thin spikes
Object (A) Structure element (Bx)
x
Opening
𝑨 ∘ 𝑩 = (𝑨⊝𝑩)⊕𝑩
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CSE554 Binary Pictures Slide 32
Mathematical Morphology
• Closing (dilate, then erode)
– Complement of union of all B that can fit in the complement of A
• Fills concave corners and thin tunnels
Closing
Object (A)
Structure element (Bx)
x
𝑨 ⋅ 𝑩 = (𝑨⊕𝑩)⊝𝑩
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CSE554 Binary Pictures Slide 33
Mathematical Morphology
• Duality, again! (for symmetric structuring elements)
– Opening (closing) object is equivalent to closing (opening) background
ClosingOpening
𝑨 ∘ 𝑩 = 𝑨 ⋅ 𝑩 𝑨 ⋅ 𝑩 = 𝑨 ∘ 𝑩
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CSE554 Binary Pictures Slide 34
Digital Morphology
• Structuring elements (symmetric)
– 2D pixels: square or cross
– 3D voxels: cube or cross
xx
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CSE554 Binary Pictures Slide 35
Digital Morphology
• Structuring element:
– Erosion
• e: an object pixel with some
background pixel in its square
neighborhood
– Dilation
• d: a background pixel with some
object pixel in its square
neighborhood
e
e
e
e
e
ee e
e
d
dd d
d d
d
d
d
d
d
d
d d
d d
d d
x
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CSE554 Binary Pictures Slide 36
Digital Morphology
• Structuring element: 3x3
square
– Opening
– Closing
e
e
e
e
e
ee e
e dd d
dd d
d d
Erosion Dilation
dd d
d dd d
d d
d d
d d
d ddd d
d
ee e
e ee e
e e
e e
e e
e eee e
Dilation Erosion
Union of white
squares within the
object
Union of black
squares within the
background
x
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CSE554 Binary Pictures Slide 37
Digital Morphology
• Increasing the size of the structuring element
– Leads to more growing/shrinking and more significant smoothing
– Equivalent to repeated applications with a small structuring element
• E.g.: k erosions (dilations) followed by k dilation (erosions) with a 3x3 square is
equivalent to opening (closing) with a (2k+1)x(2k+1) square.
Original Opening by 3x3 square Opening by 5x5 square
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CSE554 Binary Pictures Slide 38
Digital Morphology
• Implementation tips
– Using duality of erosion and dilation, you only need to implement one
function to do both morphological operations (for symmetric structure
elements).
• Dilation is same as erosion of the background
– When performing multiple-round opening, make sure you first do k times
erosion then k times dilation
• What happens if you alternate erosion and dilation for k times?
– Handle image boundary in a graceful way (not crashing the program…)
• For example, treat outside of the image as background
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CSE554 Binary Pictures Slide 39
• A simple 2D segmentation routine
– Initial segmentation using thresholding (using your code from Lab 0)
– Using connected components and opening/closing to “clean up” the
segmentation.
Lab Module 1