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ImageVisLab - Formula Reference

Complete mathematical reference for all image processing operations

Point Operations (Intensity Transformations)

Point operations transform each pixel independently based only on its intensity value.

Negative Transform

Formula:

s = (L - 1) - r

Variables:

Symbol Meaning
s Output intensity
r Input intensity
L Number of gray levels (256 for 8-bit)

Effect: Creates a photographic negative. Dark becomes light, light becomes dark.

Power-Law (Gamma) Transformation

Formula:

s = c * r^gamma

Variables:

Symbol Meaning Typical Range
s Output intensity [0, 255]
r Normalized input [0, 1]
c Scaling constant 1.0
gamma Gamma exponent [0.1, 5.0]

Effects:

  • gamma < 1: Brightens image (expands dark tones)
  • gamma = 1: No change
  • gamma > 1: Darkens image (compresses dark tones)

Applications: Monitor calibration, contrast enhancement

Logarithmic Transformation

Formula:

s = c * log(1 + r)

Variables:

Symbol Meaning
s Output intensity
r Input intensity
c Scaling constant (255/log(256))

Effect: Compresses high dynamic range. Maps narrow dark range to wide output range.

Application: Displaying Fourier spectra, HDR images

Quantization (Bit Depth Reduction)

Formula:

s = floor(r / k) * k + (k / 2)

Where k = 256 / levels

Variables:

Symbol Meaning Example
r Input intensity 127
levels Number of output levels 8
k Step size 32
s Quantized output 112

Effect: Posterization effect, reduces unique colors.

Subsampling (Spatial Resolution Reduction)

Method:

For each block of size n x n:
    output = average(all pixels in block)

Variables:

Symbol Meaning Range
n Block size [1, 32]

Effect: Pixelation effect, reduces image detail.

Histogram Equalization

Formula:

s_k = round((L - 1) * CDF(r_k))

Where:

CDF(r_k) = Sum p(r_j)  for j = 0 to k
p(r_j) = n_j / n

Variables:

Symbol Meaning
s_k Output intensity for level k
L Number of gray levels
CDF Cumulative Distribution Function
p(r_j) Probability of intensity j
n_j Number of pixels with intensity j
n Total number of pixels

Effect: Spreads histogram to use full dynamic range, enhances contrast.

Spatial Filters (Convolution)

Spatial filters use pixel neighborhoods to compute output values.

General Convolution

Formula:

g(x,y) = Sum Sum f(x+i, y+j) * h(i,j)
         i   j

Variables:

Symbol Meaning
g(x,y) Output pixel at (x,y)
f(x,y) Input pixel at (x,y)
h(i,j) Kernel/mask coefficient

Box Blur (Mean Filter)

Kernel (3x3):

    1/9  1/9  1/9
    1/9  1/9  1/9
    1/9  1/9  1/9

Formula:

g(x,y) = (1/n^2) * Sum f(x+i, y+j)

Effect: Simple smoothing, reduces noise but also blurs edges.

Gaussian Blur

Kernel Generation:

G(x,y) = (1 / 2*pi*sigma^2) * exp(-(x^2 + y^2) / 2*sigma^2)

3x3 Kernel (sigma=1):

    0.075  0.124  0.075
    0.124  0.204  0.124
    0.075  0.124  0.075

Variables:

Symbol Meaning Range
sigma Standard deviation [0.5, 3.0]
n Kernel size 3, 5, 7

Effect: Smooth blur that preserves edges better than box blur.

Sharpening (Unsharp Mask)

Formula:

sharpened = original + k * (original - blurred)

Laplacian Kernel:

     0  -1   0
    -1   5  -1
     0  -1   0

Effect: Enhances edges and fine details.

Laplacian (Edge Detection)

Formula:

Nabla^2 f = d^2f/dx^2 + d^2f/dy^2

Kernel:

     0  -1   0
    -1   4  -1
     0  -1   0

Effect: Detects edges (zero-crossings indicate edges).

Morphological Operations

Binary operations using structuring elements.

Binarization (Thresholding)

Formula:

s = { 0,   if r < T
    { 255, if r >= T

Variables:

Symbol Meaning Range
r Input intensity [0, 255]
T Threshold [0, 255]
s Output (binary) 0 or 255

Erosion

Set Definition:

A - B = { z | B_z subset A }

Meaning: Output pixel is white only if ALL pixels under the structuring element are white.

3x3 Square Element:

    1  1  1
    1  1  1
    1  1  1

Effect: Shrinks white regions, removes small white noise.

Dilation

Set Definition:

A + B = { z | B_z intersect A != empty }

Meaning: Output pixel is white if ANY pixel under the structuring element is white.

Effect: Expands white regions, fills small black holes.

Opening

Formula:

A o B = (A - B) + B

Process: Erosion followed by Dilation

Effect:

  • Removes small white objects
  • Smooths object boundaries
  • Separates weakly connected objects

Closing

Formula:

A . B = (A + B) - B

Process: Dilation followed by Erosion

Effect:

  • Fills small black holes
  • Connects nearby objects
  • Smooths object boundaries

Distance Metrics

Used in the pixel inspector for neighborhood analysis.

D4 (City Block / Manhattan)

Formula:

D4(p, q) = |x_p - x_q| + |y_p - y_q|

4-connected neighborhood: Shares edge with center pixel.

D8 (Chessboard / Chebyshev)

Formula:

D8(p, q) = max(|x_p - x_q|, |y_p - y_q|)

8-connected neighborhood: Shares edge OR corner with center pixel.

DE (Euclidean)

Formula:

DE(p, q) = sqrt[(x_p - x_q)^2 + (y_p - y_q)^2]

True geometric distance between pixels.

References

  1. Gonzalez, R.C. & Woods, R.E. (2018). Digital Image Processing (4th ed.). Pearson.
  2. Szeliski, R. (2022). Computer Vision: Algorithms and Applications (2nd ed.). Springer.

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