Any 2-dimensional representation of a 3-dimensional object must undergo some kind of projection or mapping to form the planar image. Within the context of imaging of the retina, the eye can be approximated as a sphere and mathematical cartography can be used to understand the impact of projecting a spherical retina on to a planar image. When projecting a spherical geometry on a planar geometry, not all metric properties can be retained at the same time; some distortion will be introduced. However, if the projection is known it may be possible to perform calculations "in the background" that can compensate for these distortions.
The example in Figure UUU.1-1 shows an ultra-wide field image of the human retina. The original image has been remapped to a stereographic projection according to an optical model of the scanning laser ophthalmoscope it was captured on. Two circles have been annotated with an identical pixel count. The circle focused on the fovea (A) has an area of 4.08 mm2 whereas the circle nasally in the periphery (B) has an area of 0.97 3mm2, both as measured with the Area Measurement using the Stereographic Projection method. The difference in measurement is more than 400%, which indicates how measurements on large views of the retina can be deceiving.
The fact that correct measurement on the retina in physical units is difficult to do is acknowledged in the original DICOM OP SOP Classes in the description of the Pixel Spacing (0028,0030) tag.
Note
These values are specified as nominal because the physical distance may vary across the field of the images and the lens correction is likely to be imperfect.
The following use cases are examples of how the DICOM Wide Field Ophthalmology Photography objects may be used.
On routine wide-field imaging for annual surveillance for diabetic retinopathy a patient is noted to have no retinopathy, but demonstrates a pigmented lesion of the mid-periphery of the right eye. Clinically this appears flat or minimally elevated, irregularly pigmented without lacunae, indistinct margins on two borders, and has a surface that is stippled with orange flecks. The lesion is approximately 3 X 5 DD. This lesion appears clinically benign, but requires serial comparison to rule out progression requiring further evaluation. Careful measurements are obtained in 8 cardinal positions using a standard measurement tool in the reading software that calculates the shortest distance in mm between these points. The patient was advised to return in six months for repeat imaging and serial comparison for growth or other evidence of malignant progression.
A patient with a history of high myopia has noted recent difficulties descending stairs. She believes this to be associated with a new onset blind spot in her inferior visual field of both eyes, right eye greater than left. On examination she shows a bullous elevation of the retina in the superior periphery of both eyes due to retinoschisis, OD>OS. There is no evidence of inner or outer layer breaks, and the maculae are not threatened, so a decision is made to follow closely for progression suggesting a need for intervention. Wide field imaging of both fundi is obtained, with clear depiction of the posterior extension of the retinoschisis. Careful measurements of the shortest distance in mm between the posterior edge of the retinal splitting and the fovea is made using the diagnostic display measurement tool, and the patient was advised to return in four months for repeat imaging and serial comparison of the posterior location of the retinoschisis.
Patients with diabetes are enrolled in a randomized clinical trial to prospectively test the impact of disco music on the progression of capillary drop out in the retinal periphery. The retinal capillary drop-out is demonstrated using wide-field angiography with expanse of this drop-out determined serially using diagnostic display measurement tools, and the area of the drop-out reported in mm2. Regional areas of capillary drop out are imaged such that the full expanse of the defect is captured. In some cases this involves eccentric viewing with the fovea positioned in other than the center of the image. Exclusion criteria for patient enrollment include refractive errors greater than 8D of Myopia and 4D of hyperopia.
Patients with ARMD and subfoveal subretinal neovascular membranes but refusing intravitreal injections are enrolled in a randomized clinical trial to test the efficacy of topical anti-VEGF (Vascular Endothelial Growth Factor) eye drops on progression of their disease. The patients are selected such that there is a wide range of lesion size (area measured in mm2) and retinal thickening. This includes patients with significant elevation of the macula due to subretinal fluid.
Every 2-dimensional image that represents the back of the eye is a projection of a 3-dimensional object (the retina) into a 2-dimensional space (the image). Therefore, every image acquired with a fundus camera or scanning laser ophthalmoscope is a particular projection. In ophthalmoscopy, part of the spherical retina (the back of the eye can be approximated by a sphere) is projected to a plane, i.e., a 2-dimensional image.
The projection used for a specific retinal image depends on the ophthalmoscope; its optical system comprising lenses, mirrors and other optical elements, dictates how the image is formed. These projections are not well-characterized mathematical projections, but they can be reversed to return to a sphere. Once in spherical geometry, the image can then be projected once more. This time any mathematical projection can be used, preferably one that enables correct measurements. Many projections are described in the literature, so which one should be choosen?
Certain projections are more suitable for a particular task than others. Conformal projections preserve angle, which is a property that applies to points in the plane of projection that are locally distortion-free. Practically speaking, this means that the projected meridian and parallel intersect through a point at right angles and are equiscaled. Therefore, measuring angles on the 2-dimensional image yields the same results as measuring these on the spherical representation, i.e., the retina. Conformal projections are particularly suitable for tasks where the preservation of shapes is important. Therefore, the stereographic projection explained in Figure UUU.1.2-1 can be used for images on which to perform anatomically-correct measurements. The stereographic projection has the projection plane intersect with the equator of the eye where the fovea and cornea are poles. The points Fovea, p and q on the sphere (retina) are projected onto the projection plane (image in stereographic projection) along lines through the cornea where they intersect with the project plane creating points F′, p′and q′respectively.
Note that in the definition of stereographic projection the fovea is conceptually in the center of the image. For the mathematics below to work correctly, it is critical that each image is projected such that conceptually the fovea is in the center, even if the fovea is not in the image. This is not difficult to achieve as a similar result is achieved when creating a montage of fundus images; each image is re-projected relative to the area it covers on the retina. Most montages place the fovea in the center. An example of two images of the same eye in Figure UUU.1.2-2 and Figure UUU.1.2-3 taken from different angles and then transformed to adhere to this principle are in Figure UUU.1.2-4 and Figure UUU.1.2-5 respectively.
Furthermore the mathematical "background calculations" are well known for images in stereographic projection. Given points (pixels) on a retinal image, these can be directly located as points on the sphere and geometric measurements, i.e., area and distance measurements, performed on the sphere to obtain the correct values. The mathematical details behind the calculations for locating points on a sphere are presented in Section C.8.17.11.1.1 “Center Pixel View Angle” in PS3.3 .
The shortest distance between two points on a sphere lies on a "great circle", which is a circle on the sphere's surface that is concentric with the sphere. The great circle section that connects the points (the line of shortest distance) is called a geodesic. There are several equations that approximate the distance between two points on the back of the eye along the great circle through those points (the arc length of the geodesic), with varying degrees of accuracy. The simplest method uses the "spherical law of cosines". Let λs, ϕs; λf, ϕf be the longitude and latitude of two points s and f, and ∆λ ≡ |λf−λs| the absolute difference of the longitudes, then the central angle is defined as
where the central angle is the angle between the two points via the center of the sphere, e.g., angle a in Figure UUU.1.2-6. If the central angle is given in radians, then the distance d, known as arc length, is defined as
where R is the radius of the sphere.
This equation leads to inaccuracies both for small distances and if the two points are opposite each other on the sphere. A more accurate method that works for all distances is the use of the Vincenty formulae. Now the central angle is defined as
Figure UUU.1.2-6 is an example of a polygon made up of three geodesic Ga , Gb , Gc , describing the shortest distances on the sphere between the polygon vertices x1 , x2 , x 3. Angleγ is the angle on the surface between geodesics Ga and Gb . Angle a is the central angle (angle via the sphere's center) of geodesic Ga .
If the length of a path on the image (e.g., tracing of a blood vessel) is needed, this can be easily implemented using the geodesic distance defined above, by dividing the traced path into sections with lengths of the order of 1-5 pixels, and then calculating and summing the geodesic distance of each section separately. This works because for short enough distances, the geodesic distance is equal to the on-image distance. Note that sub-pixel accuracy is required.
To measure an area A defined by a polygon on the surface of the sphere where surface angle (such as γ in Figure UUU.1.2-6) αi for i=1,…,n for n angles internal to the polygon and R the radius of the sphere, we use the following formula, which makes use of the "angle excess".
This yields a result in physical units (e.g., mm2 if R was given in mm), but if R2 is omitted in the above formula, a result is obtained in units relative to the sphere, in steradians (sr), the unit of solid angle.
In practice, if the length of the straight arms of the calipers used to measure surface angle (such as γ in Figure UUU.1.2-6) are short then the angle measured on the image is equivalent to its representation on the sphere, which is a direct result of using the stereographic projection as it is conformal.
A 2D to 3D map includes 3D coordinates of all or a subset of pixels (namely coordinate points) to the 2D image. Implementations choose the interpolation type used, but it is recommended to use a spline based interpolation. See Figure UUU.1.3-1.
Pixels' 3D coordinates could be used for different analyses and computations e.g., measuring the length of a path, and calculating the area of region of interest, 3D computer graphics, registration, shortest distance computation, etc. Some examples of methods using 3D coordinates are listed in the following subsections.
Let the path between points A, and B be represented by set of N following pixels P={pi} and p0=A and pN=B. The length of this path can be computed from the partial lengths between path points by:
and where xi, yi, zi are the 3D coordinates of the point pi which is either available in the 2D to 3D map if pi is a coordinate point or it is computed by interpolation. Here it is assumed that the sequence of path points is known and the path is 4- or 8-connected (i.e., the path points are neighbors with no more than one pixel distance in horizontal, vertical, or diagonal direction). It is recommendable to support sub-pixel processing by using interpolation.
Shortest distance between two points along the surface of a sphere, known as the great circle or orthodromic distance, can be computed from:
Where r is the radius of the sphere and the central angle (Δσ) is computed from the Cartesian coordinate of the two points in radians. Here n1 and n2 are the normals to the ellipsoid at the two positions. The above equations can also be computed based on longitudes and latitudes of the points.
However, the shortest distance in general can be computed by algorithms such as Dijkstra, which computes the shortest distance on graphs. In this case the image is represented as a graph in which the nodes refer to the pixels and the weight of edges is defined based on the connectivity of the points and their distance.
Let R be the region of interest on the 2D image and it is tessellated by set of unit triangles T={Ti}. By unit triangle we refer to isosceles right triangle that the two equal sides have one pixel distance (4-connected neighbors). The area of the region of interest can be computed as the sum of partial areas of the unit triangles in 3D. Let { ai , bi , ci } be the 3D coordinates of the three points of unit triangle Ti . The 3D area of this triangle is
Where (‖ … ‖) and ( x ) refer to the magnitude and cross product, respectively.
Consider that ai , bi and ci are the 3D coordinates not the 2D indices of the unit triangle points on the image.
If Transformation Method Code Sequence (0022,1512) is (111791, DCM, "Spherical projection") is used then all coordinates in the Two Dimensional to Three Dimensional Map Sequence (0022,1518) are expected to lie on a sphere with a diameter that is equal to Ophthalmic Axial Length (0022,1019).
The use of this model for representing the 3D retina enables the calculation of the shortest distance between two points using great circles as per section UUU.1.3.2.