Algebraic Operations

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Algebraic Operations

• Prof. K. J. Hintz
• Department of Electrical and Computer Engineering
• George Mason University

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Algebraic Operations

• Add, Subtract, Multiply or Divide Images on a
  Pixel-by-Pixel Basis
• Non-invertible
• 223113
• Information is lost due to data processing
  theorem
• Addition for White Noise Reduction
• If additive noise is uncorrelated, then S/N
  increases by square root of the number of samples
  added

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Algebraic Operations

• Subtraction of Displaced Images Yields Derivative
• dI/dx by shifting one pixel horizontal
• dI/dy by shifting one pixel vertical
• Subtraction of Transmission X-ray
• If image is of clutter and target, image of
  clutter without target can be subtracted from CT
  to yield uncluttered target

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Algebraic Operations

• Multiplication for Masking
• A region of an image can be zeroed by multiplying
  by an array of ones with the region to be masked
  replaced by zeroes
• Color Ratios for Segmentation
• Pixels of multi-band images can be replaced by a
  statistic based on the band values at that pixel,
  e.g.

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Byproducts of Addition

• Sum of Two Pixel Values May Exceed Range of
  Output Variable
• If range of output variable is exceeded, then
  clipping occurs,
• e.g., if unsigned byte, 230 80 310 gt 255
• Insure data type is sufficient to handle expected
  range
• Rescale after addition to the desired type
• General topic is scaled integer arithmetic

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Effect of Image Correlation

• Histogram of Images Which Have Been Operated on
  by Algebraic Operation Depends on Correlation of
  Input Images

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Histogram of Summed Images

• If Two Images are Uncorrelated, then

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Convolution

• Convolution in One Domain is Multiplication in
  the Fourier Transform of that Domain
• Isomorphism exists between spatial and spatial
  frequency domain and spatial convolution and
  spatial frequency multiplication
• Histogram of Sum can be Computed by
• Histogram of each image
• Fourier transform of each histogram
• Multiply transformed histograms
• Inverse Fourier transform the product

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Expectation Operator
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Summation for Noise Reduction

• If Statistically Independent Noise is Added to
  Each of Several Identical Images, the Sum Image
  has a Higher SNR
• SNR can be Computed by Taking Ratio of Expected
  Values of Signal and Noise

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Amplitude SNR Improvement

• SNR Improves as the Square Root of the Number of
  Uncorrelated Images Summed

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Edge Detection Using Subtraction

• If an Image Is Displaced (Translated) Relative to
  Another Image, Then the Difference Between Them
  Approximates the First Derivative

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Generalization of Edge Detection, Gradient
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Gradient Properties

• Points in Direction of Maximum Upward Slope
• Magnitude is Equal to Value of Slope
• Useful Scalar Function of Gradient is the
  Gradient Magnitude

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Gradient Magnitude Approx.

• Square Root is Computationally Expensive, so
  Sometimes Simpler Approximation is Used

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Gradiant Operators

• Subimages can be multiplied by operators which
  are discrete approximation of gradiant
• Sobel
• -1 2 1 -1 0 1
• 0 0 0 -2 0 2
• 1 2 1 -1 0 1
• Prewitt
• Isotropic
• Laplacian

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Gradiant Operators
Sobel
Roberts
Isotropic
Prewitt
khoros
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Multiplication for Masking

• Use Feature for Deciding Pixels to Include
• Generate Mask Image of 1s and 0s
• Multiply Original Image by Mask
• Result is Image of Unmasked Pixels