ECSE6963, BMED 6961 Cell

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ECSE-6963, BMED 6961Cell Tissue Image Analysis

• Lecture 11 Blob Segmentation Basics (Contd)
• Badri Roysam
• Rensselaer Polytechnic Institute, Troy, New York
  12180.

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Recap

• Pre-processing can improve thresholding results
• Smoothing by averaging in space and/or time
• Average, weighted average, median, weighted
  median
• Rank filters (min, median, max) can help in a
  variety of ways
• Median filtering (smoothing), Grayscale Erosion,
  Dilation, Opening, Closing are basic cleanup
  operators that result from rank filters
• Choice of pixel neighbors makes a difference
• Today
• Adaptive Smoothing Algorithms

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Recap Basic Steps to Blob Segmentation

• Pre-process the image to improve Step 2
• Separate the foreground from the background
• We looked at intensity-based segmentation methods
• Gives us a set of connected blobs
• Break up the connected blobs into individuals
  using every available hint
• Shape, appearance (intensity, texture),
  fluorescence
• Make measurements for each individual blob
• Classify individual blobs into cell types
• Figure out how much confidence we have in our
  results
• Present the low confidence cases to the user
  for editing
• Present the edited results for acceptance!

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Introducing the Idea of Iterative Smoothing

• Smoothing is performed in a series of small
  steps, instead of just one step
• One simple approach is to apply the same
  algorithm repeatedly
• Repeated median gives an unchanging pattern
  called the median root
• Consists of flat regions with sloped edges

Median root, radius 1
Smoothing
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Adaptive Smoothing

• Basic Idea Apply different algorithms and/or
  settings to different pixels based on some local
  criteria.
• No universal algorithm, need to build a custom
  algorithm for each application/modality/object
  type
• More computation, but often well worth it.
• Example 1
• Smooth the background, but not the objects of
  interest.
• Pixels brighter than threshold T are left
  untouched, while others are replaced by
  neighborhood averages (i.e., smoothed over)
• Example 2
• Suppress the averaging across known object
  boundaries

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Hints from Object Boundaries

• Local Object Boundary
• Local Intensity Gradient
• No smoothing in this direction please!!
• Local Perpendicular to Gradient
• Smoothing in this direction is good!!

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Simple-minded idea Directional 3x3 Smoothing

• Replace pixel value with weighted average for
  each kernel
• Pick the one that results in the least change
• Implicitly selects the kernel that is
  perpendicular to the boundary
• The type of averaging can vary
• E.g., center-weighted average
• E.g., median instead of mean

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Practical Considerations

• This algorithm is cute but naïve
• Every little wrinkle in the image appears like
  an edge to this algorithm
• It has a limited 3?3 view of the image
• Need to distinguish real object boundaries, and
  avoid false internal ones
• Unfortunately, smoothing can change that answer
• We need ways to prevent premature decisions about
  object boundaries
• Iterative algorithms can help!

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Edges, Edgels, Boundaries

• Edge Locations of abrupt change in image
  properties
• Intensity, Texture, Color (spectral signature)
• Edge also used for connected chains of edge
  points, i.e., fragments of object contours
• Edge points also called edgels (for edge
  elements)
• The term boundary is used for the real thing
• Edges are a fragmentary estimate of the boundary

Biggest Problem Separating the edges that really
correspond to object boundaries from others
associated with texture and/or noise
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Broader Indications of Boundaries
Noisy observation
True intensity
Step
Ramp
Roof
Ridge
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Basic Data Structures

• Segmentation Labels
• Basis for choosing a custom averaging formula,
  or a parameter, at each pixel
• Edge Sites
• Basis for suppressing averaging across object
  boundaries
• Better segmentation helps boundary detection, and
  vice versa
• Classic chicken and egg situation

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Local Intensity Gradient
is a maximum when
Object boundaries in fluorescence images commonly
occur at locations of high intensity gradients
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Edge Operators
Central Difference Approximation

• Finite-difference approximations to gradient
• Inherently, a noise enhancing operation
• Responds to every little change in I(x,y)

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Example
y
x
Gradient Magnitude
Angle
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Ways to Combat Noise
Averaging along the boundary (Prewitt operator)
Weighted Averaging (Sobel operator)
Another idea Fit a plane
to a local neighborhood of pixels
Then,
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Another Noise Combating Method

• Just suppress weak edges!
• Non-maximum suppression
• At each pixel (x,y), look within a small
  neighborhood
• Find the maximum gradient magnitude
• If the gradient magnitude at (x,y) is less than
  the maximum, simply attenuate it, or set it to
  zero!
• Better
• Find the local gradient direction
• Do the suppression just along this direction

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Thresholding

• Really, another noise combating method
• Non-maximum suppression still leaves noise-caused
  local maxima
• If the gradient magnitude is greater than a
  threshold T, set edgel to 1
• Else, set edgel to 0.
• Choosing a threshold
• T too low ? Pick up too many noisy edgels
• T too high ? Miss real edgels, making boundaries
  discontinuous.

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A Fresh, Look at Gaussian Smoothing
Hot
Pretend for now that pixel intensities I(x,y)
describe the temperature profile over a 2-D
region. With time t, this profile spreads out
according to the following formula
Solution for this differential equation
convolution
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A Fresh Look at Gaussian Smoothing
One way to perform Gaussian smoothing by a kernel
of standard width ? is to solve the heat
diffusion equation, with
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Discrete Scale Space
Consider the 1-D case for simplicity
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Lindebergs Result
Our two scale space requirements are met if and
only if
Limiting case ??1/4 leads to a very simple
update equation
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Lindebergs Result
Recall Approximate Gaussian Smoothing Filter
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Practical Observations

• The ¼ ½ ¼ filter preserves the summed
  intensity of the image
• N iterations give us a variance of N2
• Computation of this filter is extremely efficient
• Lends itself to parallel computation nicely
• Bottom Line
• The Gaussian can be realized by simulating a
  diffusion process

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Smoothing Before Differentiation
Approximate Gaussian Smoothing Filter
Differentiator
Single operation that combines both
(scale factor)
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Smoothing Along a Boundary
Note If image brightness goes up from left to
right, the result response is positive ? The
sign of the response can be put to good use
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Anisotropic Diffusion
Isotropic heat diffusion
constant
Anisotropic version
Analogous to edge sites
Interpret edge sites as conduction coefficients
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The conduction coefficients
Better Use a smoothed version of I(x,y)
instead People have proposed other ways to choose
D(x,y) Basically, any good edge indicator in the
range (0,1) usually works.
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Discrete Approximation
Discrete Approximation
Iterative Solution
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Anisotropic Diffusion
At each iteration, we also update the edge sites
according to
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Example
original
Gaussian, radius 0.7
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Example
Anisotropic diffusion
Gaussian, radius 0.7
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Example
Anisotropic diffusion
Median
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Summary

• Iterative Adaptive Smoothing Algorithms
• Exploiting edges allows effective image smoothing
• Term Projects
• Need a 1-page proposal
• Cross-disciplinary team formation preferred
• Next Class
• Post-processing of thresholding results
• Connected components labeling
• Distance transforms separation of connected
  components

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Instructor Contact Information

• Badri Roysam
• Professor of Electrical, Computer, Systems
  Engineering
• Office JEC 7010
• Rensselaer Polytechnic Institute
• 110, 8th Street, Troy, New York 12180
• Phone [phone]
• Fax [phone]
• Email roysam_at_ecse.rpi.edu
• Website http//www.ecse.rpi.edu/roysam
• Course website http//www.ecse.rpi.edu/roysam/CT
  IA
• Secretary Laraine Michaelides, JEC 7012, (518)
  276 8525, michal_at_.rpi.edu
• Grader Piyushee Jha (jhap_at_rpi.edu)

Center for Sub-Surface Imaging Sensing