The digital image is a collection of values of sensors arranged in a grid. These sensors have a field of view in the original world. Then it generates a value based on the phenomena that it measures.
This phenomena can be visible light(for conventional digital images), infrared light(for infrared images), Positron emissions scattering(PET scan), secondary electrons (Scanning electron microscope), and scattered microwave(Synthetic aperture radar SAR).
In conventional image processing, a matrix is used to represent these values. For inverse problems we convert this matrix to a vector and then use the normal equations or regularization based methods.
Some PDE based methods use kernels to process images using 2D convolution. Some times we use the notion of images as point clouds and then construct a graph based on some distance measure. Then we study the Laplacians(Diffusion maps) or the Hamiltonian operators of these graphs. The eigenvalues of these operators lead to the notion of heat kernel signatures and wave kernel signatures.
We have these alternate views of an image based on the application.
What is the most fundamental view of an image which can bring all these different notions together?
Does such a notion exist in literature?
Dear Muhammad, I think there is not a universally accepted definition of digital image; at present, each of your applicative methodologies must acquire and compute informations from data memorized in file image. See proposal of UPDIG Org at link bellow.
http://www.updig.org/guidelines/ph_file_formats.html
There are many other techniques rather than matrix representation of image.
First one is FFT, (Fast Fourier Transform)
link https://homepages.inf.ed.ac.uk/rbf/HIPR2/fourier.htm
Second is DCT (Discrete cosine transform)
Third is DWT (Discrete wavelet transform)
they all represent image in mathematical formula
@Amin Ullah: I agree with what you say but my question is a bit more fundamental. I am raising the question as to what mathematical structure (set, function, etc.) should be used to define an image.
Regards,
Zubair
Well Muhammad,
I must say this is the beauty of #s , they let you use statistics /probability /group & set theory/s & z- transforms / Wavelets /PDE / ...
The story starts with digitizing real world analog signal (either thru ADC or pixelated sensors) .
The hidden story is : material vs light interaction , say in terms of IR /PET where , the incident light /frequency interact . Again , images have reflectance data that is they capture surface property or go deeper (depending on the wavelength) and come out (diffusion) and as they do that their scattering centers needs some analysis to be done ...
This is an important question. In addition to the helpful answers already given, there is a bit more to add.
Going back to basics, a digit image represents a mapping on a visual scene recorded by an optical sensor into an array of picture element intensities. There are many schools of thought on how this plays out in the production of a digital image.
Pixel model: RGB optical sensors map to a vector of discrete intensities which are integers. Let pi: RxGxB --> Z x Z x Z defined by p(r,g,b) = (zR,zG,zB), where r, g, b are reals and zR,zG,zB are integer-valued color intensities in the range from 0 to 255. In that case, a pixel is represented by a 1x3 column vector used to "paint" a tiny square in an image display.
For more about this, see
https://www.researchgate.net/publication/315103795_A_Conceptual_Abstraction_for_Human-Computer_Interaction_Data_Notes_for_the_Development_of_New_Techniques_for_Magnetic_Resonance_Imaging/figures?lo=1
Vector model: RGB optical sensor values are mapped to vectors and the displayed image conforms to a form of triangulated image space instead of a collection of pixels. In this model, a vector image is a collection vertices and edges. The end result of vector images is the formation of a basis for applications such as Adobe Photoshop and video games.
For more about this, see
https://www.researchgate.net/publication/221100768_Evaluation_of_VISTO_A_new_vector_image_search_TOol
and
https://www.researchgate.net/publication/315103795_A_Conceptual_Abstraction_for_Human-Computer_Interaction_Data_Notes_for_the_Development_of_New_Techniques_for_Magnetic_Resonance_Imaging/figures?lo=1
Conference Paper Evaluation of VISTO: A new vector image search TOol
Article A Conceptual Abstraction for Human-Computer Interaction: Dat...
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