Equivalence of the persistence landscape diagram and the barcode? I am studying persistent homology

Equivalence of the persistence landscape diagram and the barcode?
I am studying persistent homology for the first time. I was reading Peter Bubenik's paper "Statistical Topological Data Analysis using Persistence Landscapes" from 2015 introducing persistent landscapes. I am quite confused on the approach on finding the values of the persistence landscape function using a barcode/persistence diagram. I feel like I have a naive misunderstanding of this topic as I shall attempt to explain.
Suppose X is a finite set of points in Euclidean space. From my understanding, if we consider the (finite length) persistence vector space given by the simplicial complex homology for a fixed dimension l, $\{H_l(X_k){\}}_{k=1}^n$ with maps $\{{\mathrm{\delta <!--\; \delta \; -->}}_{k,k^{\prime }}\}$, for $1\le <!--\; \le \; -->k,k^{\prime }\le <!--\; \le \; -->n$,$1\le <!--\; \le \; -->k,k^{\prime }\le <!--\; \le \; -->n$, we have
$\{H_l(X_k){\}}_{k=1}^n\cong <!--\; \cong \; -->{\oplus <!--\; \oplus \; -->}_{i=1}^m\mathbb{I}(b_i,d_i)$
(Theorem 4.10 of this paper) for some multiset $\{(b_i,d_i){\}}_{i=1}^m$, where $\mathbb{I}(b,d)$ gives the persistence vector space of length n,
$0\to <!--\; \to \; -->...\to <!--\; \to \; -->0\to <!--\; \to \; -->\mathbb{R}\to <!--\; \to \; -->\mathbb{R}\to <!--\; \to \; -->\mathbb{R}\to <!--\; \to \; -->...\to <!--\; \to \; -->0...\to <!--\; \to \; -->0$
with non-zero vector spaces at values of the specified interval.
This multiset corresponds to the persistence diagram/barcode so that the k-th Betti number can be identified by finding the number of lines of the barcode that intersect the line x=k in ${\mathbb{R}}^2$
Now Bubenik defines the Betti number of the persistence vector space for an interval [a,b] by ${\mathrm{\beta <!--\; \beta \; -->}}^{a,b}=\dim <!--\; \; -->(\text{im}({\mathrm{\delta <!--\; \delta \; -->}}_{a,b}))$, and the persistence landscape functions ${\mathrm{\lambda <!--\; \lambda \; -->}}_k:\mathbb{R}\to <!--\; \to \; -->[-<!--\; -\; -->\mathrm{\infty <!--\; \infty \; -->},\mathrm{\infty <!--\; \infty \; -->}]$, for $k\in <!--\; \in \; -->\mathbb{N}$by
${\mathrm{\lambda <!--\; \lambda \; -->}}_k(t)=sup\{m\ge <!--\; \ge \; -->0\mid <!--\; \mid \; -->{\mathrm{\beta <!--\; \beta \; -->}}^{t-<!--\; -\; -->m,t+m}\ge <!--\; \ge \; -->k\}$
Shouldn't ${\mathrm{\beta <!--\; \beta \; -->}}^{t-<!--\; -\; -->m,t+m}$ then correspond to the number of lines on the barcode that contain the interval $[t-<!--\; -\; -->m,t+m]$], so that ${\mathrm{\lambda <!--\; \lambda \; -->}}_k(t)$ is the largest value of m that has at least k lines of the barcode intersecting $[t-<!--\; -\; -->m,t+m]$?
I am confused on how the triangle construction is equivalent to the persistence landscape function instead. Any help would be much appreciated!