Proof of Correctness and Time Complexity Analysis of a Maximum Distance Transform Algorithm

TL;DR

This paper presents a new average time O(N) algorithm for maximizing the distance transform, analyzing its correctness, runtime, and duality with the minimum transform in computer vision applications.

Contribution

It introduces an average time O(N) algorithm for maximum distance transform, with a proof of correctness and analysis of duality with the minimum transform.

Findings

The algorithm achieves average linear time complexity.

Proof of correctness for the maximum distance transform algorithm.

Discussion of duality between minimum and maximum transforms.

Abstract

The distance transform algorithm is popular in computer vision and machine learning domains. It is used to minimize quadratic functions over a grid of points. Felzenszwalb and Huttenlocher (2004) describe an O(N) algorithm for computing the minimum distance transform for quadratic functions. Their algorithm works by computing the lower envelope of a set of parabolas defined on the domain of the function. In this work, we describe an average time O(N) algorithm for maximizing this function by computing the upper envelope of a set of parabolas. We study the duality of the minimum and maximum distance transforms, give a correctness proof of the algorithm and its runtime, and discuss potential applications.

Equations

$$D(x)=\underset{p}{\min }I(p)+\alpha (p-x{)}^2+\beta (p-x).$$

$$D(x,y)=\underset{p,q}{\min }I(p,q)+\alpha (p-x{)}^2+\beta (p-x)+\gamma (q-y{)}^2+\delta (q-y).$$

$$D(x)$$

$$D(x,y)$$

$${\Phi }_p\left(x\right)=I(p)+\alpha (p-x{)}^2+\beta (p-x).$$

$$\underset{p}{\max }I(p)+\alpha (p-x{)}^2+\beta (p-x)=\underset{p}{\min }-I(p)-\alpha (p-x{)}^2-\beta (p-x).$$

$$s_{p,q}=\frac{(I(q)+\alpha q^2+\beta q)-(I(p)+\alpha p^2+\beta p)}{2\alpha (p-q)}.$$

$$\frac{d}{dx}{\Phi }_p\left(x\right)=-2\alpha (p-x)-\beta$$

$$D(x,y)=\underset{p}{\max }\alpha (p-x{)}^2+\beta (p-x)+\underset{q}{\max }I(p,q)+\gamma (q-y{)}^2+\delta (q-y).$$

$$D(x,y)=\underset{p}{\max }\alpha (p-x{)}^2+\beta (p-x)+I^{\prime }(p,q).$$

Full text

1 Introduction

The distance transform algorithm is frequently used for solving graph inference problems in computer vision, machine learning and other domains, as well as in image processing applications. It has gained popularity in recent years with the proliferation of works based on deformable part models (DPMs) (Felzenszwalb et al., 2010; Zhu and Ramanan, 2012).

The minimum distance transform algorithm is used to minimize functions of specific forms on $1D$ or $2D$ grids. In this work, we assume the form of the function to be quadratic. This assumption is a design choice made by most deformable part models (Felzenszwalb et al., 2010; Zhu and Ramanan, 2012). More precisely, the minimization problem on a $1D$ grid can be expressed as

[TABLE]

The problem on a $2D$ grid is given by the equation

[TABLE]

Conventionally, distance transforms have been used for minimizing functions. However, maximizing these functions is an equivalent problem. We discuss this duality in section 2. We define maximum distance transforms by the following equations for the $1D$ and $2D$ cases respectively:

[TABLE]

Kindly note that Euclidean distance transforms are a special case ($\alpha,\gamma=1;\beta,\delta=0$) of the problems in equations 1-4. These quadratic functions can be seen as parabolas centered at the grid points, as is shown in Figure 1(a). Each grid point $p$ in equation 1 is associated with a parabolic function of the following form:

[TABLE]

Therefore, finding the minimum distance transform can be understood as finding the minimum out of these functions at each point $x$ in the domain. In short, the minimum distance transform is given by the lower envelope of these parabolas. Equivalently, the maximum distance transform is given by the upper envelope of these parabolas. We will henceforth discuss the maximum distance transform in this paper. We also use the term upper envelope to describe the curve defining the maximum distance transform for all points in the domain.

We now define the range of a parabola as follows.

Definition 1** (Range of a parabola).**

If $\Phi_{p}\left(x\right)$ is a parabola in the upper envelope centred at grid point $p$, its range, $\mathcal{R}_{p}$, is the interval $(\sigma_{1},\sigma_{2}]$ in which $\Phi_{p}\left(x\right)$ forms the upper envelope.

The ranges of several parabolas are shown in Figure 1(c). Felzenszwalb and Huttenlocher (Felzenszwalb and Huttenlocher, 2004) describe an $O(N)$ algorithm for computing the minimum distance transform for Equations 1 and 2. Their algorithm works by finding the lower envelope of these parabolas. In this work, we describe an $O(N)$ algorithm for computing the maximum distance transform (Equations 3 and 4). Our algorithm works by finding the upper envelope of the parabolas.

We begin by showing the duality between maximum and minimum distance transforms, followed by our algorithm in detail. We then give a correctness proof and analyze the complexity of our algorithm. Finally, we discuss applications of our algorithm.

2 Duality between Minimum and Maximum Distance Transforms

The duality between the Minimum and Maximum distance transforms can be trivially explained by the equation

[TABLE]

Thus, if we know how to solve the minimum distance transform we can also solve the maximum distance transform simply by changing the sign of the function. The algorithm in (Felzenszwalb and Huttenlocher, 2004) solves the minimum distance transform for upward opening parabolas. Upward opening parabolas are characterized by positive quadratic terms, or more precisely $\alpha,\gamma>0$ (Equations 1-4). Thus, the algorithm in (Felzenszwalb and Huttenlocher, 2004) can be used to find the upper envelope / maximum distance transform when $\alpha,\gamma<0$.

This is better illustrated in Figure 1(c). Upon inverting the signs of the quadratic terms, the upward opening parabolas become downward opening parabolas: the algorithm which gave the lower envelope now gives us the upper envelope. However, as mentioned before, the algorithm in (Felzenszwalb and Huttenlocher, 2004) solves the minimum distance transform only for upward opening parabolas. In the next section, we describe an algorithm which finds the upper envelope for upward opening parabolas, and therefore can be used to find the lower envelope of downward opening parabolas.

3 Algorithm

In this section we describe an algorithm for computing the maximum distance transform. We begin with the $1D$ grid case (Equation 3), and extend it for arbitrary dimensions. Our algorithm begins by computing the upper envelope of the parabolas. The distance transform at a point $x$ is simply the height of the upper envelope at $x$. This is illustrated in Figure 1(a).

3.1 Intersection of two parabolas

Kindly note that all parabolas, albeit centered at different points in space, have the same shape. This is due to the fact that they have the same $\alpha$ parameter. The parabola at a grid point $p$ is defined in Equation 5.

Lemma 1**.**

Parabolas $\Phi_{p}\left(x\right)$ and $\Phi_{q}\left(x\right)$ at two grid points $p$ and $q$, respectively, intersect at exactly one point. This point of intersection is given by

[TABLE]

Proof.

The proof follows from algebra. The point of intersection of the parabolas $\Phi_{p}\left(x\right)$ and $\Phi_{q}\left(x\right)$ can be obtained by setting $\Phi_{p}\left(x\right)=\Phi_{q}\left(x\right)$ . Since the coefficient of $x^{2}$ in both of them is independent of $p$ and $q$, it cancels out, giving us a unique solution. ∎

3.2 Upper Envelope

Lemma 2**.**

If $p<q$ are two grid points with $s_{p,q}$ being the point of intersection of $\Phi_{p}\left(x\right)$ and $\Phi_{q}\left(x\right)$, then $\Phi_{p}\left(x\right)>\Phi_{q}\left(x\right)$ for $x>s_{p,q}$, and $\Phi_{p}\left(x\right)<\Phi_{q}\left(x\right)$ for $x<s_{p,q}$.

Proof.

To prove this, we carefully examine the three conditions in which two parabolas $\Phi_{p}\left(x\right)$ and $\Phi_{q}\left(x\right)$ can intersect. Each parabola $\Phi_{p}\left(x\right)$ is centered at the grid point $p$. The left arm of a parabola is defined to be the part of the parabola to the left of $p$. Similarly, the right arm of the parabola is defined to be the part of the parabola to the right of $p$. There are three possibilities:

The right arm of $\Phi_{p}\left(x\right)$ intersects the left arm of $\Phi_{q}\left(x\right)$.  This corresponds to $p<s_{p,q}<q$. Since $\Phi_{p}\left(x\right)$ and $\Phi_{q}\left(x\right)$ have positive and negative gradients, respectively, at $s_{p,q}$, the Lemma is trivially satisfied. See Figure 1(b), part i. 2. 2.

The left arm of $\Phi_{p}\left(x\right)$ intersects the right arm of $\Phi_{q}\left(x\right)$. This situation can never occur since $p<q$, and the left arm of $\Phi_{p}\left(x\right)$ lies entirely to the left of $p$ whereas the right arm of $\Phi_{q}\left(x\right)$ lies entirely to the right of $q$. 3. 3.

The left (right) arm of $\Phi_{p}\left(x\right)$ intersects the left (right) arm of $\Phi_{q}\left(x\right)$.   This corresponds to $s_{p,q}\leq p<q$ and $p<q\leq s_{p,q}$, respectively. The expression for the gradient of the parabola $\Phi_{p}\left(x\right)$ is given by

[TABLE]

The gradient of $\Phi_{p}\left(x\right)$ at $x$, therefore, depends on $p-x$. In this case, since $|s_{p,q}-p|$ is less (greater) than $|s_{p,q}-q|$, the absolute value of the gradient of $\Phi_{p}\left(x\right)$ at $s_{p,q}$ will be less (greater) than that of $\Phi_{q}\left(x\right)$. It follows that $\Phi_{p}\left(x\right)>\Phi_{q}\left(x\right)$ for $x>s_{p,q}$, and $\Phi_{p}\left(x\right)<\Phi_{q}\left(x\right)$ for $x<s_{p,q}$. See Figure 1(b), parts ii, iii.

The proof of the Lemma follows from the three exhaustive cases. Figure 1(b) gives visual confirmation to this Lemma. ∎

For simplicity, we will define an order relation over the ranges of parabolas.

Definition 2** ($\succ$ and $\prec$ relations on ranges of parabolas).**

We say $\mathcal{R}_{p}=(\sigma_{1},\sigma_{2}]\prec\mathcal{R}_{q}$ if and only if $\sigma_{2}\leq\inf\left\{x|x\in\mathcal{R}_{q}\right\}$. $\succ$ is defined to be the inverse relation to $\prec$ so that $\mathcal{R}_{p}\prec\mathcal{R}_{q}\Leftrightarrow\mathcal{R}_{q}\succ\mathcal{R}_{p}\,$. Intuitively, $\mathcal{R}_{p}\prec\mathcal{R}_{q}$ signifies that $\mathcal{R}_{p}$ lies completely to the left of $\mathcal{R}_{q}\,$.

Definition 3** (Adjacency of ranges).**

Two ranges $\mathcal{R}_{p}$ and $\mathcal{R}_{q}$ are said to be adjacent if and only if $\sup\left\{x|x\in\mathcal{R}_{1}\right\}=\inf\left\{x|x\in\mathcal{R}_{2}\right\}$, where $\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ are $\mathcal{R}_{p}$ and $\mathcal{R}_{q}$ in no particular order.

Lemma 3**.**

If $\Phi_{p}\left(x\right)\neq\Phi_{q}\left(x\right)$ are two parabolas in the upper envelope such that $\mathcal{R}_{p}$ and $\mathcal{R}_{q}$ are adjacent, with $\mathcal{R}_{p}\prec\mathcal{R}_{q}$, then $p>q$, and vice versa.

Proof.

Since $\mathcal{R}_{p}$ and $\mathcal{R}_{q}$ are adjacent and $\mathcal{R}_{p}\prec\mathcal{R}_{q}$, we can assume their ranges are $\mathcal{R}_{p}=(\sigma_{1},s_{p,q}]$, and $\mathcal{R}_{q}=(s_{p,q},\sigma_{3}]$, for some $\sigma_{1}$ and $\sigma_{3}\,$. If $p<q$, Lemma 2 is violated. Hence, $p>q$. Conversely, if $p>q$ and $\Phi_{p}\left(x\right)$ and $\Phi_{q}\left(x\right)$ have adjacent ranges in the upper envelope, then $\mathcal{R}_{p}\prec\mathcal{R}_{q}$ from Lemma 2. ∎

Corollary 4**.**

(to Lemma 3) If a parabola $\Phi_{p}\left(x\right)$ is maximum over $\mathcal{R}_{p}=(\sigma_{1},\sigma_{2}]$, $\sigma_{2}$ being finite, then $\exists q<p$ such that $\mathcal{R}_{q}\succ\mathcal{R}_{p}$.

Proof.

The finiteness of $\sigma_{2}$ guarantees the presence of another parabola which is maximum over an interval in $(\sigma_{2},\infty)\,$. Lemma 3 tells us that a parabola $\Phi_{q}\left(x\right)$ that is maximum over an interval $\mathcal{R}_{q}$ adjacent to $\mathcal{R}_{p}$ with $\mathcal{R}_{p}\prec\mathcal{R}_{q}$ implies $q<p$. This guarantees the existence of one such $q$. Further, applying this argument repeatedly, we conclude that for all parabolas $\Phi_{t}\left(x\right)$ whose ranges satisfy $\mathcal{R}_{t}\succ\mathcal{R}_{p}$, $t<p\,$. ∎

Corollary 5**.**

(To Lemma 3) If $\mathcal{R}_{p}$ and $\mathcal{R}_{q}$ are ranges of two parabolas in the upper envelope such that $\mathcal{R}_{p}\succ\mathcal{R}_{q}$, then $p<q$, and vice versa.

Proof.

We know that $\mathcal{R}_{p}\succ\mathcal{R}_{q}$. Let $t$ be the parabola in the upper envelope so that $\mathcal{R}_{t}\prec\mathcal{R}_{p}$ and is adjacent to it. By Lemma 3, $p<t$. If we keep applying this argument repeatedly to $t$, we will eventually reach $q$. The proof follows.

Conversely, if $p<q$ and $\Phi_{p}\left(x\right)$ and $\Phi_{q}\left(x\right)$ are part of the upper envelope, then $\mathcal{R}_{p}\succ\mathcal{R}_{q}$. by a similar repeated application of Lemma 3. ∎

3.3 Algorithm

Before describing the algorithm, we describe the data-structures employed.

$k$ is an integer counting the number of parabolas in the upper envelope. 2. 2.

$v[\cdot]$ is an array which holds the indices of the parabolas currently forming the upper envelope. At any stage of the algorithm $v[\cdot]$ holds $k+1$ elements. 3. 3.

$z[\cdot]$ is an array holding the ranges of the parabolas in decreasing order, i.e. from $+\infty\rightarrow-\infty$. Thus, the range of $p$-th parabola, given by $\Phi_{v[p]}\left(x\right)$ is given by $\mathcal{R}_{v[p]}=(z[p+1],z[p]]$.

We assume that all parabolas are ordered according to the horizontal locations of their grid points. We start by initialising the upper envelope to be the parabola at the first grid point. The range is initialised to be $(+\infty,-\infty)$. We compute the upper envelope by iteratively scanning the grid points from left to right. Each time we encounter a new grid point $q$, we compute its intersection with the existing parabolas in the upper envelope. If $\Phi_{q}\left(x\right)$ intersects a parabola $\Phi_{p}\left(x\right)$ in the upper envelope inside $\mathcal{R}_{p}$, we update the upper envelope: all parabolas at grid points to the right of $p$ are removed from the upper envelope, the parabola $q$ is added to the upper envelope, and the ranges of $p$ and $q$ are updated. It is guaranteed that $\Phi_{q}\left(x\right)$ will intersect with at least one parabola $\Phi_{p}\left(x\right)$ inside $\mathcal{R}_{p}$. This is demonstrated in Figure 2 and formally proven in Lemma 7.

After computing the upper envelope, we scan the grid points from left to right filling in the values of the distance transform by investigating the range array. A pseudo-code is described in Algorithm 1. We now prove the correctness of Algorithm 1.

Theorem 6**.**

Algorithm 1 correctly computes the maximum distance transform.

Proof.

We use the principle of mathematical induction on the number of parabolas, $N$, for our proof.

Base case. For the base case, $N=1$, the algorithm does not enter the loop on line 7 at all. Since the arrays $v[\cdot]$ and $z[\cdot]$ are already initialised accordingly, the algorithm returns the correct envelope—a single parabola having the range $(-\infty,\infty)\,$.

Inductive step. Let us assume that the algorithm gives the correct upper envelope for the $q-1$ grid-points considered so far. We now consider the $q$-th parabola. The algorithm then computes $s_{v[p],q}$—the point of intersection of the $p$-th parabola in the envelope, $\Phi_{v[p]}\left(x\right)$, with $\Phi_{q}\left(x\right)$. (Line 9 in Algorithm 1). Recall $\mathcal{R}_{p}$ from Definition 1. There are two possibilities:

Case 1: $s_{v[p],q}\in\mathcal{R}_{v[p]}$.

It follows from Lemma 2 that $\Phi_{q}\left(x\right)>\Phi_{v[p]}\left(x\right)\,\forall\,x<s_{v[p],q}$, and subsequently, $\Phi_{q}\left(x\right)>\Phi_{t}\left(x\right)$ in the same interval $\forall t<q$. Hence, $\mathcal{R}_{q}=(-\infty,s_{v[p],q}]$. Furthermore, parabolas in the upper envelope whose ranges fall in $\mathcal{R}_{q}$ are no longer maximum in their respective ranges. These parabolas correspond to all the grid points between $v[p]$ and $q$, and they are removed by readjusting the value of $k$ in the algorithm. Furthermore, no more parabolas can be added or removed from the envelope in this iteration, so we break the for loop. Lines 11 to 15 in Algorithm 1 handle precisely this case.

Case 2: $s_{v[p],q}\notin\mathcal{R}_{v[p]}$.

This says that the point of intersection of $\Phi_{v[p]}\left(x\right)$ and $\Phi_{q}\left(x\right)$ lies outside $\mathcal{R}_{v[p]}$. In this case, the presence of $\Phi_{q}\left(x\right)$ in no way affects $\mathcal{R}_{v[p]}$. To see why, let us say $\mathcal{R}_{v[p]}=(\sigma_{1},\sigma_{2}]$. If $s_{v[p],q}\leq\sigma_{1}$, then $\Phi_{q}\left(x\right)<\Phi_{v[p]}\left(x\right)\forall x>s_{v[p],q}$ from Lemma 2, and $\Phi_{q}\left(x\right)<\Phi_{v[p]}\left(x\right),\,\forall x\in\mathcal{R}_{v[p]}$.

On the other hand, $s_{v[p],q}>\sigma_{2}$ is impossible (assuming $\sigma_{2}$ is finite). We will show this by contradiction. Let us say $s_{v[p],q}>\sigma_{2}$ holds. It follows from Corollary 4 that $\exists m<v[p]$ such that $\Phi_{m}\left(x\right)>\Phi_{v[p]}\left(x\right)\forall x\in(\sigma_{2},s_{v[p],q})$ and $s_{m,q}\in\mathcal{R}_{m}$. Now the algorithm scans parabolas in $v[\cdot]$ (in the second for loop) from left to right, and so the parabola $\Phi_{v[p]}\left(x\right)$, which comes in the list after $\Phi_{m}\left(x\right)$, would be removed from $v[\cdot]$ due to the intersection of $\Phi_{m}\left(x\right)$ and $\Phi_{q}\left(x\right)$ (line 10 in Algorithm 1). This is a contradiction as this implies the intersection of $\Phi_{v[p]}\left(x\right)$ and $\Phi_{q}\left(x\right)$ is never computed.

As this requires no modifications to the ranges of any parabolas in the envelope or the envelope itself, algorithm 1 does nothing if $s_{v[p],q}\notin\mathcal{R}_{v[p]}$ holds.

The second for loop (line 8) compares a new parabola with existing parabolas in $v[\cdot]$. The discussion above shows that this iteration adds the $q$-th parabola to the already existing upper envelope. It follows from the preceeding analysis that this addition is indeed done correctly. A parabola is added only when the condition in case 1, above, is satisfied. By Lemma 2, we know that the new parabola does not dominate the current upper envelope $\forall x\in(s_{v[p],q},\infty)$, and hence doesn’t affect the part of the envelope in this range. ∎

The following lemma follows as a result of the proof of Theorem 6.

Lemma 7**.**

Each parabola, when first considered, becomes part of the upper envelope.

Proof.

The algorithm initialises the limits of the upper envelope to $(-\infty,\infty)$. Throughout the algorithm, these limits are never changed. Hence, a new parabola will always intersect the upper envelope at some point. Therefore, each parabola being considered will always be added to the upper envelope (Figure 2). It may cause the deletion of previously considered parabolas from the upper envelope, and it may be deleted from the upper envelope by a subsequent parabola. Thus, in each outer loop over $q$ (line 7) the new parabola being considered modifies the upper envelope. ∎

We can state some further properties of the upper envelope in the following Lemmas.

Lemma 8**.**

The first grid point is always a part of the upper envelope.

Proof.

Let $p>0$ be the left-most grid point included in the upper envelope (assuming grid points start at [math]). Since there are no grid points $q<p$ which form a part of the upper envelope, Corollary 5 tells us that $\mathcal{R}_{p}$ must go till $+\infty$. However, as as $0<p$, $\Phi_{0}\left(x\right)>\Phi_{p}\left(x\right)\,\forall x>s_{0,p}$ (from Lemma 2). This implies $\mathcal{R}_{p}$ cannot go on until $+\infty$. Since this argument breaks down only when $p=0$, we conclude that the first grid point must be included in the upper envelope. ∎

Lemma 9**.**

The first and last grid points are always part of the upper envelope.

Proof.

The proof for the first grid point follows from Lemma 8. Furthermore, note that each time a new parabola is scanned, it is “temporarily” the last parabola, until the remaining are scanned as well. Since the last parabola scanned will always be a part of the upper envelope even if there are parabolas after it which can potentially remove it from the upper envelope, the proof is complete. ∎

3.4 Runtime Complexity Analysis

To ascertain the runtime complexity of our algorithm, we consider lines 7,8,16 of algorithm 1. The loop in line 7 iterates once over each of the $N-1$ grid points. The inner loop (line 8) iterates over the parabolas in the upper envelope until the condition in line 10 is satisfied. In the worst case, the inner loop (line 8) would always iterate through all the parabolas in the upper-envelope, which are bounded by $q$ in line 7. This is a strict upper bound because the number of parabolas in the upper envelope will always be less than or equal to the number of grid points we have considered at any iteration. Therefore, the sum of inner loop iterations over the $N-1$ outer loop iterations is $1+2+3+\cdots+N-1=\frac{N\times(N-1)}{2}$. The loop in line 16 iterates once over each grid point $N$. The worst case complexity is therefore $O(N^{2}+N)=O(N^{2})$.

Even though the worst case runtime complexity of the algorithm is the same as that of the brute force solution, it is significantly faster in practice. We now show that the average case runtime complexity of the algorithm is $O(N)$. To achieve this, we enumerate all possibilities that the inner loop in line 8 of the algorithm will see. Consider the tree in Figure 3. The nodes in the tree represent the number of parabolas in the upper envelope at any iteration. The edges represent the number of iterations that the inner loop iterates for. For instance, at the beginning of the first outer loop $N=1$, we have a single parabola in the upper envelope, hence the node at $N=1$ holds the value $1$. The inner loop can only iterate once, after which it will add a second parabola to the upper envelope; therefore the value of the node at $N=2$ is $2$. Now we have $2$ parabolas in the upper envelope. The inner loop can either (a) iterate once, in which case the upper envelope will be modified such that the parabola at the new grid point will replace the old parabola, and the number of parabolas in the upper envelope will remain $2$, or (b) iterate twice, and add one parabola to the upper envelope. We construct this tree, enumerating all possibilities. At any iteration $N$, assuming that each of these enumerated situations is equally likely, we compute the average number of iterations of the inner loop by summing over the edge weights, and dividing by the number of edges. For instance, when $N=4$, the average number of iterations of the inner loop is $\frac{1+2+1+2+3+1+2+1+2+3+1+1+3+4}{14}$.

To compute the average number of inner loop iterations for an arbitrary $N$, we need to know (a) the sum of edge weights and (b), the number of edges at the $N$-th level of the tree. Our tree construction is closely related to a well known construction in combinatorial mathematics called the Catalan Family Tree (Šunić, 2003). Using results from (Šunić, 2003), (a) the number of edges at the $N$-th level of the tree is a Catalan number, given by $\frac{(2N)!}{N!(N+1)!}$, and (b) the sum of edges at the $N$-th level is number of $(N+1)$-th generation vertices in the tree of sequences with unit increase labeled by 2, given by $\frac{3(2N)!}{(N+2)!(N-1)!}$ The average number of inner loop iterations for an arbitrary $N$ is therefore given by $\frac{3N}{N+2}$ which is $O(1)$. The overall runtime is, therefore, $O(N\cdot 1+N)=O(N)$.

3.5 Arbitrary Dimensions

For the $2D$ grid case, we rewrite Equation 4 as

[TABLE]

Since the last two terms do not depend on $p$, we can first solve the $1D$ grid problem over $q$, reducing the problem to

[TABLE]

which is a problem on the $1D$ grid $p$.

Therefore, a maximum distance transform in $2D$ can be computed by first computing the maximum distance transform along a column of the grid, and then computing the maximum distance transform along each row of the result. This can be extended to arbitrary dimensions, by processing the dimensions in any order. Kindly note that the solution does not depend on the order in which we process the dimensions, as is evident from Equations 9-10.

4 Applications

The maximum distance transform can be used to maximize quadratic functions on grids of arbitrary dimensions. It can be employed, for instance, in DPMs for finding the configurations where a score function with quadratic pairwise terms is maximized.

As described in section 2, the maximum distance transform algorithm can be used to find the minimum distance transform for downward opening parabolas. In this capacity, it can be used for efficient message passing in graph inference problems wherever the energy function has negative quadratic pairwise terms.

Our algorithm, and the one in (Felzenszwalb and Huttenlocher, 2004) together allow us to find the global optimal solution for any inference problem on quadratic functions. This would allow us to have fewer constraints on our model parameters, and would lead to a better optimal solution.

5 Conclusion

In this work, we propose an average time $O(N)$ algorithm for maximizing quadratic functions. We give the proof of correctness and analyze the runtime complexity of our algorithm. Given the minimum and maximum distance transform algorithms together, we can efficiently optimise quadratic functions of any form (without regard to the sign of the quadratic terms), and hopefully this ability will allow us more freedom in the choice of our model parameters in optimisation problems where efficiency is indispensable.