Department of Data Communication Networks and Systems
Lecturer Shukhrat Palvanov
Continuous Fourier analysis,
Fourier transform
What is color?
Color and light
Color of light arriving at camera depends on
– Spectral reflectance of the surface light is leaving
– Spectral radiance of light falling on that patch
Color perceived depends on
– Physics of light
– Visual system receptors
– Brain processing, environment
Color and light
White light:
composed of about equal energy in all wavelengths of the visible spectrum
The Eye
The human eye is a camera!
– Iris - colored annulus with radial muscles
– Pupil - the hole (aperture) whose size is controlled by the iris
– Lens - changes shape by using ciliary muscles (to focus on objects at different distances)
– Retina - photoreceptor cells
Types of light-sensitive receptors
Cones
cone-shaped less sensitive operate in high light color vision
Rods
rod-shaped highly sensitive operate at night gray-scale vision
Types of cones
• React only to some wavelengths, with different sensitivity (light fraction absorbed)
• Brain fuses responses from local neighborhood of several cones for perceived color
• Sensitivities vary per person, and with age
• Color blindness: deficiency in at least one type of cone
Color mixing
Cartoon spectra for color names:
Additive color mixing
Colored lights are mixed using additive color properties. Light colors are combining two or more additive colors together which creates a lighter color that is closer to white.
Examples of additive color systems
Subtractive color mixing
When we mix colors using paint, or through the printing process, we are using the subtractive color method. Subtractive color mixing means that one begins with white and ends with black; as one adds color, the result gets darker and tends to black
RGB color space
HSV color space
A cylindrical coordinate representation of points in an RGB color model
– Hue, Saturation, Value
– Nonlinear – reflects topology of colors by coding hue as an angle
RGB to HSV
HSV to RGB
RGB to Gray
Color-based image retrieval
Given collection (database) of images:
– Extract and store one color histogram per image
Given new query image:
– Extract its color histogram
– For each database image:
Compute intersection between query histogram and database histogram
– Sort intersection values (highest score = most similar)
– Rank database items relative to query based on this sorted order
What is edge?
Edges in images are areas with strong intensity contrasts. Infact edge is a jump in intensity from one pixel/region to the next.
What is edge detection?
Edge detection is a terminology in image processing and computer vision, particularly in the areas of feature detection and feature extraction, to refer to algorithms which aim at identifying points in a digital image at which the image brightness changes sharply or more formally has discontinuities.
Why we do need edge detection?
What is Fourier Transform (FT)?�
A powerful mathematical tool that converts a signal or image from the spatial (time/space) domain into the frequency domain.
Instead of analyzing the values of pixels directly, FT focuses on how often intensity values change.
Basic Idea
Importance of FT in Image Processing�
Mathematical Background�Fourier Series vs Fourier Transform�
Fourier Series: Represents periodic signals as a sum of sinusoids.
Fourier Transform (FT): Extends the concept to non-periodic signals.
In image processing, signals (images) are generally non-periodic → FT is used.
2D Fourier Transform�
2D Fourier Transform
Interpretation
Why Important?
Fourier Transform Properties
1. Linearity
FT of a sum = Sum of FTs
Useful for combining signals.
2. Shift Property
A shift in spatial domain → phase change in frequency domain.
Image translation does not affect magnitude spectrum.
3. Scaling (Dilation)
Enlarging an image in spatial domain shrinks its spectrum in frequency domain.
Compression in spatial domain expands the spectrum.
4. Convolution Theorem
Convolution in spatial domain = Multiplication in frequency domain.
Very useful for image filtering (blurring, sharpening).
5. Symmetry
For real-valued images, the Fourier spectrum is symmetric.
Frequency Representation of Images
Key Concept
An image can be expressed as a combination of low-frequency and high-frequency components.
Fourier Transform separates these components clearly.
Low Frequencies
Located near the center of Fourier spectrum.
Represent smooth intensity variations (background, lighting, gradual changes).
Carry the overall structure of the image.
High Frequencies
Located at the edges of the spectrum.
Represent rapid intensity changes (edges, fine details, noise).
Important for sharpness and texture.
Discrete Fourier Transform (DFT)
Why DFT?
Visualization of Fourier Spectrum
Magnitude and Phase
Log Transformation
Thank you