If a minor contains only one row from some view, the image coordinate corresponding to this row can be factored out (using Laplace expansion along the corresponding column).

In the weakly calibrated case, i.e., when point correspondences are the only information available, a projective reconstruction can be obtained.

In the noise-free case, D=diag⁢σ1,σ2,σ3,σ4,0,…⁢0, thus, only the first 4 columns of U (V) contribute to this matrix product. Let U3×m×4 (Vn×4) the matrix of the first 4 columns of U (V). Then:

This is the trifocal constraint, that links (via a trilinear form) m1, s2 (any line through m2) and s3 (any line through m3).

In conclusion, the Modulation Transfer Function is a pivotal parameter for evaluating and optimizing optical systems. By understanding resolution, contrast, and how MTF combines these factors, optical designers can make informed decisions to select the right components and achieve superior image quality for their applications. MTF data serves as a powerful tool in the hands of those seeking precision and excellence in optical system design.

This indicates that no interesting constraints can be written for more than four views5Actually, it can be proven that also the quadrifocal constraints are not independent (Ma et al.,2003)..

Hence, at least one row has to be taken from each view to obtain a meaningful constraint, plus another row from each camera to prevent the constraint to be trivially factorized.

numerische Apertur {f} optics. numerical {adj} · in Zahlen ausgedrückt · zahlenmäßig · numerisch. Werbung. numerical reasoning · Zahlenverständnis {n} ...

If the intrinsic parameters of the cameras are known, we can obtain a Euclidean reconstruction, that differs from the true reconstruction by a similarity transformation. This is composed by a rigid displacement (due to the arbitrary choice of the world reference frame) plus a a uniform change of scale (due to the well-known depth-speed ambiguity).

The minors that does not contain at least one row from each camera are identically zero, since they contain a zero column.

If m>4, there is no way to avoid that the minors contain only one row from some views. Hence, constraints involving more than 4 cameras can be factorised as product of the two-, three-, or four-views constraints and image point coordinates.

An elegant method for multi-image reconstruction was described in Sturm and Triggs (1996), based on the same idea of the factorization method of Tomasi and Kanade (1992).

Epipolar transfer fails when the three optical rays are coplanar, because the epipolar lines are coincident. This happens:

Please note that given two (arbitrary) lines in two images, they can be always seen as the image of a 3-D line L, because two planes always define a line, in projective space (this is why there is no such thing as the epipolar constraint between lines.)

Hence, s3=HT⁢s1 since s1 and s3 are both projection of the same line, that belongs to the plane 3The reader can verify that if H is the homography induced by a plane between two views, such that conjugate points are related by m2=H⁢m1, conjugate lines are related by s2=HT⁢s1..

The difference between both lenses comes from their shape, while a spherical lens shape can be defined from a virtual center and a fix radius of curvature, an ...

This allows for point transfer or prediction. Indeed, m3 belongs simultaneously to the epipolar line of m1 and to the epipolar line of m2, hence:

MTFOptics

MTF is a powerful tool to quantify the overall imaging performance of a system in terms of resolution and contrast. Understanding the MTF curves of each imaging lens and camera sensor within a system allows designers to make informed choices when optimizing for specific resolutions.

Please note that Eq. (95) can be also used to triangulate one point M in multiple views, by solving the homogeneous linear system for M,-ζ1,-ζ2,⋯,-ζmT.

In this formula the mij are known, but all the other quantities are unknown, including the projective depths ζij. Equation (89) tells us that W can be factored into the product of a 3×m×4 matrix P and a 4×n matrix M. This also means that W has rank four.

If one applies the method of Section 4.5.3 to consecutive pairs of views, she would obtain, in general, a set of reconstructions linked to each other by an unknown projective transformation (because each camera pair defines its own projective frame).

This technique is fast, requires no initialization, and gives good results in practice, although there is no guarantee that the iterative process will converge. A provably convergent iterative method has been presented by Mahamud et al. (2001).

Modulation transferfunctionRadiology

If we assume for a moment that the projective depths ζij are known, then matrix W is known too and we can compute its singular value decomposition:

Modulation transferfunctionimage processing

For instance, the imaging lens, camera sensor, image capture boards, and video cables each have their associated MTF. By analyzing the system MTF curve, designers can determine which combination of components will provide sufficient performance for a given application, considering factors like contrast requirements and resolution.

This is the point transfer equation: if m1 and m2 are conjugate points in the first and second view respectively, the position of the conjugate point m3 in the third view is computed by means of the trifocal matrix.

This is the trifocal constraint for lines, which also allows direct line transfer: if s3 and s2 are two lines in the third and second view respectively, the image s1 in the first view of the line in space determined by s2 and s3 is obtained by means of the trifocal matrix.

Resolution and contrast are fundamental factors in achieving sharp and clear images. Resolution pertains to an imaging system’s ability to distinguish fine object details and is typically expressed in line-pairs per millimeter (lp/mm), where each line-pair consists of a black line followed by a white line. Contrast, on the other hand, measures an optical system’s ability to distinguish between light and dark areas in an image.

If the trifocal geometry is known, given two conjugate points m1 and m2 in view 1 and 2 respectively, the position of the conjugate point m3 in view 3 is completely determined (Figure 12).

This geometry could be described in terms of fundamental matrices linking pairs of cameras, but a more compact and elegant description is provided by a suitable trilinear form, in the same way as the epipolar (bifocal) geometry is described by a bilinear form.

The Kronecker notation and the tensorial notation are deeply related, as both represents multilinear forms. To draw this relationship in the case of the trifocal geometry, let us expand the trifocal matrix into its columns T=t1⁢t2⁢t3 and m1 into its components m1=u,v,wT. Then, thanks to the linearity of the vector transposition:

A third equivalent formulation of the trifocal constraint is derived if we look at the vector T⁢m1 in Eq. (73) as the vectorization of a suitable matrix. This is easy to write thanks to the vector transposition2The vector transposition operator Apgeneralizes the transpose of a matrix A by operating on vectors of p entries at a time.:

This reconstruction is unique up to a (unknown) projective transformation. Indeed, for any non singular projective transformation T, T⁢P and T-1⁢M is an equally valid factorization of the data into projective motion and structure.

Double Concave Lens (aslo named Bi-Concave Lens) is used to divergent light. The Bi-Concave is best used when the input beam is converging.

If m=4, choosing one row from each of four different views gives a quadrilinear four-view constraint, expressed by the quadrifocal tensor.

Consider a line L in space projecting to s1, s2 and s3 in the three cameras. The trifocal constraint must hold for any point m1 contained in the line s1:

Denoting the cameras by 1,2,3, there are now three fundamental matrices, F1,2, F1,3, F2,3, and six epipoles, ei,j, as in Figure 11. The three fundamental matrices describe completely the trifocal geometry (Faugeras and Robert,1994).

An equivalent formulation of the trifocal constraint that generalizes the expression of a bilinear form (Cfr. pg. 4.5.2) is obtained by applying once again the property v⁢e⁢c⁢A⁢X⁢B=BT⊗A⁢v⁢e⁢c⁢X:

High-quality optics excel in transferring contrast at higher spatial frequencies, which translates to higher resolution. To assess this ability, MTF comes into play. MTF quantifies a lens’s capacity to transfer the contrast of a sample to an image using spatial frequency (resolution). Spatial frequency is defined as the number of line pairs per millimeter (lp/mm). Typically, MTF is determined using test charts featuring alternating black and white lines.

Mtf functionin optical

Hence we can write a total of nine constraints similar to Eq. (72), only four of which are independent (two for each point):

Writing Eq. (45) for each camera pair (taking the centre of the third camera as the point M) results in three epipolar constraints:

The trifocal geometry could be used to link together consistently triplets of views. In Section 4.5.3 we saw how a camera pair can be extracted from the fundamental matrix. Likewise, a triplet of consistent cameras can extracted from the trifocal matrix (or tensor). The procedure is more tricky, though.

MTFimage quality

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A half waveplate, sometimes equivalently called half wave retarder or half-wavelength plate, induces a retardation of 1/2 lambada between the fast component and ...

A line s2 in the second view defines (by back-projection) a 3-D plane, which induces a homography H between the first and the third view.

Modulation transferfunctionin Ophthalmology

In traditional system integration, the resolution is often estimated based on the principle of the weakest link, assuming that the system’s resolution is solely limited by the component with the lowest resolution. However, this approach is flawed as every component within the system contributes to image quality, and the overall MTF of the system is the product of all the MTF curves of its components.

The trifocal constraint represents the trifocal geometry (nearly) without singularities. It only fails is when the cameras are collinear and the 3-D point is on the same line.

This description of the trifocal geometry fails when the three cameras are collinear, and the trifocal plane reduces to a line.

Therefore, every triplet {m1, m2, m3} of corresponding points gives four linear independent equations. Seven triplets determine the 27 entries of T.

Depth of focus characterizes how much tip and tilt is tolerated between the lens image plane and the sensor plane itself. As f/# decreases, the depth of focus ...

This equation can be used to recover T (likewise we did for F). The coefficient matrix is a 9×27 matrix; its rank is four, being the Kronecker product of a vector by a rank-2 matrix by a rank-2 matrix.

Mtf functionformula

This implies that the transpose of the leftmost term in parentheses (which is a 3-D vector) belongs to the kernel of m3×, which is equal to m3 (up to a scale factor) by construction. Hence

This implies that T⁢m13 can be seen as the linear combination of the matrices t13,t23,t33 with the components of m1 as coefficients. Therefore, the action of the trilinear form Eq. (81) is to first combine matrices t13,t23,t33 according to m1, then combine the columns of the resulting matrix according to s3 and finally to combine the elements of the resulting vector according to s2, to obtain a scalar.

Starting from an initial guess for ζij (typically ζij=1), the following iterative procedure4Whilst this procedure captures the main idea of Sturm and Triggs, it is not exactly the algorithm proposed in (Sturm and Triggs,1996). To start with, the original algorithm (Sturm and Triggs,1996) was not iterative and used the epipolar constraint (Eq.45) to fix the ratio of the projective depths of one point in successive images. It was Triggs (1996) who made the scheme iterative. Moreover in (Sturm and Triggs,1996) the normalization of W is performed by normalizing rows and columns of W. The Frobenius norm was used by (Oliensis,1999). A similar scheme was also proposed by Heyden (1997). is used:

If one applies the method of Section 4.4.2 to view pairs 1-2, 1-3 and 2-3 one obtains three displacements R12,t⌃12,R13,t⌃13 and R23,t⌃23 known up a scale factor, as the norm of translation cannot be recovered, (the symbol ⌃ indicates a unit vector).

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The plane containing the three optical centres is called the trifocal plane. It intersects each image plane along a line which contains the two epipoles.

We also discover that three views are all we need, in the sense that additional views do not allow us to compute anything we could not already compute (Section 5.4).

View synthesisLaveau and Faugeras (1994); Avidan and Shashua (1997); Boufama (2000), exploit the trifocal geometry to generate novel (synthetic) images starting from two reference views. A related topic is image-based rendering(Lengyel,1998; Zhang and Chen,2003; Isgrò et al.,2004).

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Geometrically, the trifocal constraint imposes that the optical rays of m1 intersect the 3-D line L that projects onto s2 in the second image and s3 in the third image.

This implies that the 3×m×m+4 matrix L is rank-deficient, i.e., r⁢a⁢n⁢k⁢L

In presence of noise, σ5 will not be zero. By forcing D=diag⁢σ1,σ2,σ3,σ4,0,…⁢0 one computes the solution that minimizes the following error:

Three fundamental matrices include 21 free parameters, less the 3 constraints above; the trifocal geometry is therefore determined by 18 parameters.

In addition, by generalizing the case of two views, one might conjecture that the trifocal geometry should be represented by a trilinear form in the coordinates of three conjugate points.

MTF, as its name suggests, measures a lens’s capability to transfer contrast at specific resolutions from the object to the image. It combines both resolution and contrast into a single specification. As the line spacing decreases (frequency increases) on the test target, it becomes progressively challenging for the lens to efficiently transfer this decrease in contrast, resulting in a decrease in MTF.

If m=3, choosing two rows from one view, one row from another view and one row from a third view gives a trilinear three-view constraint, expressed by the trifocal tensor.

Modulation transferfunctionformula

The Modulation Transfer Function (MTF) is a vital parameter used to assess the performance of optical systems, ranging from simple lenses to complex imaging lens assemblies. It serves as a standardized quantitative measure for optical designers and microscopists to evaluate and compare lenses for various applications such as DNA sequencers, cell analyzers, slide scanners, and industrial inspection equipment. In this article, we will delve into the details of MTF, exploring its components, significance, and applications.

Oct 7, 2020 — In order to be exact in solving for focal length, you would need to know the aspect ratio of the sensor. If exact is not necessary the diagonal ...

In this section we study the relationship that links three or more views of the same 3-D scene, known in the three-view case as trifocal geometry.

If m=2 choosing two rows from one view and two rows from another view gives a bilinear two-view constraint, expressed by the bifocal tensor i.e., the fundamental matrix.

In both cases, the solution is not a straightforward generalization of the two view case, as the problem of global consistency comes into play (i.e., how to relate each other the local reconstructions that can be obtained from view pairs).

Aug 26, 2024 — One of the most common optical structures is the achromatic doublet. It is used to reduce chromatic aberrations. At its most basic, ...

We outline here an alternative and elegant way to derive all the meaningful multi-linear constraints on N views, based on determinants, described in (Heyden,1998). Consider one image point viewed by m cameras: