Digital image processing is the use of a
digital computer
A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations automatically. Modern computers can perform generic sets of operations known as Computer program, programs. These ...
to process
digital image
A digital image is an composed of s, also known as ''pixels'', each with ', ' of numeric representation for its or that is an output from its fed as input by its denoted with ''x'', ''y'' on the x-axis and y-axis, respectively. Depending on ...
s through an
algorithm
In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as specifications for performing s, , , and other ...

.
As a subcategory or field of
digital signal processing
Digital signal processing (DSP) is the use of digital processing
Digital data, in information theory and information systems, is information represented as a string of discrete symbols each of which can take on one of only a finite number of ...
, digital image processing has many advantages over
analog image processing
Analog or analogue may refer to:
Computing and electronics
* Analog signal, in which information is encoded in a continuous variable
** Analog device, an apparatus that operates on analog signals
*** Analog electronics, circuits which use analog s ...
. It allows a much wider range of algorithms to be applied to the input data and can avoid problems such as the build-up of
noise
Noise is unwanted sound
In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid.
In human physiology and psychology, sound is the ''reception'' of such waves and t ...
and
distortion
Distortion is the alteration of the original shape (or other characteristic) of something. In communications
Communication (from Latin ''communicare'', meaning "to share" or "to be in relation with") is "an apparent answer to the painful di ...

during processing. Since images are defined over two dimensions (perhaps more) digital image processing may be modeled in the form of
multidimensional systems
In mathematical systems theory, a multidimensional system or m-D system is a system in which not only one independent variable exists (like time), but there are several independent variables.
Important problems such as factorization and Stability t ...
. The generation and development of digital image processing are mainly affected by three factors: first, the development of computers; second, the development of mathematics (especially the creation and improvement of discrete mathematics theory); third, the demand for a wide range of applications in environment, agriculture, military, industry and medical science has increased.
History
Many of the techniques of
digital image
A digital image is an composed of s, also known as ''pixels'', each with ', ' of numeric representation for its or that is an output from its fed as input by its denoted with ''x'', ''y'' on the x-axis and y-axis, respectively. Depending on ...
processing, or digital picture processing as it often was called, were developed in the 1960s, at
Bell Laboratories
Nokia Bell Labs (formerly named Bell Labs Innovations (1996–2007), AT&T Bell Laboratories (1984–1996) and Bell Telephone Laboratories (1925–1984)) is an American industrial research and scientific development company
A company, ab ...
, the
Jet Propulsion Laboratory
The Jet Propulsion Laboratory (JPL) is a federally funded research and development center
Federally funded research and development centers (FFRDCs) are public-private partnerships which conduct research and development
Research is " c ...
,
Massachusetts Institute of Technology
Massachusetts Institute of Technology (MIT) is a private
Private or privates may refer to:
Music
* "In Private
"In Private" was the third single in a row to be a charting success for United Kingdom, British singer Dusty Springfield, aft ...
,
University of Maryland
The University of Maryland, College Park (University of Maryland, UMD, or simply Maryland) is a public
In public relations
Public relations (PR) is the practice of managing and disseminating information from an individual or an org ...
, and a few other research facilities, with application to
satellite imagery
Satellite images (also Earth observation imagery, spaceborne photography, or simply satellite photo) are images of Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. About 29% of Earth's su ...
,
wire-photo standards conversion,
medical imaging
Medical imaging is the technique and process of imaging
Imaging is the representation or reproduction of an object's form; especially a visual representation (i.e., the formation of an image).
Imaging technology is the application of materi ...
,
videophone
Videotelephony, sometimes also referred to as video teleconference or videoconferencing, comprises the technologies for the reception and transmission of audio
Audio most commonly refers to sound
In physics
Physics (from grc, φυ ...
,
character recognition
Optical character recognition or optical character reader (OCR) is the electronic or mechanical conversion of image
An SAR radar imaging, radar image acquired by the SIR-C/X-SAR radar on board the Space Shuttle Endeavour shows the Teide ...
, and photograph enhancement. The purpose of early image processing was to improve the quality of the image. It was aimed for human beings to improve the visual effect of people. In image processing, the input is a low-quality image, and the output is an image with improved quality. Common image processing include image enhancement, restoration, encoding, and compression. The first successful application was the American Jet Propulsion Laboratory (JPL). They used image processing techniques such as geometric correction, gradation transformation, noise removal, etc. on the thousands of lunar photos sent back by the Space Detector Ranger 7 in 1964, taking into account the position of the sun and the environment of the moon. The impact of the successful mapping of the moon's surface map by the computer has been a huge success. Later, more complex image processing was performed on the nearly 100,000 photos sent back by the spacecraft, so that the topographic map, color map and panoramic mosaic of the moon were obtained, which achieved extraordinary results and laid a solid foundation for human landing on the moon.
The cost of processing was fairly high, however, with the computing equipment of that era. That changed in the 1970s, when digital image processing proliferated as cheaper computers and dedicated hardware became available. This led to images being processed in real-time, for some dedicated problems such as
television standards conversion
Television standards conversion is the process of changing a television transmission or recording from one television
Television, sometimes shortened to TV or telly, is a telecommunication Media (communication), medium used for transmitt ...
. As
general-purpose computer
A computer is a machine
A machine is a man-made device that uses power to apply forces and control movement to perform an action. Machines can be driven by animals and people
A people is a plurality of person
A person (plural ...
s became faster, they started to take over the role of dedicated hardware for all but the most specialized and computer-intensive operations. With the fast computers and signal processors available in the 2000s, digital image processing has become the most common form of image processing, and is generally used because it is not only the most versatile method, but also the cheapest.
Image sensors
The basis for modern
image sensors
An image sensor or imager is a sensor that detects and conveys information used to make an image. It does so by converting the variable attenuation of light waves (as they refraction, pass through or reflection (physics), reflect off objects) int ...
is
metal-oxide-semiconductor (MOS) technology,
which originates from the invention of the
MOSFET
The metal–oxide–semiconductor field-effect transistor (MOSFET, MOS-FET, or MOS FET), also known as the metal–oxide–silicon transistor (MOS transistor, or MOS), is a type of insulated-gate field-effect transistor that is fabricated by th ...

(MOS field-effect transistor) by
Mohamed M. Atalla
Mohamed M. Atalla ( ar, محمد عطاالله; August 4, 1924 – December 30, 2009) was an Egyptian-American engineer, physical chemist
Physical chemistry is the study of macroscopic
The macroscopic scale is the length scale on which ...

and
Dawon Kahng
Dawon Kahng ( ko, 강대원; May 4, 1931 – May 13, 1992) was a Korean-American electrical engineer and inventor, known for his work in solid-state electronics
Solid-state electronics means semiconductor
A semiconductor material has an el ...

at
Bell Labs
Nokia Bell Labs (formerly named Bell Labs Innovations (1996–2007), AT&T Bell Laboratories (1984–1996) and Bell Telephone Laboratories (1925–1984)) is an American industrial research and scientific development company
A company, ab ...
in 1959.
This led to the development of digital
semiconductor
A semiconductor material has an electrical conductivity
Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric curre ...
image sensors, including the
charge-coupled device
A charge-coupled device (CCD) is an integrated circuit containing an array of linked, or coupled, capacitors. Under the control of an external circuit, each capacitor can transfer its electric charge to a neighboring capacitor. CCD sensors are a ...
(CCD) and later the
CMOS sensor An active-pixel sensor (APS) is an image sensor
An image sensor or imager is a that detects and conveys information used to make an . It does so by converting the variable of light s (as they or objects) into , small bursts of that convey t ...
.
The charge-coupled device was invented by
Willard S. Boyle
Willard Sterling Boyle, (August 19, 1924May 7, 2011) was a Canadian
Canadians (french: Canadiens) are people identified with the country of Canada. This connection may be residential, legal, historical or cultural. For most Canadians, many ...
and
at Bell Labs in 1969. While researching MOS technology, they realized that an electric charge was the analogy of the magnetic bubble and that it could be stored on a tiny
MOS capacitor
The metal–oxide–semiconductor field-effect transistor (MOSFET, MOS-FET, or MOS FET), also known as the metal–oxide–silicon transistor (MOS transistor, or MOS), is a type of insulated-gate field-effect transistor
The field-effect trans ...
. As it was fairly straightforward to
fabricate a series of MOS capacitors in a row, they connected a suitable voltage to them so that the charge could be stepped along from one to the next.
The CCD is a semiconductor circuit that was later used in the first
digital video camera
A video camera is a camera
A camera is an optical
Optics is the branch of physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior ...

s for
television broadcasting
A television network or broadcaster is a telecommunications network for distribution of television program content, where a central operation provides programming to many television stations or multichannel video programming distributor, pay tel ...

.
The
NMOS active-pixel sensorAn active-pixel sensor (APS) is an image sensor where each pixel sensor unit cell has a photodetector
Photodetectors, also called photosensors, are sensors of light
Light or visible light is electromagnetic radiation within the portion o ...
(APS) was invented by
Olympus in Japan during the mid-1980s. This was enabled by advances in MOS
semiconductor device fabrication
Semiconductor device fabrication is the process used to manufacture semiconductor devices
A semiconductor
A semiconductor material has an Electrical resistivity and conductivity, electrical conductivity value falling between that of a El ...
, with
MOSFET scaling
The metal–oxide–semiconductor field-effect transistor (MOSFET, MOS-FET, or MOS FET), also known as the metal–oxide–silicon transistor (MOS transistor, or MOS), is a type of insulated-gate field-effect transistor
The field-effect tran ...
reaching smaller
micron and then sub-micron levels.
The NMOS APS was fabricated by Tsutomu Nakamura's team at Olympus in 1985. The
CMOS
Complementary metal–oxide–semiconductor (CMOS, pronounced "see-moss"), also known as complementary-symmetry metal–oxide–semiconductor (COS-MOS), is a type of metal–oxide–semiconductor field-effect transistor
The metal–oxide–se ...
active-pixel sensor (CMOS sensor) was later developed by
Eric Fossum
Eric R. Fossum (born October 17, 1957) is an American physicist and engineer known for developing the CMOS sensor, CMOS image sensor. He is currently a professor at Thayer School of Engineering in Dartmouth College.
Early years and education
Fo ...
's team at the
NASA
The National Aeronautics and Space Administration (NASA; ) is an independent agency
A regulatory agency or regulatory authority, is a Public benefit corporation
Public-benefit corporation is a term that has different meanings in differen ...

Jet Propulsion Laboratory
The Jet Propulsion Laboratory (JPL) is a federally funded research and development center
Federally funded research and development centers (FFRDCs) are public-private partnerships which conduct research and development
Research is " c ...
in 1993.
By 2007, sales of CMOS sensors had surpassed CCD sensors.
Image compression
An important development in digital
image compression
Image compression is a type of data compression
In signal processing
Signal processing is an electrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, ...
technology was the
discrete cosine transform
A discrete cosine transform (DCT) expresses a finite sequence of data points
In statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data
Data (; ) are ...
(DCT), a
lossy compression
In information technology, lossy compression or irreversible compression is the class of data compression, data encoding methods that uses inexact approximations and partial data discarding to represent the content. These techniques are used to r ...
technique first proposed by
Nasir Ahmed in 1972.
DCT compression became the basis for
JPEG
JPEG ( ) is a commonly used method of lossy compression
In information technology, lossy compression or irreversible compression is the class of data encoding methods that uses inexact approximations and partial data discarding to represe ...

, which was introduced by the
Joint Photographic Experts Group
The Joint Photographic Experts Group (JPEG) is the joint committee between ISO
The International Organization for Standardization (ISO ) is an international standard
An international standard is a technical standard
A technical standard is an ...
in 1992.
JPEG compresses images down to much smaller file sizes, and has become the most widely used
image file format
Image file formats are standardized means of organizing and storing digital image
A digital image is an composed of s, also known as ''pixels'', each with ', ' of numeric representation for its or that is an output from its fed as input ...
on the
Internet
The Internet (or internet) is the global system of interconnected computer networks that uses the Internet protocol suite (TCP/IP) to communicate between networks and devices. It is a ''internetworking, network of networks'' that consist ...

. Its highly efficient DCT compression algorithm was largely responsible for the wide proliferation of
digital images
Digital usually refers to something using digits, particularly binary digits.
Technology and computing Hardware
*Digital electronics
Digital electronics is a field of electronics
Electronics comprises the physics, engineering, technology a ...
and
digital photo
Digital photography uses cameras
A camera is an optical
Optics is the branch of physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the ...
s,
with several billion JPEG images produced every day as of 2015.
Digital signal processor (DSP)
Electronic
signal processing
Signal processing is an electrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetis ...

was revolutionized by the wide adoption of
MOS technology
MOS Technology, Inc. ("MOS" being short for Metal Oxide Semiconductor
The metal–oxide–semiconductor field-effect transistor (MOSFET, MOS-FET, or MOS FET), also known as the metal–oxide–silicon transistor (MOS transistor, or MOS), i ...
in the 1970s.
MOS integrated circuit
file:MOSFET Structure.png, upright=1.6, MOSFET, showing metal gate, gate (G), body (B), source (S) and drain (D) terminals. The gate is separated from the body by an gate oxide, insulating layer (pink).
The metal–oxide–semiconductor field-eff ...
technology was the basis for the first single-chip
microprocessors
A microprocessor is a computer processor where the data processing logic and control is included on a single integrated circuit
An integrated circuit or monolithic integrated circuit (also referred to as an IC, a chip, or a microchi ...
and
microcontrollers
A microcontroller (MCU for ''microcontroller unit'') is a small computer on a single MOSFET, metal-oxide-semiconductor (MOS) integrated circuit (IC) chip. A microcontroller contains one or more Central processing unit, CPUs (processor cores) alo ...
in the early 1970s,
and then the first single-chip
digital signal processor
A digital signal processor (DSP) is a specialized microprocessor
A microprocessor is a computer processor where the data processing logic and control is included on a single integrated circuit
An integrated circuit or monolithic i ...
(DSP) chips in the late 1970s.
DSP chips have since been widely used in digital image processing.
The
discrete cosine transform
A discrete cosine transform (DCT) expresses a finite sequence of data points
In statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data
Data (; ) are ...
(DCT)
image compression
Image compression is a type of data compression
In signal processing
Signal processing is an electrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, ...
algorithm has been widely implemented in DSP chips, with many companies developing DSP chips based on DCT technology. DCTs are widely used for
encoding
In communication
Communication (from Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', mean ...

, decoding,
video coding
A video coding format (or sometimes video compression format) is a content representation format for storage or transmission of digital
Digital usually refers to something using digits, particularly binary digits.
Technology and computing Ha ...
,
audio coding
An audio coding format (or sometimes audio compression format) is a content representation format for storage or transmission of digital audio
Digital audio is a representation of sound recorded in, or converted into, Digital signal (signal ...
,
multiplexing
In telecommunications
Telecommunication is the transmission of information by various types of technologies over , radio, , or other systems. It has its origin in the desire of humans for communication over a distance greater than that ...
, control signals,
signaling
In signal processing
Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image processing, images, and scientific measurements. Sig ...
,
analog-to-digital conversion, formatting
luminance
Luminance is a photometricPhotometry can refer to:
* Photometry (optics), the science of measurement of visible light in terms of its perceived brightness to human vision
* Photometry (astronomy), the measurement of the flux or intensity of an ...

and color differences, and color formats such as
YUV444 and
YUV411. DCTs are also used for encoding operations such as
motion estimation
Motion estimation is the process of determining ''motion vector
In video compression
In signal processing
Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as a ...
,
motion compensation
Motion compensation is an algorithmic technique used to predict a frame in a video, given the previous and/or future frames by accounting for motion of the camera and/or objects in the video. It is employed in the encoding of video data for video ...
,
inter-frame
An inter frame is a frame in a video compression
In signal processing, data compression, source coding, or bit-rate reduction is the process of encoding information using fewer bits than the original representation. Any particular compression i ...
prediction,
quantization, perceptual weighting,
entropy encoding
In information theory, an entropy coding (or entropy encoding) is a lossless compression , lossless data compression scheme that is independent of the specific characteristics of the medium.
One of the main types of entropy coding creates and assi ...
, variable encoding, and
motion vector
In video compression
In signal processing
Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image processing, images, and scienti ...
s, and decoding operations such as the inverse operation between different color formats (
YIQ
An image along with its Y, I, and Q components.
YIQ is the color space
A color space is a specific organization of colors. In combination with color profiling supported by various physical devices, it supports reproducible representations of co ...
,
YUV
YUV is a color encoding system typically used as part of a color image pipelineAn image pipeline or video pipeline is the set of components commonly used between an image source (such as a camera, a scanner, or the rendering engine in a compute ...
and
RGB
s
The RGB color model is an additive color, additive color model
A color model is an abstract mathematical model describing the way colors can be represented as tuples of numbers, typically as three or four values or color components. When ...

) for display purposes. DCTs are also commonly used for
high-definition television
High-definition television (HD or HDTV) describes a television system providing a substantially higher image resolution
Image resolution is the detail an holds. The term applies to s, film images, and other types of images. Higher resolution m ...
(HDTV) encoder/decoder chips.
Medical imaging
In 1972, the engineer from British company EMI Housfield invented the X-ray computed tomography device for head diagnosis, which is what is usually called CT (computer tomography). The CT nucleus method is based on the projection of the human head section and is processed by computer to reconstruct the cross-sectional image, which is called image reconstruction. In 1975, EMI successfully developed a CT device for the whole body, which obtained a clear tomographic image of various parts of the human body. In 1979, this diagnostic technique won the Nobel Prize.
Digital image processing technology for medical applications was inducted into the
Space Foundation
Space Foundation is an American nonprofit organization that space advocacy, advocates for all sectors of the global space industry through space awareness activities, educational programs and major industry events. Founded in 1983.
Locations
Loc ...
Space Technology Hall of Fame in 1994.
Tasks
Digital image processing allows the use of much more complex algorithms, and hence, can offer both more sophisticated performance at simple tasks, and the implementation of methods which would be impossible by analogue means.
In particular, digital image processing is a concrete application of, and a practical technology based on:
*
Classification Classification is a process related to categorization
Categorization is the human ability and activity of recognizing shared features or similarities between the elements of the experience
Experience refers to conscious
, an English Paracels ...
*
Feature extractionIn machine learning, pattern recognition, and image processing, feature extraction starts from an initial set of measured data and builds derived values (Feature (machine learning), features) intended to be informative and non-redundant, facilitatin ...
*
Multi-scale signal analysis
*
Pattern recognition
Pattern recognition is the automated recognition of pattern
A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A geometric pattern is a kind of ...
*
Projection
Some techniques which are used in digital image processing include:
*
Anisotropic diffusion
In image processing and computer vision, anisotropic diffusion, also called Perona–Malik diffusion, is a technique aiming at reducing image noise without removing significant parts of the image content, typically edges, lines or other details tha ...
*
Hidden Markov model
Hidden Markov Model (HMM) is a statistical Markov model in which the system being mathematical model, modeled is assumed to be a Markov processcall it Xwith unobservable ("''hidden''") states. HMM assumes that there is another process Y whose beh ...
s
*
Image editing
Image editing encompasses the processes of altering images, whether they are Digital photography, digital photographs, traditional Photographic processing, photo-chemical photographs, or illustrations. Traditional analog image editing is kno ...
*
Image restoration
Image restoration is the operation of taking a corrupt/noisy image and estimating the clean, original image. Corruption may come in many forms such as motion blur
Motion blur is the apparent streaking of moving objects in a photograph or a sequen ...
*
Independent component analysis
In signal processing
Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image processing, images, and scientific measurements. Signal p ...
*
Linear filter
Linearity is the property of a mathematical relationship (''function (mathematics), function'') that can be graph of a function, graphically represented as a straight Line (geometry), line. Linearity is closely related to Proportionality (mathema ...
ing
*
Neural networks#REDIRECT Artificial neural network
Artificial neural networks (ANNs), usually simply called neural networks (NNs), are computing systems vaguely inspired by the biological neural networks that constitute animal brain
A brain is an organ ( ...
*
Partial differential equations
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
*
Pixelation
In computer graphics, pixelation (or pixellation in British English) is caused by displaying a bitmap or a section of a bitmap at such a large size that individual pixels, small single-colored square display elements that comprise the bitmap, ar ...
*
Point feature matching
*
Principal components analysis
*
Self-organizing map
A self-organizing map (SOM) or self-organizing feature map (SOFM) is an unsupervised machine learning
Machine learning (ML) is the study of computer algorithms that can improve automatically through experience and by the use of data. It is ...
s
*
Wavelet
A wavelet is a wave
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other wor ...

s
Digital image transformations
Filtering
Digital filters are used to blur and sharpen digital images. Filtering can be performed by:
*
convolution
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
with specifically designed
kernels (filter array) in the spatial domain
* masking specific frequency regions in the frequency (Fourier) domain
The following examples show both methods:
Image padding in Fourier domain filtering
Images are typically padded before being transformed to the Fourier space, the
highpass filtered images below illustrate the consequences of different padding techniques:
Notice that the highpass filter shows extra edges when zero padded compared to the repeated edge padding.
Filtering code examples
MATLAB example for spatial domain highpass filtering.
img=checkerboard(20); % generate checkerboard
% ************************** SPATIAL DOMAIN ***************************
klaplace= -1 0; -1 5 -1; 0 -1 0 % Laplacian filter kernel
X=conv2(img,klaplace); % convolve test img with
% 3x3 Laplacian kernel
figure()
imshow(X,[]) % show Laplacian filtered
title('Laplacian Edge Detection')
Affine transformations
Affine transformations enable basic image transformations including scale, rotate, translate, mirror and shear as is shown in the following examples:
To apply the affine matrix to an image, the image is converted to matrix in which each entry corresponds to the pixel intensity at that location. Then each pixel's location can be represented as a vector indicating the coordinates of that pixel in the image,
, y
The comma is a punctuation
Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
where x and y are the row and column of a pixel in the image matrix. This allows the coordinate to be multiplied by an affine-transformation matrix, which gives the position that the pixel value will be copied to in the output image.
However, to allow transformations that require translation transformations, 3 dimensional
homogeneous coordinates
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
are needed. The third dimension is usually set to a non-zero constant, usually 1, so that the new coordinate is
, y, 1 This allows the coordinate vector to be multiplied by a 3 by 3 matrix, enabling translation shifts. So the third dimension, which is the constant 1, allows translation.
Because matrix multiplication is associative, multiple affine transformations can be combined into a single affine transformation by multiplying the matrix of each individual transformation in the order that the transformations are done. This results in a single matrix that, when applied to a point vector, gives the same result as all the individual transformations performed on the vector
, y, 1in sequence. Thus a sequence of affine transformation matrices can be reduced to a single affine transformation matrix.
For example, 2 dimensional coordinates only allow rotation about the origin (0, 0). But 3 dimensional homogeneous coordinates can be used to first translate any point to (0, 0), then perform the rotation, and lastly translate the origin (0, 0) back to the original point (the opposite of the first translation). These 3 affine transformations can be combined into a single matrix, thus allowing rotation around any point in the image.
Image denoising with Morphology
Mathematical morphology
Mathematical morphology (MM) is a theory and technique for the analysis and processing of Geometry, geometrical structures, based on set theory, lattice theory, topology, and random functions. MM is most commonly applied to digital images, but it ca ...
is suitable for denoising images.
Structuring element are important in
Mathematical morphology
Mathematical morphology (MM) is a theory and technique for the analysis and processing of Geometry, geometrical structures, based on set theory, lattice theory, topology, and random functions. MM is most commonly applied to digital images, but it ca ...
.
The following examples are about Structuring elements. The denoise function, image as I, and structuring element as B are shown as below and table.
e.g.
Define Dilation(I, B)(i,j) =
. Let Dilation(I,B) = D(I,B)
D(I', B)(1,1) =
Define Erosion(I, B)(i,j) =
. Let Erosion(I,B) = E(I,B)
E(I', B)(1,1) =
After dilation
After erosion
An opening method is just simply erosion first, and then dilation while the closing method is vice versa. In reality, the D(I,B) and E(I,B) can implemented by
Convolution
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
In order to apply the denoising method to an image, the image is converted into grayscale. A mask with denoising method is logical matrix with