Point spread function - AbsoluteAstronomy.com

The point spread function (PSF) describes the response of an imaging system to a point source

Point source

A point source is a localised, relatively small source of something.Point source may also refer to:*Point source , a localised source of pollution**Point source water pollution, water pollution with a localized source...

or point object. A more general term for the PSF is a system's impulse response

Impulse response

In signal processing, the impulse response, or impulse response function , of a dynamic system is its output when presented with a brief input signal, called an impulse. More generally, an impulse response refers to the reaction of any dynamic system in response to some external change...

, the PSF being the impulse response of a focused optical system. The PSF in many contexts can be thought of as the extended blob in an image that represents an unresolved object. In functional terms it is the spatial domain version of the modulation transfer function. It is a useful concept in Fourier optics

Fourier optics

Fourier optics is the study of classical optics using Fourier transforms and can be seen as the dual of the Huygens-Fresnel principle. In the latter case, the wave is regarded as a superposition of expanding spherical waves which radiate outward from actual current sources via a Green's function...

, astronomical imaging

Astronomy

Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

, electron microscopy

Electron microscope

An electron microscope is a type of microscope that uses a beam of electrons to illuminate the specimen and produce a magnified image. Electron microscopes have a greater resolving power than a light-powered optical microscope, because electrons have wavelengths about 100,000 times shorter than...

and other imaging techniques such as 3D

Dimension

In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

microscopy

Microscopy

Microscopy is the technical field of using microscopes to view samples and objects that cannot be seen with the unaided eye...

(like in Confocal laser scanning microscopy

Confocal laser scanning microscopy

Confocal laser scanning microscopy is a technique for obtaining high-resolution optical images with depth selectivity. The key feature of confocal microscopy is its ability to acquire in-focus images from selected depths, a process known as optical sectioning...

) and fluorescence microscopy. The degree of spreading (blurring) of the point object is a measure for the quality of an imaging system. In incoherent

Coherence (physics)

In physics, coherence is a property of waves that enables stationary interference. More generally, coherence describes all properties of the correlation between physical quantities of a wave....

imaging systems such as fluorescent microscopes, telescopes or optical microscopes, the image formation process is linear in power and described by linear system

Linear system

A linear system is a mathematical model of a system based on the use of a linear operator.Linear systems typically exhibit features and properties that are much simpler than the general, nonlinear case....

theory. This means that when two objects A and B are imaged simultaneously, the result is equal to the sum of the independently imaged objects. In other words: the imaging of A is unaffected by the imaging of B and vice versa, owing to the non-interacting property of photons. The image of a complex object can then be seen as a convolution

Convolution

In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...

of the true object and the PSF. However, when the detected light is coherent

Coherence (physics)

In physics, coherence is a property of waves that enables stationary interference. More generally, coherence describes all properties of the correlation between physical quantities of a wave....

, image formation is linear in the complex

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

field

Electric field

In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

. Recording the intensity image then can lead to cancellations or other non-linear effects.

Introduction

By virtue of the linearity property of optical imaging systems, i.e.,

Image(Object₁ + Object₂) = Image(Object₁) + Image(Object₂)

the image of an object in a microscope or telescope can be computed by expressing the object-plane field as a weighted sum over 2D impulse functions, and then expressing the image plane field as the weighted sum over the images of these impulse functions. This is known as the superposition principle, valid for linear systems. The images of the individual object-plane impulse functions are called point spread functions, reflecting the fact that a mathematical point of light in the object plane is spread out to form a finite area in the image plane (in some branches of mathematics and physics, these might be referred to as Green's functions or impulse response

Impulse response

functions).

When the object is divided into discrete point objects of varying intensity, the image is computed as a sum of the PSF of each point. As the PSF is typically determined entirely by the imaging system (that is, microscope or telescope), the entire image can be described by knowing the optical properties of the system. This process is usually formulated by a convolution

Convolution

equation. In microscope image processing

Microscope image processing

Microscope image processing is a broad term that covers the use of digital image processing techniques to process, analyze and present images obtained from a microscope. Such processing is now commonplace in a number of diverse fields such as medicine, biological research, cancer research, drug...

and astronomy

Astronomy

Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

, knowing the PSF of the measuring device is very important for restoring the (original) image with deconvolution

Deconvolution

In mathematics, deconvolution is an algorithm-based process used to reverse the effects of convolution on recorded data. The concept of deconvolution is widely used in the techniques of signal processing and image processing...

Theory

The point spread function may be independent of position in the object plane, in which case it is called shift invariant. In addition, if there is no distortion in the system, the image plane coordinates are linearly related to the object plane coordinates via the magnification

Magnification

Magnification is the process of enlarging something only in appearance, not in physical size. This enlargement is quantified by a calculated number also called "magnification"...

M as:

.

If the imaging system produces an inverted image, we may simply regard the image plane coordinate axes as being reversed from the object plane axes. With these two assumptions, i.e., that the PSF is shift-invariant and that there is no distortion, calculating the image plane convolution integral is a straightforward process.

Mathematically, we may represent the object plane field as:

i.e., as a sum over weighted impulse functions, although this is also really just stating the shifting property of 2D delta functions (discussed further below). Rewriting the object transmittance function in the form above allows us to calculate the image plane field as the superposition of the images of each of the individual impulse functions, i.e., as a superposition over weighted point spread functions in the image plane using the same weighting function as in the object plane, i.e.,

. Mathematically, the image is expressed as:

in which PSF(x_i − Mu,y_i − Mv) is the image of the impulse function δ(x_o − u,y_o − v).

The 2D impulse function may be regarded as the limit (as side dimension w tends to zero) of the "square post" function, shown in the figure below (click to enlarge).

We imagine the object plane as being decomposed into square areas such as this, with each having its own associated square post function. If the height, h, of the post is maintained at 1/w², then as the side dimension w tends to zero, the height, h, tends to infinity in such a way that the volume (integral) remains constant at 1. This gives the 2D impulse the sifting property (which is implied in the equation above), which says that when the 2D impulse function, δ(x − u,y − v), is integrated against any other continuous function, f(u,v), it "sifts out" the value of f at the location of the impulse, i.e., at the point (x,y).

Since the concept of a perfect point source object is so central to the idea of PSF, it's worth spending some time on that before proceeding further. First of all, there is no such thing in nature as a perfect mathematical point source radiator; the concept is completely non-physical and is nothing more than a mathematical construct used to model and understand optical imaging systems. The utility of the point source concept comes from the fact that a point source in the 2D object plane can only radiate a perfect uniform-amplitude, spherical wave — a wave having perfectly spherical, outward travelling phase fronts with uniform intensity everywhere on the spheres (see Huygens-Fresnel principle

Huygens-Fresnel principle

The Huygens–Fresnel principle is a method of analysis applied to problems of wave propagation both in the far-field limit and in near-field diffraction.-History:...

). Such a source of uniform spherical waves is shown in the figure below (click to enlarge). We also note that a perfect point source radiator will not only radiate a uniform spectrum of propagating plane waves, but a uniform spectrum of exponentially decaying (evanescent

Evanescent wave

An evanescent wave is a nearfield standing wave with an intensity that exhibits exponential decay with distance from the boundary at which the wave was formed. Evanescent waves are a general property of wave-equations, and can in principle occur in any context to which a wave-equation applies...

) waves as well, and it is these which are responsible for resolution finer than one wavelength (see Fourier optics

Fourier optics

). This follows from the following Fourier transform

Fourier transform

In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

expression for a 2D impulse function,

The quadratic lens

Lens (optics)

A lens is an optical device with perfect or approximate axial symmetry which transmits and refracts light, converging or diverging the beam. A simple lens consists of a single optical element...

intercepts a portion of this spherical wave, and refocuses it onto a blurred point in the image plane. For a single lens

Lens (optics)

A lens is an optical device with perfect or approximate axial symmetry which transmits and refracts light, converging or diverging the beam. A simple lens consists of a single optical element...

, an on-axis point source in the object plane produces an Airy disc

Airy disc

In optics, the Airy disk and Airy pattern are descriptions of the best focused spot of light that a perfect lens with a circular aperture can make, limited by the diffraction of light....

PSF in the image plane. This comes about in the following way. It can be shown (see Fourier optics

Fourier optics

, Huygens-Fresnel principle

Huygens-Fresnel principle

The Huygens–Fresnel principle is a method of analysis applied to problems of wave propagation both in the far-field limit and in near-field diffraction.-History:...

, Fraunhofer diffraction

Fraunhofer diffraction

In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is viewed at the focal plane of an imaging lens....

) that the field radiated by a planar object (or, by reciprocity, the field converging onto a planar image) is related to its corresponding source (or image) plane distribution via a Fourier transform

Fourier transform

(FT) relation. In addition, a uniform function over a circular area (in one FT domain) corresponds to the Airy function, J₁(x)/x in the other FT domain, where J₁(x) is the first-order Bessel function

Bessel function

In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...

of the first kind. That is, a uniformly-illuminated circular aperture that passes a converging uniform spherical wave yields an Airy function image at the focal plane. A graph of a sample 2D Airy function is shown in the adjoining figure (click to enlarge).

Therefore, the converging (partial) spherical wave shown in the figure above produces an Airy disc

Airy disc

In optics, the Airy disk and Airy pattern are descriptions of the best focused spot of light that a perfect lens with a circular aperture can make, limited by the diffraction of light....

in the image plane. The argument of the Airy function is important, because this determines the scaling of the Airy disc (in other words, how big the disc is in the image plane). If Θ_max is the maximum angle that the converging waves make with the lens axis, r is radial distance in the image plane, and wavenumber

Wavenumber

In the physical sciences, the wavenumber is a property of a wave, its spatial frequency, that is proportional to the reciprocal of the wavelength. It is also the magnitude of the wave vector...

k = 2π/λ where λ = wavelength, then the argument of the Airy function is: kr tan(Θ_max). If Θ_max is small (only a small portion of the converging spherical wave is available to form the image), then radial distance, r, has to be very large before the total argument of the Airy function moves away from the central spot. In other words, if Θ_max is small, the Airy disc is large (which is just another statement of Heisenberg's uncertainty principle

Uncertainty principle

In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...

for FT pairs, namely that small extent in one domain corresponds to wide extent in the other domain, and the two are related via the space-bandwidth product. By virtue of this, high magnification

Magnification

Magnification is the process of enlarging something only in appearance, not in physical size. This enlargement is quantified by a calculated number also called "magnification"...

systems, which typically have small values of Θ_max (by the Abbe sine condition

Abbe sine condition

The Abbe sine condition is a condition that must be fulfilled by a lens or other optical system in order for it to produce sharp images of off-axis as well as on-axis objects...

), can have more blur in the image, owing to the broader PSF. The size of the PSF is proportional to the magnification

Magnification

Magnification is the process of enlarging something only in appearance, not in physical size. This enlargement is quantified by a calculated number also called "magnification"...

, so that the blur is no worse in a relative sense, but it is definitely worse in an absolute sense.

In the figure above, illustrating the truncation of the incident spherical wave by the lens, we may note one very significant fact. In order to measure the point spread function — or impulse response function — of the lens, we do not need a perfect point source that radiates a perfect spherical wave in all directions of space. This is because our lens has a only a finite (angular) bandwidth, or finite intercept angle. Therefore any angular bandwidth contained in the source, which extends past the edge angle of the lens (i.e., lies outside the bandwidth of the system), is essentially wasted source bandwidth because the lens can't intercept it in order to process it. As a result, a perfect point source is not required in order to measure a perfect point spread function. All we need is a light source which has at least as much angular bandwidth as the lens being tested (and of course, is uniform over that angular sector). In other words, we only require a point source which is produced by a convergent (uniform) spherical wave whose half angle is greater than the edge angle of the lens.

History and methods

The diffraction theory of point-spread functions was first studied by Airy

George Biddell Airy

Sir George Biddell Airy PRS KCB was an English mathematician and astronomer, Astronomer Royal from 1835 to 1881...

in the nineteenth century. He developed an expression for the point-spread function amplitude and intensity of a perfect instrument, free of aberrations (the so-called Airy disc

Airy disc

In optics, the Airy disk and Airy pattern are descriptions of the best focused spot of light that a perfect lens with a circular aperture can make, limited by the diffraction of light....

). The theory of aberrated point-spread functions close to the optimum focal plane was studied by the Dutch physicists Fritz Zernike and Nijboer in the 1930–40s. A central role in their analysis is played by Zernike’s circle polynomials

Zernike polynomials

In mathematics, the Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk. Named after Frits Zernike, they play an important role in beam optics.-Definitions:There are even and odd Zernike polynomials...

that allow an efficient representation of the aberrations of any optical system with rotational symmetry. Recent analytic results have made it possible to extend Nijboer and Zernike’s approach for point-spread function evaluation to a large volume around the optimum focal point. This Extended Nijboer-Zernike (ENZ) theory is instrumental in studying the imperfect imaging of three-dimensional objects in confocal microscopy

Confocal microscopy

Confocal microscopy is an optical imaging technique used to increase optical resolution and contrast of a micrograph by using point illumination and a spatial pinhole to eliminate out-of-focus light in specimens that are thicker than the focal plane. It enables the reconstruction of...

or astronomy under non-ideal imaging conditions. The ENZ-theory has also been applied to the characterization of optical instruments with respect to their aberration by measuring the through-focus intensity distribution and solving an appropriate inverse problem

Inverse problem

An inverse problem is a general framework that is used to convert observed measurements into information about a physical object or system that we are interested in...

PSF in microscopy

In microscopy, experimental determination of a PSF's shape is usually tricky, due to the difficulty of finding sub-resolution (point-like) radiating sources. Quantum dot

Quantum dot

A quantum dot is a portion of matter whose excitons are confined in all three spatial dimensions. Consequently, such materials have electronic properties intermediate between those of bulk semiconductors and those of discrete molecules. They were discovered at the beginning of the 1980s by Alexei...

s and fluorescent bead

Bead

A bead is a small, decorative object that is usually pierced for threading or stringing. Beads range in size from under to over in diameter. A pair of beads made from Nassarius sea snail shells, approximately 100,000 years old, are thought to be the earliest known examples of jewellery. Beadwork...

s are usually considered for this purpose. Theoretical models, on the other hand, allow the detailed calculation of the PSF for various imaging conditions. The most compact diffraction limited shape of the PSF is usually preferred. However by using appropriate optical elements (e.g., a spatial light modulator

Spatial light modulator

A spatial light modulator is an object that imposes some form of spatially-varying modulation on a beam of light. A simple example is an overhead projector transparency. Usually when the phrase SLM is used, it means that the transparency can be controlled by a computer. In the 1980s, large SLMs...

) the shape of the PSF can be engineered towards different applications.

The PSF in astronomy

In observational astronomy

Observational astronomy

Observational astronomy is a division of the astronomical science that is concerned with getting data, in contrast with theoretical astrophysics which is mainly concerned with finding out the measurable implications of physical models...

the experimental determination of a PSF is often very straightforward due to the ample supply of point sources (star

Star

A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...

s or quasars). The form and source of the PSF may vary widely depending on the instrument and the context in which it is used.

For radio telescopes and diffraction-limited space telescopes the dominant terms in the PSF may be inferred from the configuration of the aperture in the Fourier domain. In practice there may be multiple terms contributed by the various components in a complex optical system. A complete description of the PSF will also include diffusion of light (or photo-electrons) in the detector, as well as tracking errors in the spacecraft or telescope.

For ground based optical telescopes, atmospheric turbulence (known as astronomical seeing

Astronomical seeing

Astronomical seeing refers to the blurring and twinkling of astronomical objects such as stars caused by turbulent mixing in the Earth's atmosphere varying the optical refractive index...

) dominates the contribution to the PSF. In high-resolution ground-based imaging, the PSF is often found to vary with position in the image (an effect called anisoplanatism). In ground based adaptive optics

Adaptive optics

Adaptive optics is a technology used to improve the performance of optical systems by reducing the effect of wavefront distortions. It is used in astronomical telescopes and laser communication systems to remove the effects of atmospheric distortion, and in retinal imaging systems to reduce the...

systems the PSF is a combination of the aperture of the system with residual uncorrected atmospheric terms.

Point spread functions in ophthalmology

PSFs have recently become a useful diagnostic tool in clinical ophthalmology

Ophthalmology

Ophthalmology is the branch of medicine that deals with the anatomy, physiology and diseases of the eye. An ophthalmologist is a specialist in medical and surgical eye problems...

. Patients are measured with a wavefront

Wavefront

In physics, a wavefront is the locus of points having the same phase. Since infrared, optical, x-ray and gamma-ray frequencies are so high, the temporal component of electromagnetic waves is usually ignored at these wavelengths, and it is only the phase of the spatial oscillation that is described...

sensor, and special software calculates the PSF for that patient’s eye. In this manner a physician can "see" what the patient sees. This method also allows a physician to simulate potential treatments on a patient, and see how those treatments would alter the patient’s PSF. Additionally, once measured the PSF can be minimized using an adaptive optics system. This, in conjunction with a CCD, can be used to visualize anatomical structures not otherwise visible in vivo, such as cone photoreceptors.

Introduction

Theory

History and methods

PSF in microscopy

The PSF in astronomy

Point spread functions in ophthalmology

See also