ece 420-embedded dsp laboratory...spatial filteringis a predefined operation that is performed on...
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ECE 420- Embedded DSP LaboratoryLecture 5 – Image Processing
Thomas MoonFebruary 24, 2020
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• Lab4 - pitch detection
• Lab5 - pitch synthesis• Resampling• TD-PSOLA
• Lab6 – image processing
Lab Summary So Far
2
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𝑡
1-dimensional signal
𝑢 = 𝑓(𝑡)
𝑥𝑦
𝑢 = 𝑓(𝑥, 𝑦)
2-dimensional signal
à spatial-domain
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Digital Image Acquisition
González, R. Callejas and Richard E. Woods. “Digital Image Processing, Third Edition.” (2009).
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• Geometric Transformation
• Intensity Transforming• Spatial Filtering
Spatial Domain Process𝑔 𝑥, 𝑦 = 𝑇[𝑓 𝑥, 𝑦 ]
scaling
rotation
translation
original image
𝑥
𝑦
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Intensity
255 255 255 255 … 255255255 0 0 0. 0 0 0. 255 0 0. 255 0 0
255
𝑓 𝑥, 𝑦 =
Pixel values can be• binary (0/1)• grayscale (single intensity value, e.g. 8bit intà[0,255]) • color (e.g. 3-channel RGB image)
pixel
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• Change the value of pixels. Basic Intensity Transformation
𝑠 = 𝑇(𝑟)
𝑠 = 𝑟 + 𝐶
𝑠, 𝑟 ∈ [0, 𝐿 − 1]
Enhance image
𝐶 = 100 𝐶 = 100
𝑟
𝑠
𝐶
𝐿 − 1
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• Change the value of pixels. Basic Intensity Transformation
𝑠 = 𝑇(𝑟)
𝑠 = 𝐿 − 1 − 𝑟
𝑠, 𝑟 ∈ [0, 𝐿 − 1]
𝐿 − 1
𝐿 − 1 𝑟
𝑠
Negative image
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• Change the value of pixels. Basic Intensity Transformation
𝑠 = 𝑇(𝑟) 𝑠, 𝑟 ∈ [0, 𝐿 − 1]𝑠
𝐿 − 1
𝐿 − 1 𝑟Intensity-level Slicing
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• Histogramà Represents distribution of numerical data
• Each bin denotes a particular outcome
• Number assigned to a bin is the count of observed occurrencesof values for that bin
Intensity Transformation –Histogram Processing
Histogram of 1,000 two Dice Rolls
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original image: low-contrast
dark
bright
small dynamic range
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low-contrast, small dynamic range
high-contrast, wide dynamic range
Histogram Equalizationàmake your histogram (pdf) uniform distribution
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Histogram Equalization
𝑐𝑑𝑓(𝑠)
𝑠𝐿 − 1
1𝑐𝑑𝑓(𝑟)
𝑟𝐿 − 1
1
𝑝𝑑𝑓(𝑟)
𝑟𝐿 − 1
before𝑝𝑑𝑓(𝑠)
𝑠𝐿 − 1
1𝐿 − 1
after
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Histogram Equalization
𝑐𝑑𝑓 𝑥 =:;
<
ℎ𝑖𝑠𝑡[𝑘] ℎ𝑖𝑠𝑡[𝑘] = number of pixels with value 𝑘
𝑠 = ℎ 𝑟 = 𝑟𝑜𝑢𝑛𝑑𝑐𝑑𝑓 𝑟 − 𝑐𝑑𝑓BCD𝑀𝑁 − 𝑐𝑑𝑓BCD
(𝐿 − 1)
𝑀,𝑁: width, height of image𝐿: total number of gray levels
𝑐𝑑𝑓BCD: minimum non-zero value of cdf
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Spatial Filteringorigin
𝑥
𝑦
spatial domainimage 𝑓
(𝑥H, 𝑦H)
3X3 neighborhood of (𝑥H, 𝑦H)
Spatial filtering is a predefined operation that is performed on the image pixels surrounded by the neighborhood.
Intensity transformation is a special case of spatial filtering with 1X1 neighborhood size.
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2D Convolution
𝑔 𝑥, 𝑦 = 𝑓 𝑥, 𝑦 ∗ 𝑤 𝑥, 𝑦 = :C
:K
𝑓 𝑖, 𝑗 𝑤(𝑥 − 𝑖, 𝑦 − 𝑗)
𝑔 𝑡 = 𝑓 𝑡 ∗ 𝑤 𝑡 = :;
𝑓 𝑘 𝑤(𝑡 − 𝑘)
• 𝑤 typically referred to as the filter kernel
• Same idea of a weighted average of elements over a sliding window
𝑤
1D convolution
2D convolution
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2D Convolution - Example
0 0 0 0 0
0 0 0 0 0
0 0 1 0 0
0 0 0 0 0
0 0 0 0 0
1 2 3
4 5 6
7 8 9
𝑓 𝑥, 𝑦
𝑤 𝑥, 𝑦
𝑔 𝑥, 𝑦 = 𝑓 𝑥, 𝑦 ∗ 𝑤 𝑥, 𝑦 = :C
:K
𝑓 𝑖, 𝑗 𝑤(𝑥 − 𝑖, 𝑦 − 𝑗)
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2D Convolution - Example
0 0 0 0 0
0 0 0 0 0
0 0 1 0 0
0 0 0 0 0
0 0 0 0 0
9 8 7
6 5 4
3 2 1
𝑔 𝑥, 𝑦 = 𝑓 𝑥, 𝑦 ∗ 𝑤 𝑥, 𝑦 = :C
:K
𝑓 𝑖, 𝑗 𝑤(𝑥 − 𝑖, 𝑦 − 𝑗)
0
Assume zero-pad outside
rotated kernel
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2D Convolution - Example
0 0 0 0 0
0 0 0 0 0
0 0 1 0 0
0 0 0 0 0
0 0 0 0 0
9 8 7
6 5 4
3 2 1
𝑔 𝑥, 𝑦 = 𝑓 𝑥, 𝑦 ∗ 𝑤 𝑥, 𝑦 = :C
:K
𝑓 𝑖, 𝑗 𝑤(𝑥 − 𝑖, 𝑦 − 𝑗)
0 0
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2D Convolution - Example
0 0 0 0 0
0 0 0 0 0
0 0 1 0 0
0 0 0 0 0
0 0 0 0 0
9 8 7
6 5 4
3 2 1
𝑔 𝑥, 𝑦 = 𝑓 𝑥, 𝑦 ∗ 𝑤 𝑥, 𝑦 = :C
:K
𝑓 𝑖, 𝑗 𝑤(𝑥 − 𝑖, 𝑦 − 𝑗)
0 0 0 0 0
0 1
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2D Convolution - Example
0 0 0 0 0
0 0 0 0 0
0 0 1 0 0
0 0 0 0 0
0 0 0 0 0
9 8 7
6 5 4
3 2 1
𝑔 𝑥, 𝑦 = 𝑓 𝑥, 𝑦 ∗ 𝑤 𝑥, 𝑦 = :C
:K
𝑓 𝑖, 𝑗 𝑤(𝑥 − 𝑖, 𝑦 − 𝑗)
0 0 0 0 0
0 1 2
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2D Convolution - Example
0 0 0 0 0
0 0 0 0 0
0 0 1 0 0
0 0 0 0 0
0 0 0 0 0
9 8 7
6 5 4
3 2 1
𝑔 𝑥, 𝑦 = 𝑓 𝑥, 𝑦 ∗ 𝑤 𝑥, 𝑦 = :C
:K
𝑓 𝑖, 𝑗 𝑤(𝑥 − 𝑖, 𝑦 − 𝑗)
0 0 0 0 0
0 1 2 3 0
0 4 5 6 0
0 7 8 9
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2D Convolution - Example
0 0 0 0 0
0 0 0 0 0
0 0 1 0 0
0 0 0 0 0
0 0 0 0 0
9 8 7
6 5 4
3 2 1
𝑔 𝑥, 𝑦 = 𝑓 𝑥, 𝑦 ∗ 𝑤 𝑥, 𝑦 = :C
:K
𝑓 𝑖, 𝑗 𝑤(𝑥 − 𝑖, 𝑦 − 𝑗)
0 0 0 0 0
0 1 2 3 0
0 4 5 6 0
0 7 8 9 0
0 0 0 0 0
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2D Correlation - Example
0 0 0 0 0
0 0 0 0 0
0 0 1 0 0
0 0 0 0 0
0 0 0 0 0
1 2 3
4 5 6
7 8 9
𝑔 𝑥, 𝑦 = 𝑓 𝑥, 𝑦 ∗ 𝑤 𝑥, 𝑦 = :C
:K
𝑓 𝑖, 𝑗 𝑤(𝑥 + 𝑖, 𝑦 + 𝑗)
0 0 0 0 0
0 9 8 7 0
0 6 5 4 0
0 3 2 1 0
0 0 0 0 0
2D-correlation
non-rotated kernel
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• Different varieties of output sets for 2D convolution
1. Valid – region where w does not go outside image boundary. Output size 𝑁 − 𝐾 + 1.
2. Same – same size as input image, requires handling of elements outside of image. Output size 𝑁.
3. Full – expanded output image by sizeof w, usually assumes zeros outsideimage, useful for ‘overlap-add’ type image block processing. Output size 𝑁 + 𝐾 − 1
Convolution Output Domain
Full
Same
Valid
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0 0 0 0 0
0 0 0 0 0
0 0 1 0 0
0 0 0 0 0
0 0 0 0 0
9 8 7
6 5 4
3 2 1
0 0 0 0 0
0 0 0 0 0
0 0 1 0 0
0 0 0 0 0
0 0 0 0 0
9 8 7
6 5 4
3 2 1
0 0 0 0 0
0 0 0 0 0
0 0 1 0 0
0 0 0 0 0
0 0 0 0 0
9 8 7
6 5 4
3 2 1
Same
Valid
Full
Convolution Output Domain
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• “Average” the neighborhood pixels• Reduce “sharp” transitions in intensities• Similar to integration• Lowpass filter in frequency domain• Bigger size filter à More blurring
Smoothing Spatial Filters
1 1 1
1 1 1
1 1 1
PQ
X
1 2 1
2 4 2
1 2 1
PPR
X
3x3 averaging filter 3x3 weighted averaging filter 2D-Gaussian filter
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original image
Gaussian filter size of 15
Gaussian filter size of 31
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• Median filter• Replace the value of a pixel by the median of the
neighborhood intensity• Effective to remove the impulse noise or salt-and-
pepper noise
Smoothing Spatial Filters- Nonlinear
Original image corrupted by salt-and-pepper noise Gaussian filter Median filter
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• Highlights transitions in intensity
• Similar to derivative
• Highpass filter in frequency domain
ØFirst-order derivatives – the GradientØSecond-order derivatives – the Laplacian
Sharpening Spatial Filters
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3 3 3 2 1 0 0 0 0 3 3 3
0
-1
3
Intensity
𝑥
𝑥
𝑥0
-1
3
-3
𝜕𝑓𝜕𝑥
𝜕T𝑓𝜕𝑥T
ramp
step
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Gradient – First Derivative
∇𝑓 ≡𝑔<𝑔W ≡
𝜕𝑓𝜕𝑥𝜕𝑓𝜕𝑦
𝑧P 𝑧T 𝑧Y
𝑧Z 𝑧[ 𝑧R
𝑧\ 𝑧] 𝑧Q
𝑔< = 𝑧\ + 2𝑧] + 𝑧Q − (𝑧P + 2𝑧T + 𝑧Y) 𝑔W = 𝑧Y + 2𝑧R + 𝑧Q − (𝑧P + 2𝑧Z + 𝑧\)
−1 −2 −1
0 0 0
1 2 1
−1 0 1
−2 0 2
−1 0 1
𝑀 𝑥, 𝑦 = 𝑔<T + 𝑔WT
𝑥
𝑦
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𝑔<
𝑔W
𝑔<T + 𝑔WT
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Laplacian – Second Derivative
∇T𝑓 =𝜕T𝑓𝜕𝑥T
+𝜕T𝑓𝜕𝑦T
𝜕T𝑓𝜕𝑥T
= 𝑓 𝑥 + 1, 𝑦 + 𝑓 𝑥 − 1, 𝑦 − 2𝑓(𝑥, 𝑦)
𝜕T𝑓𝜕𝑦T
= 𝑓 𝑥, 𝑦 + 1 + 𝑓 𝑥, 𝑦 − 1 − 2𝑓(𝑥, 𝑦)
= 𝑓 𝑥 + 1, 𝑦 + 𝑓 𝑥 − 1, 𝑦 + 𝑓 𝑥, 𝑦 + 1 + 𝑓 𝑥, 𝑦 − 1 − 4𝑓(𝑥, 𝑦)
0 −1 0
−1 4 −1
0 −1 0
−1 −1 −1
−1 8 −1
−1 −1 −1
Laplacian filter mask Laplacian filter maskwith diagonal directions
Laplacian:
0 1 0
1 −4 1
0 1 0
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Laplacian Laplacian withdiagonal directions
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• Groups of 2-3 • Groups of 3 should be larger scope
• Lead-in to the Final Project• Explore a DSP algorithm from the literature
• Implementation in Python for this stage, NOT on tablet yet• Proposal for Assigned Lab
• Overview of proposed algorithm, cite source(s)• Plan for testing and validation• Rough idea(s) for Final Project application
• Assigned Lab Report submission at end of project• Demo incorporated into Final Proposal design review
Reminder: Assigned Lab
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• Lab 5: Pitch Synthesizer Quiz/Demo
• Lab 6: Image Processing (Histogram and Filtering)
• Assigned Project Lab Proposals Due 03/09 @2pm
This week