Course 13 Curves and Surfaces ----- Surface Representation Representation Interpolation Approximation ----- Surface Segmentation Segmentation Registration

Basic Geometry 1) Curve: Parameter form: e.g., a line segment between and :   Implicit form: Explicit form:

2) Surface: Parameter form: Implicit form: Explicit form: (graph surface)

3) Planes: Assume are 3 non-collinear points in space. A plane is uniquely determined by the points. Let The equation of the plane is

where is the normal of the plane where is the normal of the plane. In parametric form of the plane, we can redefine Then the plane equation is: where are parameters,

4) Surface Curvatures: Assume a surface in form of The normal of the surface of point is where are partial derivatives of the surface.

(1)  Normal curvature: The intersection of a surface and a plane that passes through the normal at a point P of the surface is a 2D curve. The curvature of the 2D curve at point P is called normal curvature of the surface at point P. ---- At a surface point. The surface has infinite many normal curvatures. ---- The minimum and maximum normal curvatures, K1 and K2 are called principal curvatures.

(2) Gaussian and mean curvatures. Gaussian curvature: Mean curvature:

(3) Types of a surface region K(x,y)   + – peak ridge saddle ridge flat minimal surface pit valley saddle Valley

(4) Curvature Calculation: For a surface

2. Representation of curve and surface 1) Curve representations: ----- Piecewise curve segments -----Joint condition: smooth at least in the first order derivatives,i.e. continuous in curve slope. ----- Lowerest order of polynomials. This leads to cubic spline curve: x(t) = axt3 + bxt2 + cxt + dx y(t) = ayt3 + byt2 + cyt + dy z(t) = azt3 + bzt2 + czt + dz

When one uses: then a curve can be written as:

For a curve, it is in piecewise form: With connection conditions:

2) Polygonal mesh surfaces: ----- Each mesh is a planar polygon (facet) ----- How to structure data if n data points are given. ----- Easily search from facet, edge or vertex. Data structure of polygonal mesh surface has 3 records: edge record, vertex record and facet record. Each record links the other two by proper pointers.

int: edge number, other attributes; Edge Record{ int: edge number, other attributes; pointer: North-vertex, south_vertex; pointer: left_facet, right_facet pointer: NE_winged_edge, NW_winged_edge, SE_winged_edge, SW_winged_edge ; } Vertex Record{ int: vertex_number; int: position(x,y); pointer: edge1, edge2, edge3;} Facet Record{ int: facet_number; pointer: edge1,edge2, …, edgen; int: other attributes; } N S NE NW SW SE W E

3) Surface Patches Surface patch is a portion of surface. It can be modeled by polynomial or tensors-product if it is a graph surface.

(1) Polynomial surface patch: Planar patch: z = a0 + a1x + a2y Bilinear patch: z = a0 + a1x + a2y + a3xy Biquadratic patch: z = a0 + a1x + a2y + a3xy + a4x2 + a5y2 Other high-order patches. The coefficients of polynomials are determined from data points using surface fitting methods, such as least-square regression.

(2) Tensor –product surfaces: Tensor-product surface representation is formed by combination of two curve representations, one for each parametric coordinate. Let curve1 and curve2 be cubic polynomial curves:

The tensor-product surface is defined as: Denote The surface is expressed as where the parametric matrix ca be determined from data points using regression.

3. Surface Interpolation Create data of surface from existing data points, e.g. ----- rectify the coordinate grids. ----- smooth representation of a surface by planar meshes.

1) Triangular mesh interpolation: From 3 data points, a planar patch can be determined: then at position , the depth of the surface is calculated as ----- It is good for coordinate rectification. ----- It cannot improve the smoothness of the surface even more data points are created.

2) Bilinear Interpolation From 4 data points, a bilinear surface patch can be determined as z = f(x,y) = a1 + a2x + a3y + a4xy then the new surface point can be calculated from given position (xi, yi) zi = f(x,y) = a1 + a2xi + a3yi + a4xiyi

Surface Approximation ----- Estimate mathematic form of surface from enough measurement data. ----- Smoothness constraint should be enforced to obtain a unique solution.

Variational method: Let the surface be in the form , at data points we should have By variation calculus, we get a differential equation. This differential equation can be solved for with Proper boundary conditions.

2) B-spline Regression Smoothness constraint is automatically applied when we approach surface regression by combining a set of smooth functions which are nonzero over a certain intervals. B-spline is the one commonly used.

In one-Dimensional case, any function f(x) has: ----- is nonzero over interval . i.e. spans over 4 intervals, and has one interval shift in the combination. ----- is continuous up to second order derivatives in its domain.

The definition of 1-D B-spline function is

Since each B-spline function covers 4 intervals, f(x) is approximated by only 4 spline functions in each interval. In interval [i, i+1]

At curve’s data point , above approximation becomes : With m+1 data point on the curve, the coefficients a0, a1,…, am can be solved (note: scaling and shift xi to interval [0, 1]). Although SVD method can be used in solving the linear equations, spare matrix technique is preferred as there are a lot of zero elements in coefficient matrix of the equations.

In 2D case, a surface is expressed as a tensor-product form i.e., where and are 1-D B-spline functions. A surface is approximated by 2D B-spline functions. At a data point of the surface on a surface patch, the above approximation becomes

With (n+1)(m+1) data points on the surface, the coefficients aij can be solved linearly. Again, spare matrix technique can yield fast computation.

3) Weighted Spline Approximation: So far, we have discussed surface approximations only for smooth surfaces. What shall we do with surface boundaries? Solution one: If we can know boundary positions, we can approach the surfaces piecewise, and the unions of the surface pieces are the object surface.

Solution two: weighted regression:   w(x,y): Large in surface far from boundary Small in surface near to boundary e.g., Where is gradient of surface is weight control parameter

5. Surface Segmentation 1) Concept: ----- a complex surface is segmented into several sub-surfaces. For each sub-surface, approximation is made to approach an analytic form.

-----Use curvature properties of surface initially to partition the surface into core regions, then shrink the surface region to small surface patches,so-called “seeds”. -----Extend surface seeds to cover neighboring data to form subsurfaces. Surface approximation and data extending are iteratively performed in this process.

2) Algorithm Given a set of surface data zi = f(xi, yi) to segment the surface into subsurfaces. Compute the first and second partial derivatives of the surface data. Compute Gaussian and mean curvatures at each image location. Label each surface data with the formula T[i,j] = 1 + 3(1 + sgn(H[i,j])) + (1 + sgn(K[i,j])) The formula numerates surface types:

T[i,j]=1: saddle ridge; T[i,j]=2: ridg T[i,j]=3: peak; T[i,j]=4: minimal surface T[i,j]=5: Flat; T[i,j]=6: No definition T[i,j]=7: Saddle valley; T[i,j]=8: Valley T[i,j]=9: Pit d) Identify core regions by finding connected components (clustering) e) Shrink the core region to eliminate false labels near region boundaries. f) Remove the core regions that are too small. g) Fit a bi-variable patch to each core region

h) Find a set of neighboring pixels of the regions with the similar data values to the neighboring pixels in the region. Include the candidate pixels in the region. i) Refit the surface with the surface data of the union of original region pixels and candidate pixels. If the fit is acceptable (residue is small enough), add the candidate data to the region. Otherwise, discard the candidates. j) Repeat step h) and step i) until no region is changed.

6. Surface Registration Registration problem is to align two shapes without knowing feature correspondences (e,g., point correspondences or line correspondences), such as align a set of surface data (e.g. range data) with a model of surface. This is to determine a transformation (rotation, translation and scaling) to surface data so that they can well align with the surface model.

Basic concept of surface registration: Given: a set of surface data (e.g. range data), and the model of the surface. Find: Rotation, translation and scaling applied to surface data to align the data with the surface model.

Step1: Apply transformation to surface data: Step2: Find distance from to surface model. d = distance (from to model). Step3: minimize the distance measurement to find R, T and s.