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Perspective Transformation

When human eyes see near things they look bigger as compare to those who are far away. This is called perspective in a general way. Whereas transformation is the transfer of an object e.t.c from one state to another.

So overall , the perspective transformation deals with the conversion of 3d world into 2d image. The same principle on which human vision works and the same principle on which the camera works.

We will see in detail about why this happens , that those objects which are near to you look bigger , while those who are far away , look smaller even though they look bigger when you reach them.

We will start this discussion by the concept of frame of reference:

Frame of reference:

Frame of reference is basically a set of values in relation to which we measure something.

5 FRAMES OF REFERENCE

In order to analyze a 3d world/image/scene, 5 different frame of references are required.

• Object
• World
• Camera
• Image
• Pixel

OBJECT COORDINATE FRAME

Object coordinate frame is used for modeling objects. For example , checking if a particular object is in a proper place with respect to the other object. It is a 3d coordinate system.

WORLD COORDINATE FRAME

World coordinate frame is used for co-relating objects in a 3 dimensional world. It is a 3d coordinate system.

CAMERA COORDINATE FRAME

Camera co-ordinate frame is used to relate objects with respect of the camera. It is a 3d coordinate system.

IMAGE COORDINATE FRAME

It is not a 3d coordinate system , rather it is a 2d system. It is used to describe how 3d points are mapped in a 2d image plane.

PIXEL COORDINATE FRAME

It is also a 2d coordinate system. Each pixel has a value of pixel co ordinates.

Transformation between these 5 frames

Thats how a 3d scene is transformed into 2d , with image of pixels.

Now we will explain this concept mathematically.

Where

Y = 3d object

y = 2d Image

f = focal length of the camera

Z = distance between image and the camera

Now there are two different angles formed in this transform which are represented by Q.

The first angle is

Where minus denotes that image is inverted. The second angle that is formed is:

Comparing these two equations we get

From this equation, we can see that when the rays of light reflect back after striking from the object , passed from the camera , an invert image is formed.

We can better understand this, with this example.

For example

Calculating the size of image formed

Suppose an image has been taken of a person 5m tall, and standing at a distance of 50m from the camera, and we have to tell that what is the size of the image of the person , with a camera of focal length is 50mm.

SOLUTION:

Since the focal length is in millimeter , so we have to convert every thing in millimeter in order to calculate it.

So,

Y = 5000 mm.

f = 50 mm.

Z = 50000 mm.

Putting the values in the formula , we get

= -5 mm.

Again, the minus sign indicates that the image is inverted.

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Concept of Pixel

Pixel

Pixel is the smallest element of an image. Each pixel correspond to any one value. In an 8-bit gray scale image, the value of the pixel between 0 and 255. The value of a pixel at any point correspond to the intensity of the light photons striking at that point. Each pixel store a value proportional to the light intensity at that particular location.

PEL

A pixel is also known as PEL. You can have more understanding of the pixel from the pictures given below.

In the above picture, there may be thousands of pixels, that together make up this image. We will zoom that image to the extent that we are able to see some pixels division. It is shown in the image below.

In the above picture, there may be thousands of pixels, that together make up this image. We will zoom that image to the extent that we are able to see some pixels division. It is shown in the image below.

Relation ship with CCD array

We have seen that how an image is formed in the CCD array. So a pixel can also be defined as

The smallest division the CCD array is also known as pixel.

Each division of CCD array contains the value against the intensity of the photon striking to it. This value can also be called as a pixel

CALCULATION OF TOTAL NUMBER OF PIXELS

We have define an image as a two dimensional signal or matrix. Then in that case the number of PEL would be equal to the number of rows multiply with number of columns.

This can be mathematically represented as below:

Total number of pixels = number of rows ( X ) number of columns

Or we can say that the number of (x,y) coordinate pairs make up the total number of pixels.

We will look in more detail in the tutorial of image types , that how do we calculate the pixels in a color image.

Gray level

The value of the pixel at any point denotes the intensity of image at that location , and that is also known as gray level.

We will see in more detail about the value of the pixels in the image storage and bits per pixel tutorial, but for now we will just look at the concept of only one pixel value.

PIXEL VALUE.(0)

As it has already been define in the beginning of this tutorial , that each pixel can have only one value and each value denotes the intensity of light at that point of the image.

We will now look at a very unique value 0. The value 0 means absence of light. It means that 0 denotes dark, and it further means that when ever a pixel has a value of 0, it means at that point , black color would be formed.

Have a look at this image matrix

0	0	0
0	0	0
0	0	0

Now this image matrix has all filled up with 0. All the pixels have a value of 0. If we were to calculate the total number of pixels form this matrix , this is how we are going to do it.

Total no of pixels = total no. of rows X total no. of columns

= 3 X 3

= 9.

It means that an image would be formed with 9 pixels, and that image would have a dimension of 3 rows and 3 column and most importantly that image would be black.

The resulting image that would be made would be something like this

Now why is this image all black. Because all the pixels in the image had a value of 0.

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Camera Mechansim

In this tutorial, we will discuss some of the basic camera concepts, like aperture , shutter , shutter speed , ISO and we will discuss the collective use of these concepts to capture a good image.

Aperture

Aperture is a small opening which allows the light to travel inside into camera. Here is the picture of aperture.

You will see some small blades like stuff inside the aperture. These blades create a octagonal shape that can be opened closed. And thus it make sense that , the more blades will open, the hole from which the light would have to pass would be bigger. The bigger the hole , the more light is allowed to enter.

EFFECT

The effect of the aperture directly corresponds to brightness and darkness of an image. If the aperture opening is wide , it would allow more light to pass into the camera. More light would result in more photons, which ultimately result in a brighter image.

The example of this is shown below

CONSIDER THESE TWO PHOTOS

The one on the right side looks brighter, it means that when it was captured by the camera , the aperture was wide open. As compare to the other picture on the left side , which is very dark as compare to the first one, that shows that when that image was captured, its aperture was not wide open.

SIZE

Now lets discuss the maths behind the aperture. The size of the aperture is denoted by a f value. And it is inversely proportional to the opening of aperture.

Here are the two equations , that best explain this concept.

Large aperture size = Small f value

Small aperture size = Greater f value

Pictorially it can be represented as:

Shutter

After the aperture , there comes the shutter. The light when allowed to pass from the aperture , falls directly on to the shutter. Shutter is actually a cover, a closed window , or can be thought of as a curtain. Remember when we talk about the CCD array sensor on which the image is formed. Well behind the shutter is the sensor. So shutter is the only thing that is between the image formation and the light , when it is passed from aperture.

As soon as the shutter is open , light falls on the image sensor , and the image is formed on the array.

EFFECT

If the shutter allows light to pass a bit longer , the image would be brighter. Similarly a darker picture is produced , when a shutter is allowed to move very quickly and hence, the light that is allowed to pass has very less photons , and the image that is formed on the CCD array sensor is very dark.

Shutter has further two main concepts:

1. Shutter Speed
2. Shutter time

SHUTTER SPEED

The shutter speed can be referred to as the number of times the shutter get open or close. Remember we are not talking about for how long the shutter get open or close.

SHUTTER TIME

The shutter time can be defined as

When the shutter is open , then the amount of wait time it take till it is closed is called shutter time.

In this case we are not talking about how many times , the shutter got open or close , but we are talking about for how much time does it remain wide open.

For example:

We can better understand these two concepts in this way. That lets say that a shutter opens 15 times and then get closed, and for each time it opens for 1 second and then get closed. In this example , 15 is the shutter speed and 1 second is the shutter time.

RELATIONSHIP

The relationship between shutter speed and shutter time is that they are both inversely proportional to each other.

This relationship can be defined in the equation below.

More shutter speed = less shutter time

Less shutter speed = more shutter time.

EXPLANATION:

The lesser the time required , the more is the speed. And the greater the time required , the less is the speed.

Applications

These two concepts together make a variety of applications. Some of them are given below.

FAST MOVING OBJECTS:

If you were to capture the image of a fast moving object , could be a car or anything. The adjustment of shutter speed and its time would effect a lot.

So , in order to capture an image like this, we will make two amendments:

1. Increase shutter speed
2. Decrease shutter time

What happens is , that when we increase shutter speed , the more number of times , the shutter would open or close. It means different samples of light would allow to pass in. And when we decrease shutter time , it means we will immediately captures the scene, and close the shutter gate.

If you will do this , you get a crisp image of a fast moving object.

In order to understand it , we will look at this example. Suppose you want to capture the image of fast moving water fall.

You set your shutter speed to 1 second and you capture a photo. This is what you get

Then you set your shutter speed to a faster speed and you get.

Then again you set your shutter speed to even more faster and you get.

You can see in the last picture , that we have increase our shutter speed to very fast, that means that a shutter get opened or closed in 200th of 1 second and so we got a crisp image.

ISO

ISO factor is measured in numbers. It denotes the sensitivity of light to camera. If ISO number is lowered , it means our camera is less sensitive to light and if the ISO number is high, it means it is more senstivie.

EFFECT

The higher is the ISO , the more brighter the picture would be. IF ISO is set to 1600 , the picture would be very brighter and vice versa.

SIDE EFFECT

If the ISO increases, the noise in the image also increases. Today most of the camera manufacturing companies are working on removing the noise from the image when ISO is set to higher speed.

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Image Formation on Camera

How human eye works?

Before we discuss , the image formation on analog and digital cameras , we have to first discuss the image formation on human eye. Because the basic principle that is followed by the cameras has been taken from the way , the human eye works.

When light falls upon the particular object , it is reflected back after striking through the object. The rays of light when passed through the lens of eye , form a particular angle , and the image is formed on the retina which is the back side of the wall. The image that is formed is inverted. This image is then interpreted by the brain and that makes us able to understand things. Due to angle formation , we are able to perceive the height and depth of the object we are seeing. This has been more explained in the tutorial of perspective transformation.

As you can see in the above figure, that when sun light falls on the object (in this case the object is a face), it is reflected back and different rays form different angle when they are passed through the lens and an invert image of the object has been formed on the back wall. The last portion of the figure denotes that the object has been interpreted by the brain and re-inverted.

Now lets take our discussion back to the image formation on analog and digital cameras.

Image formation on analog cameras

In analog cameras , the image formation is due to the chemical reaction that takes place on the strip that is used for image formation.

A 35mm strip is used in analog camera. It is denoted in the figure by 35mm film cartridge. This strip is coated with silver halide ( a chemical substance).

A 35mm strip is used in analog camera. It is denoted in the figure by 35mm film cartridge. This strip is coated with silver halide ( a chemical substance).

Light is nothing but just the small particles known as photon particles.So when these photon particles are passed through the camera, it reacts with the silver halide particles on the strip and it results in the silver which is the negative of the image.

In order to understand it better , have a look at this equation.

Photons (light particles) + silver halide ? silver ? image negative.

This is just the basics, although image formation involves many other concepts regarding the passing of light inside , and the concepts of shutter and shutter speed and aperture and its opening but for now we will move on to the next part. Although most of these concepts have been discussed in our tutorial of shutter and aperture.

This is just the basics, although image formation involves many other concepts regarding the passing of light inside , and the concepts of shutter and shutter speed and aperture and its opening but for now we will move on to the next part. Although most of these concepts have been discussed in our tutorial of shutter and aperture.

Image formation on digital cameras

In the digital cameras , the image formation is not due to the chemical reaction that take place , rather it is a bit more complex then this. In the digital camera , a CCD array of sensors is used for the image formation.

IMAGE FORMATION THROUGH CCD ARRAY

CCD stands for charge-coupled device. It is an image sensor, and like other sensors it senses the values and converts them into an electric signal. In case of CCD it senses the image and convert it into electric signal e.t.c.

This CCD is actually in the shape of array or a rectangular grid. It is like a matrix with each cell in the matrix contains a censor that senses the intensity of photon.

Like analog cameras , in the case of digital too , when light falls on the object , the light reflects back after striking the object and allowed to enter inside the camera.

Each sensor of the CCD array itself is an analog sensor. When photons of light strike on the chip , it is held as a small electrical charge in each photo sensor. The response of each sensor is directly equal to the amount of light or (photon) energy striked on the surface of the sensor.

Since we have already define an image as a two dimensional signal and due to the two dimensional formation of the CCD array , a complete image can be achieved from this CCD array.

It has limited number of sensors , and it means a limited detail can be captured by it. Also each sensor can have only one value against the each photon particle that strike on it.

So the number of photons striking(current) are counted and stored. In order to measure accurately these , external CMOS sensors are also attached with CCD array.

Introduction to pixel

The value of each sensor of the CCD array refers to each the value of the individual pixel. The number of sensors = number of pixels. It also means that each sensor could have only one and only one value.

Storing image

The charges stored by the CCD array are converted to voltage one pixel at a time. With the help of additional circuits , this voltage is converted into a digital information and then it is stored.

Each company that manufactures digital camera, make their own CCD sensors. That include , Sony , Mistubishi , Nikon ,Samsung , Toshiba , FujiFilm , Canon e.t.c.

Apart from the other factors , the quality of the image captured also depends on the type and quality of the CCD array that has been used.

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Concept of Dimensions

We will look at this example in order to understand the concept of dimension.

Consider you have a friend who lives on moon, and he wants to send you a gift on your birthday present. He ask you about your residence on earth. The only problem is that the courier service on moon doesnot understand the alphabetical address, rather it only understand the numerical co-ordinates. So how do you send him your position on earth?

Thats where comes the concept of dimensions. Dimensions define the minimum number of points required to point a position of any particular object within a space.

So lets go back to our example again in which you have to send your position on earth to your friend on moon. You send him three pair of co-ordinates. The first one is called longitude , the second one is called latitude, and the third one is called altitude.

These three co-ordinates define your position on the earth. The first two defines your location , and the third one defines your height above the sea level.

So that means that only three co-ordinates are required to define your position on earth. That means you live in world which is 3 dimensional. And thus this not only answers the question about dimension , but also answers the reason , that why we live in a 3d world.

Since we are studying this concept in reference to the digital image processing, so we are now going to relate this concept of dimension with an image.

Dimensions of image

So if we live in the 3d world , means a 3 dimensional world, then what are the dimensions of an image that we capture. An image is a two dimensional, thats why we also define an image as a 2 dimensional signal. An image has only height and width. An image doesnot have depth. Just have a look at this image below.

If you would look at the above figure , it shows that it has only two axis which are the height and width axis. You cannot perceive depth from this image. Thats why we say that an image is two dimensional signal. But our eye is able to perceive three dimensional objects , but this would be more explained in the next tutorial of how the camera works , and image is perceived.

This discussion leads to some other questions that how 3 dimension systems is formed from 2 dimension.

How does television works?

If we look the image above , we will see that it is a two dimensional image. In order to convert it into three dimension , we need one other dimension. Lets take time as the third dimension , in that case we will move this two dimensional image over the third dimension time. The same concept that happens in television, that helps us perceive the depth of different objects on a screen. Does that mean that what comes on the T.V or what we see in the television screen is 3d. Well we can yes. The reason is that, in case of T.V we if we are playing a video. Then a video is nothing else but two dimensional pictures move over time dimension. As two dimensional objects are moving over the third dimension which is a time so we can say it is 3 dimensional.

Different dimensions of signals

1 DIMENSION SIGNAL

The common example of a 1 dimension signal is a waveform. It can be mathematically represented as

F(x) = waveform

Where x is an independent variable. Since it is a one dimension signal , so thats why there is only one variable x is used.

Pictorial representation of a one dimensional signal is given below:

The above figure shows a one dimensional signal.

Now this lead to another question, which is, even though it is a one dimensional signal ,then why does it have two axis?. The answer to this question is that even though it is a one dimensional signal , but we are drawing it in a two dimensional space. Or we can say that the space in which we are representing this signal is two dimensional. Thats why it looks like a two dimensional signal.

Perhaps you can understand the concept of one dimension more better by looking at the figure below.

Now refer back to our initial discussion on dimension, Consider the above figure a real line with positive numbers from one point to the other. Now if we have to explain the location of any point on this line, we just need only one number, which means only one dimension.

2 dimensions signal

The common example of a two dimensional signal is an image , which has already been discussed above.

As we have already seen that an image is two dimensional signal, i-e: it has two dimensions. It can be mathematically represented as:

F (x , y) = Image

Where x and y are two variables. The concept of two dimension can also be explained in terms of mathematics as:

Now in the above figure, label the four corners of the square as A,B,C and D respectively. If we call , one line segment in the figure AB and the other CD , then we can see that these two parallel segments join up and make a square. Each line segment corresponds to one dimension , so these two line segments correspond to 2 dimensions.

3 dimension signal

Three dimensional signal as it names refers to those signals which has three dimensions. The most common example has been discussed in the beginning which is of our world. We live in a three dimensional world. This example has been discussed very elaborately. Another example of a three dimensional signal is a cube or a volumetric data or the most common example would be animated or 3d cartoon character.

The mathematical representation of three dimensional signal is:

F(x,y,z) = animated character.

Another axis or dimension Z is involved in a three dimension, that gives the illusion of depth. In a Cartesian co-ordinate system it can be viewed as:

4 dimension signal

In a four dimensional signal , four dimensions are involved. The first three are the same as of three dimensional signal which are: (X, Y, Z), and the fourth one which is added to them is T(time). Time is often referred to as temporal dimension which is a way to measure change. Mathematically a four d signal can be stated as:

F(x,y,z,t) = animated movie.

The common example of a 4 dimensional signal can be an animated 3d movie. As each character is a 3d character and then they are moved with respect to the time, due to which we saw an illusion of a three dimensional movie more like a real world.

So that means that in reality the animated movies are 4 dimensional i-e: movement of 3d characters over the fourth dimension time.

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Applications and Usage

Since digital image processing has very wide applications and almost all of the technical fields are impacted by DIP, we will just discuss some of the major applications of DIP.

Digital Image processing is not just limited to adjust the spatial resolution of the everyday images captured by the camera. It is not just limited to increase the brightness of the photo, e.t.c. Rather it is far more than that.

Electromagnetic waves can be thought of as stream of particles, where each particle is moving with the speed of light. Each particle contains a bundle of energy. This bundle of energy is called a photon.

The electromagnetic spectrum according to the energy of photon is shown below.

In this electromagnetic spectrum, we are only able to see the visible spectrum. Visible spectrum mainly includes seven different colors that are commonly term as (VIBGOYR). VIBGOYR stands for violet , indigo , blue , green , orange , yellow and Red.

But that doesnot nullify the existence of other stuff in the spectrum. Our human eye can only see the visible portion, in which we saw all the objects. But a camera can see the other things that a naked eye is unable to see. For example: x rays , gamma rays , e.t.c. Hence the analysis of all that stuff too is done in digital image processing.

This discussion leads to another question which is

why do we need to analyze all that other stuff in EM spectrum too?

The answer to this question lies in the fact, because that other stuff such as XRay has been widely used in the field of medical. The analysis of Gamma ray is necessary because it is used widely in nuclear medicine and astronomical observation. Same goes with the rest of the things in EM spectrum.

Applications of Digital Image Processing

Some of the major fields in which digital image processing is widely used are mentioned below

• Image sharpening and restoration
• Medical field
• Remote sensing
• Transmission and encoding
• Machine/Robot vision
• Color processing
• Pattern recognition
• Video processing
• Microscopic Imaging
• Others

Image sharpening and restoration

Image sharpening and restoration refers here to process images that have been captured from the modern camera to make them a better image or to manipulate those images in way to achieve desired result. It refers to do what Photoshop usually does.

This includes Zooming, blurring , sharpening , gray scale to color conversion, detecting edges and vice versa , Image retrieval and Image recognition. The common examples are:

The original image

The zoomed image

Blurr image

Sharp image

Edges

Medical field

The common applications of DIP in the field of medical is

1. Gamma ray imaging
2. PET scan
3. X Ray Imaging
4. Medical CT
5. UV imaging

UV imaging

In the field of remote sensing , the area of the earth is scanned by a satellite or from a very high ground and then it is analyzed to obtain information about it. One particular application of digital image processing in the field of remote sensing is to detect infrastructure damages caused by an earthquake.

As it takes longer time to grasp damage, even if serious damages are focused on. Since the area effected by the earthquake is sometimes so wide , that it not possible to examine it with human eye in order to estimate damages. Even if it is , then it is very hectic and time consuming procedure. So a solution to this is found in digital image processing. An image of the effected area is captured from the above ground and then it is analyzed to detect the various types of damage done by the earthquake.

The key steps include in the analysis are

1. The extraction of edges
2. Analysis and enhancement of various types of edges

Transmission and encoding

The very first image that has been transmitted over the wire was from London to New York via a submarine cable. The picture that was sent is shown below.

The picture that was sent took three hours to reach from one place to another.

Now just imagine , that today we are able to see live video feed , or live cctv footage from one continent to another with just a delay of seconds. It means that a lot of work has been done in this field too. This field doesnot only focus on transmission , but also on encoding. Many different formats have been developed for high or low bandwith to encode photos and then stream it over the internet or e.t.c.

Machine/Robot vision

Apart form the many challenges that a robot face today , one of the biggest challenge still is to increase the vision of the robot. Make robot able to see things , identify them , identify the hurdles e.t.c. Much work has been contributed by this field and a complete other field of computer vision has been introduced to work on it.

Hurdle detection

Hurdle detection is one of the common task that has been done through image processing, by identifying different type of objects in the image and then calculating the distance between robot and hurdles.

Line follower robot

Most of the robots today work by following the line and thus are called line follower robots. This help a robot to move on its path and perform some tasks. This has also been achieved through image processing.

Color processing

Color processing includes processing of colored images and different color spaces that are used. For example RGB color model , YCbCr, HSV. It also involves studying transmission , storage , and encoding of these color images.

Pattern recognition

Pattern recognition involves study from image processing and from various other fields that includes machine learning ( a branch of artificial intelligence). In pattern recognition , image processing is used for identifying the objects in an images and then machine learning is used to train the system for the change in pattern. Pattern recognition is used in computer aided diagnosis , recognition of handwriting , recognition of images e.t.c

Video processing

A video is nothing but just the very fast movement of pictures. The quality of the video depends on the number of frames/pictures per minute and the quality of each frame being used. Video processing involves noise reduction , detail enhancement , motion detection , frame rate conversion , aspect ratio conversion , color space conversion e.t.c.

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