image reconstruction

DIGITAL IMAGE PROCESSING

IMAGE RECONSTRUCTION

by Dr. K. M. Bhurchandi

Fourier Slice theorem

• Fourier slice theorem (FST) explains the reconstruction of the object from the projection data.

• It is derived by taking the one dimension Fourier transform of the parallel projections and noting that it is equal to the slices of the two dimensions Fourier transform of the object

• The projection data should estimate the object using two dimensional inverse Fourier transform


• In above figure, the (x, y) coordinate system is rotated by an angle θ.

• The FFT of the projection is equal to the 2-D FFT of the object slice along a line rotated by θ.

• Thus the FST states that, the Fourier transform of parallel projection of an image f(x, y) taken at an angle θ gives a slice of the 2-D transform, subtending an angle θ with the u-axis.

• In other words one dimensional FT of the set of projections gives the value of two dimensional FT along lines BB.

Introduction

• Image reconstruction is simple and can be explained intuitively considering an example.

a) Flat region with b) Result of object, beam backprojecting & detector Sensed strip data c) Beam & e) = b + d detectors rotated by 90˚ d) Back-projection

Introduction

• From (e) figure we can identify the object, whose amplitude is twice that of individual backprojection.

• As the number of projections increases, the strength of non-intersecting backprojections decreases relative to the strength of regions in which multiple backprojections intersect.

• Net result: - Brighter region will dominate the result, and backprojections with few or no intersections will fade into the background.

• Result from 32 backprojections is shown next

Principles of Computed Tomography (CT)

• In 1917, Johann Radon, a mathematician from Vienna derived a method for projecting a 2-D object along parallel rays as part of his work on line integrals.

• This method is known as Radon Transform.

• 45 years later, Allan Cormack, a physicist at Tufts University rediscovered these concept and applied to CT.

• Godfrey N. Hounsfield & his colleagues at EMI in London built first medical CT machine.

• Cormack & Hounsfield shared Noble Prize in 1979.


• First Generation (G1) CT Scanners: • It employ a pencil X-ray beam and a single detector.


• Second Generation (G2) CT scanners • Operates on same principle as G1 scanners, but the beam

used is in the shape of a fan.


• Third Generation (G3) scanners • They employ a bank of detectors long enough (around 1000)

to cover the entire field of view of a wider beam.


• Fourth Generation (G4) scanners • They employed a circular ring of detectors (around 5000), only

the source has to rotate.

4 Generations of CT Scanners


• Fifth Generation (G5) CT scanners a.k.a. electron beam computed tomography (EBCT) eliminate all mechanical motion by employing electron beams controlled electromagnetically.

• Sixth Generation (G6) scanners a.k.a. helical CT. The source/detector pairs rotates continuously through 360˚ while the patient is moved at a constant speed along the axis perpendicular to the scan.

• Seventh Generation (G7) scanners a.k.a. multislice CT Scanners are emerging in which thick fan beams are used in conjunction with parallel banks of detectors to collect volumetric CT data simultaneously.

Projections and the Radon Transform

• A straight line in cartesian coordinates can be described either by its slope intercept form,

y = ax + b Or x cos θ + y sin θ = ρ


• The projection of a parallel ray beam may be modeled by a set of such lines as,


• An arbitrary point in the projection signal is given by the raysum along the line x cos θk + y sin θk = ρj

• The raysum is a line integral given by:

• If all the values of ρ & θ are considered the above equation

becomes


• The above equation in discrete form becomes where, x, y, ρ & θ are discrete variables. • When the Radon transform, g(ρ, θ), is displayed as an image

with ρ & θ as rectilinear coordinates, the result is called a sinogram (like Fourier transform, however, g(ρ, θ) is always a real function.)

• Like Fourier Transform, a sinogram contains data necessary to reconstruct f(x, y).


• The key objective of CT is to obtain a 3D representation of a volume from its projections.

• The approach is to back-project each projection and then sum all the backprojections to generate one image.

• Stacking all the resulting images produces a 3D rendition of the volume.

• To obtain a formal expression for a back-projected image from the Radon Transform, begin with a single point, g(ρi, θk) of the complete projection, g(ρ, θk), for a fixed value of rotation, θk.


• In general, the Image formed from a single backprojection obtained at an angle θ is given by:

• Final image is formed by integrating all backprojected images • In discrete case, the integral becomes sum of all

backprojected images

where, x, y & θ are discrete variables.

Reconstruction using Parallel-Beam Filtered Backprojections


• Obtaining backprojections yields blurred results.

• Straightforward solution to this problem is filtering the projections before computing the backprojections.

• Using 2-D Inverse Fourier Transform of F(u, v) is:

• Taking, u = ω cosθ & v = ω sinθ & dudv = ω dω dθ, we can express above equation in polar coordinates:


• Then, using Fourier-Slice theorem,

• Splitting the integral in 2 expressions, for θ in the range 0˚ to 180˚ & 180˚ to 360˚ and using the fact G(ω,θ + 180˚) = G(-ω,θ) we get,

• In the term wrt ω, the term x cos θ + y sin θ = ρ


• The term inside the bracket is inverse Fourier Transform of the product of two frequency domain functions, which are equal to the convolution of the spatial representations of these 2 functions.

Numerical

1) Use Radon transform to obtain an analytical expression for projection of circular object shown below:

𝑓 𝑥, 𝑦 = 𝐴 𝑥2 + 𝑦2 ≤ 𝑟2

0 𝑜𝑡𝑕𝑒𝑟𝑤𝑖𝑠𝑒

where, A – constant & r – radius of the object Sol: We assume that, circle is centered on the Origin of the xy-plane. Since, the object is circularly symmetric, its projections are the same for all angles. So, we have to obtain the projection for θ = 0˚

Numerical

The equation for Radon transform is given as:

𝑔 ρ, θ = 𝑓 𝑥, 𝑦 δ 𝑥 − ρ 𝑑𝑥 𝑑𝑦∞

−∞

∞

−∞

= 𝑓 ρ, 𝑦 𝑑𝑦∞

−∞

This is a line integral along the line L(ρ, 0) in this case. Here, 𝑔 ρ, θ = 0 when ρ > r.

When ρ ≤ r ; integral is evaluated from y= − 𝑟2 − ρ2 to

y= 𝑟2 − ρ2

So, 𝑔 ρ, θ = 𝑓 ρ, 𝑦 𝑑𝑦𝑟2−ρ2

− 𝑟2−ρ2

= 𝐴 𝑑𝑦𝑟2−ρ2

− 𝑟2−ρ2

Numerical

Integrating results in,

𝑔 ρ, θ = 𝑔 ρ = 2𝐴 𝑟2 − ρ2 ρ ≤ r

0 𝑜𝑡𝑕𝑒𝑟𝑤𝑖𝑠𝑒

Here, we used the fact that, 𝑔 ρ, θ = 0 when ρ > r. Note that, 𝑔 ρ, θ = 𝑔 ρ ; that is, g is independent of θ because the object is symmetric about the origin.

Noise Models

• The noise component may be characterized by a PDF. The most common PDFs found in image processing applications are as below:

i) Gaussian Noise: Used in spatial and frequency domain. The PDF of a Gaussian random variable, z, is given by

ii) Rayleigh Noise: The PDF of Rayleigh noise is given by

Mean Variance

Noise Models

iii) Erlang (Gamma) Noise: The PDF of Erlang noise is given by: where, a > 0, b is a positive integer. Mean Variance iv) Exponential Noise: The PDF of exponential noise is given by: where, a > 0 Mean Variance This is a special case of erlang PDF with b = 1

Noise Models

v) Uniform Noise: The PDF of uniform noise is given by: Mean Variance vi) Impulse (salt and pepper) noise: The PDF of impulse (bipolar) noise is given by: If b > a, intensity b will appear as a light dot in the image. Conversely, level a will appear like a dark dot. If either Pa or Pb is zero, the impulse noise is called unipolar.

Noise Models

Model of Image Degradation/Restoration process

where, f(x, y) – input image g(x, y) – degraded image

𝑓 (x, y) – estimate of original image

The more we know about H and η, the closer 𝑓 (x, y) will be to f(x, y).

• If H is a linear, position-invariant process, then the degraded image is given in the spatial domain by:

• Where, h(x, y) is the spatial representation of the degraded function H.

• Frequency domain representation of the above equation will be:

• G

Gaussian Rayleigh Gamma

Exponential Uniform Salt & Pepper

Restoration in presence of noise only

• When the only degradation present in an image is noise g(x, y) = f(x, y) + η(x, y) And G(u, v) = F(u, v) + N(u, v) Ex. Mean Filters – Arithmetic mean, Geometric mean, Harmonic mean, etc. Order-Statistic Filter: These are the spatial filters whose response is based on ordering (ranking) the values of the pixels contained in the image area encompassed by the filter. The ranking result determines the response of the filter. - Median filter, Max and min filter, etc.

Inverse Filtering

• The simplest approach to restoration is direct inverse filtering, where we compute an estimate, 𝐹 𝑢, 𝑣 , of the transform of the original image simply by dividing the transform of the degraded image, G(u, v), by the degraded function:

𝐹 𝑢, 𝑣 = 𝐺(𝑢, 𝑣)

𝐻(𝑢, 𝑣)

Substituting the RHS of frequency domain representation of the model of image degradation/reconstruction we get:

𝐹 𝑢, 𝑣 = 𝐹 𝑢, 𝑣 +𝑁(𝑢, 𝑣)

𝐻(𝑢, 𝑣)

This equation tells that, even if we know the degradation function we cannot recover the undegraded image [F(u, v)] exactly because N(u, v) is not known.

If degradation function has zero or very small value then the ratio could easily dominate the result.

Minimum Mean Square Error (Wiener) Filtering

• Inverse filtering approach has no provision for noise handling. • This method is based on considering image and noise as

random variables, and objective is to find an estimate 𝑓 of the uncorrupted image 𝑓such that the mean square error between them is minimized. The error measure is given by:

𝑒2 = 𝐸* 𝑓 − 𝑓 2+

where, 𝐸*. + is expected value of the argument. Assumptions: i) Noise and image are uncorrelated; ii) Any one has zero mean. Based on these conditions, the minimum error function in above equation is given in the frequency domain by:

Minimum Mean Square Error (Wiener) Filtering

• Here, the fact that the product of a complex quantity with conjugate is equal to the magnitude of the complex quantity squared. This result is known as the Wiener filter.

• Mean Square Error

• Signal to Noise Ratio

image reconstruction

Engineering

generation g1 ct scanners

generation g5 ct scanners

generation g2 ct scanners

object slice

d object

d transform

computed tomography

generation g4 scanners